2021
111
Clàudia Pay a ó Bo ás
Complexi y in he en angled bank:
On he s uc u al and dynamical
p ope ies o empi ical mu ualis ic
ne wo ks
Di ec o /es
Mo eno Vega, Yami
He nández, Lau a
© Uni e sidad de Za agoza
Se icio de Publicaciones
ISSN 2254-7606
Clàudia Pay a ó Bo ás
COMPLEXITY IN THE ENTANGLED BANK: ON THE
STRUCTURAL AND DYNAMICAL PROPERTIES OF
EMPIRICAL MUTUALISTIC NETWORKS
Di ec o /es
Mo eno Vega, Yami
He nández, Lau a
Tesis Doc o al
Au o
2020
Reposi o io de la Uni e sidad de Za agoza – Zaguan h p://zaguan.uniza .es
UNIVERSIDAD DE ZARAGOZA
Escuela de Doc o ado
P og ama de Doc o ado en Física
Tesis Doc o al
Complexi y in he en angled bank:
On he s uc u al and dynamical p ope ies
o empi ical mu ualis ic ne wo ks
Clàudia Pay a ó Bo às
Di ec o es
Yami Mo eno Vega
Lau a He nández
BIFI – Depa amen o de Física Teó ica
2020

¡Qué encan o es e de las
imaginaciones de la niñez,
Pla e o, que yo no sé si ú ienes
o has enido! Todo a y iene, en
ueques delei osos; se mi a odo y
no se e, más que como es ampa
momen ánea de la an asía... Y
anda uno semiciego, mi ando
an o aden o como a ue a,
olcando, a eces, en la somb a
del alma la ca ga de imágenes de
la ida, o ab iendo al sol, como
una lo cie a, y poniéndola en
una o illa e dade a, la poesía,
que luego nunca más se encuen a,
del alma iluminada.
Pla e o y yo, J. R. Jiménez,
La ciencia siemp e ha enido pa a mí el egus o de los juegos in an iles, ese ma a illa se
que acompaña el descub imien o de lo que ocu e más allá de noso os mismos y que solo
más a de podemos o dena , con el esón y p obablemen e la sencillez, del que comple a un
puzzle, clasi ica sus igu i as o colo ea esc upulosamen e un dibujo. Es á, asimismo, en la
ascinación obs inada del que se in en a un nue o juego o el que busca, en el an iquísimo
escondi e, a sus he manos, con esa ce eza an a aigada de que les encon a á.
El ecue do de esos momen os me ha isi ado a menudo du an e la esc i u a de es a esis,
y me gus a ía pensa que he conseguido condensa algún es o de su b illo –aunque a eces
casi se apaga a– en es as páginas. El documen o que sigue no deja de se , po que así se pide
y al ez así deba se , el esul ado isible y a empe ado de las ilusiones y las desilusiones, las
aleg ías y las us aciones, al in y al cabo, lo ap endido y –muy a mi pesa – lo igno ado
du an e los cua o años que ha du ado mi esis doc o al. También es, sin duda, un e lejo de
las pe sonas que me han acompañado.
Aho a, como les hab á ocu ido a an os o os an es que a mí, da po e minado es e
eje cicio de desp endimien o despie a una melancolía p ema u a. Hay quienes de es an la
melancolía po sen imen aloide o imp oduc i a. Muy al con a io, yo c eo que del mismo
modo que una u a una ez azuca ada acaba e men ando en un sabo ag idulce, sin
melancolía no end íamos la ce eza de habe i ido, aunque sea un ins an e, algo bueno.
i
Abs ac
Mu ualis ic ela ionships, ha in he pas had been long o e looked as ascina ing bu
ma ginally ele an , a e oday known o play a c ucial ole in shaping ecosys ems. In his
hesis we look in o he complexi y o how hese ecological ela ionships a e in e wined in
na u al sys ems, o wha Da win amously called he ‘en angled bank’, om he iewpoin
o he ne wo k o malism.
In he i s pa o he hesis, we conside he o igin o he a chi ec u e o mu ualis ic
ne wo ks. In de ail, by applying concep s om in o ma ion heo y and s a is ical physics,
we add ess he ques ion o he eme gence o a widesp ead pa e n known as nes edness. By
analyzing a la ge da ase o empi ical ne wo ks, we show ha he in e play o a ew minimum
assump ions on he numbe o mu ualis ic in e ac ions pe species and he e ec o chance
a e su icien o ep oduce he obse ed s uc u e, p ecluding he in oduc ion o selec i e
p essu es o mechanis ic p ocesses. In his sense, ou esul s show ha he global s uc u e
o mu ualis ic communi ies can be explained, in s a is ical e ms, by he lowe -o de ea u es
o he sys em. Wi h hese esul a hand, we hen explo e how he di e en me ics p oposed
in he li e a u e quan i y nes ed pa e ns, e alua ing hei o e all pe o mance on eal and
syn he ic ne wo ks. Ou esul s indica e ha he anking and compa ison o nes ed pa e ns
among di e en ecosys ems is hampe ed by he p esence o undesi ed dependencies on o he
ne wo k pa ame e s.
In he second pa o his hesis, we con inue digging in o he o ganiza ion o mu ualis ic
communi ies bu ackling ano he challenge, namely ha o mo ing beyond he s ill-p e ailing
agg ega ed pa adigm. To s a wi h, we cha ac e ize a se o empi ical ne wo ks and assess
how inco po a ing in o ma ion abou he empo al a iabili y modi ies he s a ic ne wo k
desc ip ion. Nex , we p opose a g oup o models o gene a e, unde di e se assump ions,
syn he ic con igu a ions o phenology o a gi en ne wo k. We ind ha , while he adequacy o
mechanis ic models o p oduce ealis ic con igu a ions is highly sys em-dependen , a s a is ical
model based on he maximum en opy p inciple pe o ms gene ally well independen ly o he
ne wo k de ails. Elabo a ing u he upon his line o hough , we hen b ie ly explo e he
dynamical consequences o species pe sis ence o aking in o accoun he phenology. We
ind ha species wi h a sho pe iod o ac i i y ace a la ge unce ain y in hei obus ness
agains pe u ba ions. This p elimina y app oach, hough, calls o u he esea ch, specially
in he con ex o a changing clima e.
On he whole, along his hesis we analyze how he ne wo k language can be used o
disen angle he complexi y o na u al mu ualis ic sys ems, by assessing on he one hand he
minimum in o ma ion equi ed o unde s and he ‘en angled bank’, and on he o he hand,
iden i ying he limi a ions o he s a ic ep esen a ion ha s ill p edomina es in he ield.
The wo ks we will p esen and discuss he e ela e o he ollowing publica ions:
•
Pay a ó-Bo às, C., L. He nández, and Y. Mo eno (2019). B eaking he spell o
nes edness: The en opic o igin o nes edness in mu ualis ic sys ems. Physical Re iew
X 9 (3), 031024.
•
Pay a ó-Bo às, C., L. He nández, and Y. Mo eno (2020). Measu ing nes edness: A com-
pa a i e s udy o he pe o mance o di e en me ics. P ep in in a Xi :2002.00534/.
To appea in Ecology and E olu ion.
iii
Con en s
6 Dynamics one mo e ime 97
6.1 S abili y in a changing clima e, o why ime ma e s . . . . . . . . . . . . 97
6.2 A model o inco po a e phenology . . . . . . . . . . . . . . . . . . . . . . . 98
6.3 Resul s in wo empi ical da ase s . . . . . . . . . . . . . . . . . . . . . . . 100
6.4 Conclusions and pe spec i es . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7 Conclusions 105
7.1 ‘In o ming’ecology ............................... 106
7.2 F om Madagasca o Delphi . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Appendices 109
A Da ase s 111
A.1 Ecologicalne wo ks............................... 111
A.2 Economicne wo ks ............................... 111
A.3 Phenology and plan -pollina o ne wo ks . . . . . . . . . . . . . . . . . . . 112
B Compu a ional implemen a ion o he null model 115
B.1 Cons ained maximiza ion o he en opy . . . . . . . . . . . . . . . . . . . 115
B.2 Local solu ion o he sys em o equa ions . . . . . . . . . . . . . . . . . . . 116
C S a is ical measu es o s able-NODF 117
C.1 Analy ical exp essions o he i s wo momen s o s able-NODF . . . . . 117
C.2 S a is ical measu es o s able-NODF . . . . . . . . . . . . . . . . . . . . . 117
D Addi ional me hods o assess he signi icance o nes ed pa e ns 119
D.1 Signi icance es s ................................ 119
D.2 Sel o ganizing ne wo k model . . . . . . . . . . . . . . . . . . . . . . . . . 119
D.3 S a is ical measu es o deg ee asso a i i y . . . . . . . . . . . . . . . . . . 119
E Measu es o nes edness on a sampling 121
E.1 S a is ical measu es o NODF on a sampling . . . . . . . . . . . . . . . . . 121
E.2 S a is ical measu es o spec al adius on a sampling . . . . . . . . . . . . 123
F Me hods o assessing nes edness’ me ics pe o mance 125
F.1 Compu a ion o he nes edness index o each o he s udied me ics . . . . 125
F.2 Co ela ions among me ics and ne wo k ea u es . . . . . . . . . . . . . . 126
G The nullnes eposi o y 127
H S a is ical es s on phenology 129
H.1 Quali y o a i using he Kolmogo o -Smi no es . . . . . . . . . . . . . 129
H.2 Kolmogo o -Smi no wo sample es . . . . . . . . . . . . . . . . . . . . . 129
H.3 Kullback-Leible di e gence . . . . . . . . . . . . . . . . . . . . . . . . . . 130
H.4 Pea son co ela ion wi h ime lag . . . . . . . . . . . . . . . . . . . . . . . 130
I Nume ical implemen a ion o syn he ic models 131
I.1 Shi o s a ing da es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
I.2 Cons ained op imiza ions . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
I.3 Synch onized phenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
I.4 Minimiza ion o he compe i ion . . . . . . . . . . . . . . . . . . . . . . . . 133
I.5 Maximiza ion o he a iance . . . . . . . . . . . . . . . . . . . . . . . . . . 133
I.6 Maximiza ion o he en opy . . . . . . . . . . . . . . . . . . . . . . . . . . 133
J Nume ical in eg a ion o he popula ion model 135
K T ansla ion o summa ized conclusions 137
K.1 ConclusionsinF ench.............................. 137
K.2 Conclusions in Spanish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
x

Con en s
Bibliog aphy 141
xi
CHAPTER 1
In oduc ion
1.1 Some de ini ions o Complex Sys ems
Acco ding o he ‘Concise e ymological dic iona y o he English language’ (Skea Wal e ,
1980), he wo d complex comes om he ancien La in wo d complexus, composed by he
p e ix com –which means ‘ oge he ’, ‘join ly’– and he pas pa iciple o m o he oo plec e e
-which means ‘ o plai ’, ‘ o wea e’-. This o igin s ands in con as wi h he e ymology o
he wo d simple om he La in simplex: he oo emains he same bu he p e ix changes
o sim, ha ansla es in o ‘one’ o ‘as one’. In e es ingly, his no ion o complexi y as a
la ge agg ega e o elemen s in ica ely angled, and mo eo e as mul iplici y as opposed
o singleness, is su p isingly close o he mode n scien i ic concep o a Complex Sys em.
Indeed, Simon w o e (Simon, 1991):
Roughly, by a complex sys em I mean one made up o a la ge numbe o pa s
ha in e ac in a nonsimple way. In such sys ems, he whole is mo e han he
sum o he pa s [...] in he impo an p agma ic sense ha , gi en he p ope ies
o he pa s and he laws o hei in e ac ion, i is no a i ial ma e o in e
he p ope ies o he whole.
This en a i e de ini ion al eady poin s ou se e al ea u es ha a e commonly a ibu ed
o Complex Sys ems, namely: mul iplici y o componen s, non- i ial in e ac ions among
hem and eme gen beha io . The la e consis s in he ac ha as he scale o a sys em
changes (i.e. he numbe o elemen s in ol ed inc eases) no el p ope ies and phenomena
appea . Hence, quan i a i e di e ences ans o m in o quali a i e ones, o as Ande son pu
i , mo e is di e en (Ande son, 1972). Thus, we should no expec o be able o explain
he collec i e beha io o complex sys ems by solely adding up scien i ic concep s and laws
ha ha e been de i ed h ough a educ i e app oach, ha is, by educing such sys ems o a
mic oscopic desc ip ion o hei pa s.
T u h be old, howe e , he e is no ag eed-on o mal de ini ion o Complex Sys ems o
Complexi y Science (Mi chell, 2009; Kwapień and D ożdż, 2012), nei he he e exis s scien i ic
consensus abou wha is he op imal way o quan i a i ely measu e complexi y (Lloyd, 2001).
Since i is no ou aim o en e in his so o compa a i e discussions hough, we will ins ead
elabo a e a bi u he on he undamen al p ope ies and phenomena ha a e gene ally
iden i ied in Complex Sys ems. In o de o do so, we in oduce he de ini ion p oposed
by Mi chell (2009), which eads:
[...] I can p opose a de ini ion o he e m complex sys em: a sys em in which
la ge ne wo ks o componen s wi h no cen al con ol and simple ules o ope a ion
gi e ise o complex collec i e beha io , sophis ica ed in o ma ion p ocessing, and
adap a ion ia lea ning o e olu ion.
This in e p e a ion ei e a es se e al o he a o emen ioned p ope ies, bu i also
in oduces some no el ones. Fi s , he capabili y o hese sys ems o exchange in o ma ion
wi h i s en i onmen . Indeed, Siegen eld and Ba -Yam a gue ha , in o de o be e icien , a
sys em should be a leas as complex as i s en i onmen (Siegen eld and Ba -Yam, 2019).
O he wise, i would no be able o p ocess and espond acco dingly o he wide ange o a ied
1
1. In oduc ion
ex e nal s imuli i ecei es. In a sense, he second ea u e ou lined by Mi chell is ela ed
o his a iabili y: complex sys ems show gene ally an adap i e and e ol ing cha ac e . I
mus be said, none heless, ha e olu ion is no always conside ed ele an , ei he because i
does no play a majo ole in he dynamics, ei he because i s e ec s a e negligible a he
ime-scale a which we s udy he sys em (Mi chell, 2009).
Admi edly, he se o ea u es ha we ha e discussed so a a e some o he mo e
commonly e e ed, bu he e a e many mo e ha we a e no conside ing which may be o
special ele ance in a pa icula domain. In ac , he in e es ing hing abou all hese gene al
ea u es is ha hey a e sha ed by a as a ie y o sys ems o dispa a e na u es: om insec
colonies o he Wo ld Wide Web, om he human b ain o he na u al language (Mi chell,
2009; Kwapień and D ożdż, 2012). All hese examples o Complex Sys ems exhibi , despi e
he di e ences in hei mic oscopic desc ip ion, simila complex beha io s a a mac oscopic
scale.
The ques ion ha na u ally a ises is, hen, whe he i is possible o desc ibe such
phenomena using a common language. The de ini ion o Complex Sys ems sugges ed
by Mi chell also p o ides a hin in his sense: she calls he web o in e ac ions among
cons i uen s a ne wo k. Re iewing he ne wo k ep esen a ion o Complex Sys ems and how
i can con ibu e o ou unde s anding o hem is, ac ually, he subjec o he nex sec ion.
1.2 Th ough he ne wo k glass, and wha we ound he e
Alice h ough he looking glass, o iginal illus a ion by John Tenniel (Ca oll, 1872).
As discussed be o e, Ande son c i icized he no ion o a nai e scien i ic cons uc i ism
acco ding o which mac oscopic beha io s may be explained by agg ega i ely applying
mic oscopic laws. Bu wha i , in o de o explain mac oscopic pa e ns, no only no el
laws and concep s need o be in oduced, bu also some mic oscopic de ails can be sa ely
neglec ed? This is, in e y gene al e ms, he idea in which he ne wo k language is g ounded.
In sho , a ne wo k is a se o componen s whe e some o hem a e pai wise connec ed.
The componen s a e ypically e e ed o as he nodes o e exs, while he connec ions a e
called he links o edges. This simple objec can be used as an abs ac ep esen a ion o a
s ikingly as ange o sys ems. Indeed, he ew examples o Complex Sys ems we men ioned
be o e – he b ain, human language o inancial ma ke s– can all be e ec i ely ep esen ed by
ne wo ks. In wha ollows, we will b ie ly discuss he his o y and consequences o iewing
complex sys ems ‘ h ough he ne wo ks glass’.
F om g aphs o ne wo ks
The de elopmen o ne wo k science begins wi h he bi h o g aph heo y, which is gene ally
a ibu ed o he ma hema ician Leonha d Eule (1707-1783) o his solu ion o he p oblem
known as he Se en B idges o Königsbe g (Biggs e al., 1986). Königsbe g ( oday Kalining ad,
Russia) was an old Eas P ussian ci y ac oss which lew he i e P egel, spanned by a
o al o se en b idges (see Fig. 1.1). The ma hema ical iddle ha occupied Eule may be
o mula ed as succinc ly as ollows: is i possible o de ise a walk h ough he ci y ha
2
1.2. Th ough he ne wo k glass, and wha we ound he e
c osses each b idge once, and only once? Despi e he simplici y o he ques ion, answe ing
i is no a i ial ask. A b u e o ce a emp would en ail, in ac , ying e e y di e en
possible pa h ac oss he map, esul ing in an o e whelming combina o ial e o .
Figu e 1.1: Map o Königsbe g, adap ed om Joachim Be ing 1613’s o iginal eng a ing. The
se en b idges a e highlig ed in ed, he P egel i e is colou ed in blue and he island and
he di e en mainland po ions a e named by capi al le e s.
None heless, Eule showed ha he e is an al e na i e, less pain ul, app oach. In an
a icle published in 1736 (Eule , 1741), he p oposed o look a his puzzle om a abs ac
pe spec i e, o e looking he indi idual de ails o he b idges and po ions o lands in o de
o wo k, ins ead, wi h an s ylized po ayal. In such g aph ep esen a ion (see Fig. 1.2), land
po ions a e depic ed by nodes while he b idges a e ep esen ed by links. In e es ingly, he
puzzle may be sol ed using one o he key p ope ies o a node: i s numbe o connec ions,
o in he g aph language, he deg ee. In o de o he non- edundan walk o exis , i mus
be possible o a i e o a node h ough one link and lea e i om a di e en one. In o he
wo ds, all nodes mus ha e an e en deg ee. This necessa y condi ion is, in ac , no e i ied
in he Königsbe g map. As can be seen in Fig. 1.2, all ou nodes ha e an odd numbe o
connec ions, hus implying ha a walk ha isi s each link exac ly once, ac ually called an
Eule ian pa h, does no exis .
Figu e 1.2: G aph ep esen a ion o he Königsbe g’ b idges p oblem. Each node ep esen s
a po ion o land, as named abo e, while he b idges a e depic ed by links.
A e he dea h o Eule , he ield o G aph Theo y con inued g owing hanks o he
con ibu ions o se e al ma hema icians, like Ki kman, Hamil on o Cayley (Ha a y, 1969).
Al hough he bulk o p oduc ion in he ield was undoub edly p ima y heo e ical, i is s ill
possible o ind some ins ances o empi ical inspi a ion, as illus a e he Ki chho ’s s udies
on elec ic ne wo ks, he wo k o Cayley on molecula g aphs o ep esen chemical isome s
o he de elopmen s o Syl es e and B unel on g aph chemis y (Ha a y, 1969). All hese
3

1. In oduc ion
examples may be ega ded as p edecesso s o he explosion o ne wo k science ha s a ed a
he beginning o he wen ie h cen u y wi hin he ield o Sociology, and hen ex ended in he
mos ecen decades o an as onishing ange o a eas as dispa a e as economics, neu oscience,
epidemiology, gene ics o ecology, o men ion jus a ew o a much longe lis . Indeed, he
concep behind he e m ne wo k di e s om ha o g aph in ha , in a s ic sense, he
g aph e e s o he pu e ma hema ical objec while he ne wo k is a ep esen a ion (o model)
o a eal sys em (Ba abási e al., 2016). As we will now see, his common ne wo k language
has pe mi ed un eiling some uni e sal p ope ies and beha io s o , o he wise, ai ly dis inc
sys ems.
Looking a eal sys ems
Acco ding o Ba -Yam (2000), mac oscopic sys ems may be cha ac e ized by a ‘complexi y
p o ile’ ha indica es how he amoun o in o ma ion needed o desc ibe i changes a a ious
le els o de ail, ha is, a di e en scales. He a gues ha , b oadly speaking, h ee dis inc
ypes o sys ems may be dis inguished: andom, cohe en and co ela ed. The andom ype
equi es he la ges amoun in o ma ion a he smalles scale, bu his quan i y apidly
declines as he scale g ows. On he o he hand, desc ibing cohe en s uc u es demands a
cons an and ela i ely small le el o in o ma ion ac oss all scales, since he beha io o each
o he pa s is exac ly known. Finally, co ela ed sys ems lie somewhe e in be ween: hey
a e nei he en i ely andom, nei he comple ely egula , bu ins ead hey exhibi signi ican
eme gen complexi y a all scales.
In e es ingly enough, his s ylized classi ica ion ecalls he seminal pape by Wa s and
S oga z abou he small wo ld phenomenon, whe e hey show how a complex ne wo k
a chi ec u e a ises in he middle-way among a andom g aph and a egula g aph (Wa s
and S oga z, 1998). In sho , he small wo ld s uc u e implies high clus e ing oge he
wi h a small cha ac e is ic pa h leng h. This combina ion leads o he peculia i y ha any
pai o nodes a e ypically sepa a ed by jus a ew s eps, a si ua ion ha ecalls he ‘se en
deg ees o sepa a ion’ hypo hesis, acco ding o which any pai o pe sons in he wo ld a e
sepa a ed, in a e age, by only se en links. Such pa icula s uc u e has singula e ec s on
he dynamics, as accele a ing he sp ead o dynamical p ocesses like epidemics o umo s.
Wa s and S oga z no only p oposed a simple ne wo k model ha ep oduces his e ec ,
bu hey also e ealed ha i na u ally a ises in a as my iad o sys ems, om he powe
g id o he neu onal ne wo k o C. Elegans (Ama al e al., 2000).
The his o ical ele ance o he Wa s and S oga z’s a icle s ems om he ac ha
i is one o he i s e idences o how simple models o ne wo ks may ail o desc ibe he
complex a chi ec u e o eal ne wo ks. Ano he example o a a he uni e sal ea u e ha
has been ex ensi ely explo ed ac oss disciplines is he amily o scale- ee ne wo ks, whose
deg ee dis ibu ion ollows a powe law (Ba abási and Albe , 1999). The consequen deg ee
he e ogenei y ansla es in o a conside able di e si y in he oles o he ne wo k’s componen s:
many nodes ha e ew links, while a small po ion o nodes hold many connec ions, ac ing as
hubs. This special opology esul s in o a ‘ obus ye agile’ e ec on he dynamics, such
ha he ne wo k’s connec i i y is esilien agains andom ailu e bu ex emely ulne able
o a ge ed a acks (Albe e al., 2000). Once again, his pa icula s uc u al p ope y is
exhibi ed by a wide a ie y o sys ems, al hough ecen ly some deba e ook place ques ioning
whe he he powe law deg ee dis ibu ion is indeed as ubiqui ous as some claim, o ins ead
empi ical deg ee dis ibu ions a e me ely hea y- ailed (B oido and Clause , 2019). In any
case, high inhomogenei y in he deg ees is, undoub edly, a key and widely obse ed ea u e
o eal sys ems.
In he ecen yea s, he s udy o eal ne wo ks has ce ainly e ol ed well beyond hese
pa adigma ic examples. Howe e , e iewing i is ai ly beyond he scope o his hesis.
Ins ead, by sho ly in oducing he small wo ld and scale ee a chi ec u es, my aim is o
p o ide a quick glimpse o how ne wo k language can cap u e some complex ea u es o eal
sys ems. I is as well an a emp o con eying he idea ha , al hough he p esen wo k is
mainly conce ned wi h ecological communi ies, ou conclusions will be some imes ansla able
o o he sys ems and disciplines, since he s uc u al pa e ns and dynamical ea u es we
will s udy he e a e mo e han once sha ed by a b oad ange o di e en sys ems.
4
1.2. Th ough he ne wo k glass, and wha we ound he e
Basic quan i ies, ypes o ne wo ks and no a ion
Be o e con inuing, we will in oduce, o he sake o cla i y, some o he basic concep s and
no a ion in ne wo k heo y. We will also summa ize, as concisely as possibly, he main ypes
o ne wo ks ha can be dis inguished, gene ally speaking, in e ms o he link’s p ope ies
on he one hand and he o e all ne wo k s uc u e one he o he hand.
Fo a gi en ne wo k, he in o ma ion o he connec i i y among nodes is ypically enclosed
in an objec called adjacency ma ix. F om now on, we will depic his quan i y by he le e
A. In he mos simple case, i is de ined as a squa e ma ix ha ul ills ha :
Aij = 1 i node iand node jsha e a link, (1.1)
Aij = 0 o he wise. (1.2)
The second mos basic quan i y ha we may de ine in a ne wo k is he numbe o
in e ac ions pe node, so-called he deg ee o a node, commonly ep esen ed by he le e
k
.
Using he adjacency ma ix, he deg ee o a node iis s aigh o wa dly calcula ed as:
ki=X
j
Aij,(1.3)
while he se o deg ees o each o he ne wo k’s nodes
{k1, ..., ki, ...}
is wha we call he
deg ee sequence. A closely ela ed measu e is he deg ee dis ibu ion, namely he p obabili y
dis ibu ion ob ained by calcula ing he ac ion o nodes
pk
whose deg ee is equal o
k
.
Despi e hei elemen a iness, he deg ee and he deg ee dis ibu ion can be highly in o ma i e
quan i ies bo h a an indi idual and a a global le el, and indeed along his hesis we will
pay g ea a en ion o hem and hei ela ion wi h o he s uc u al pa e ns.
The pic u e we ha e d awn up o now co esponds o he mos basic scena io, in which
links only in o m abou which node is connec ed o which bu no abou how hey a e
connec ed. A leas wo undamen al modi ica ions exis ha pe mi inco po a ing mo e
complex p ope ies o he link: (i) di ec ion and (ii) weigh . When in oducing (i), he links
a e p o ided wi h a di ec ion, which u ns he symme ic in e ac ion in o an asymme ic
one such ha one node ac s as he ‘sende ’ and he o he as he ‘ ecei e ’. By adding (ii),
he connec ions a e ponde ed by hei ela i e impo ance o s eng h in a g adual scale, in
opposi ion o he bina y case whe e in e ac ions a e me ely exis ing (a one in he adjacency
ma ix) o no (a ze o).
In oducing hese wo p ope ies o links al eady b ings up ou possible ypes o ne wo ks:
di ec ed/undi ec ed and bina y/weigh ed. By mo ing in o a mo e gene al pe spec i e on he
s uc u al con igu a ion o he whole ne wo k, we may dis inguish he ollowing classes o
ne wo ks:
•Bipa i e ne wo ks
. A ne wo k is said o be bipa i e i i can be pa i ioned in o
wo sepa a e se s (no mally called guilds), such ha links exis only among he nodes o
dis inc guilds, bu no among he membe s o he same guild (see Fig. 1.3). Mo eo e ,
he sepa a ion o nodes in o wo g oups usually ep esen s as well he exis ence o
wo kinds o nodes, dis inguished by ei he hei iden i y, unc ion o ole. A ypical
example o a bipa i e ne wo k is ha o ilms and ac o s: a link is placed be ween
a mo ie and an ac o i he o she appea ed on i (Newman, 2010). Gi en ha no
in a-guilds connec ions exis , he adjacency ma ix o bipa i e ne wo ks –in he case
o undi ec ed and bina y in e ac ions– has he ollowing pa icula o m:
A=0B
B|0,(1.4)
whe e
B
is wha we call he bi-adjacency ma ix. In he mos gene al case,
B
is
a ec angula ma ix o size
n×m
, whe e
n
and
m
ep esen he numbe o nodes
belonging o each guild. As be o e,
Bkl
= 1 when node
k
and node
l
sha e a link
5
1. In oduc ion
and
Bkl
= 0 when hey don’ , wi h he peculia i y ha he nodes
k
and
l
belong
o di e en guilds. Ne wo ks which can no be pa i ioned in o wo guilds a e called
monopa i e.
•Mul ilaye ne wo ks.
The idea behind mul ilaye ne wo ks is o di ide he ne wo k
in o a ious laye s, in o de o inco po a e in o a sole ne wo k di e en kinds o nodes
and links. Each single laye , in isola ion, may be ega ded as a ne wo k whe e nodes
a e connec ed be ween hem ia in a-laye links, while he nodes in di e se laye s
a e connec ed h ough in e -laye links (see Fig. 1.3 o an example). Such inc ease
o complexi y in he ep esen a ion pe mi s ypi ying di e en kinds o in e ac ions
and nodes h ough he ob ious dis inc ion among laye s and in e /in a-laye links, as
well as modeling how he pe u ba ions and dynamics in one laye ge s ans e ed
o he o he s. A classic example o his class o ne wo ks is he ne wo k o u ban
anspo , whe e each laye ep esen s a di e en sys em o public anspo a ion (i.e.
bus, ain, subway, e c) and he links ep esen he connec ions among lines (Gallo i
and Ba helemy, 2015; Ale a e al., 2017). A pa icula case o mul ilaye ne wo ks a e
he so-called mul iplex ne wo ks, cha ac e ized by he ac ha he same se o nodes
is ep esen ed in all he laye s, ye he ype o in e ac ions ep esen ed in each laye
di e s.
Figu e 1.3: Illus a ion o di e en ne wo k ypes. In
a)
, bipa i e ne wo k, whe e one guild
is ep esen ed in eal and he o he in ed. In
b)
, mul ilaye ne wo k composed by h ee
laye s. The in e -laye links a e d awn in g ey, while he in a-laye links a e depic ed in
whi e.
This classi ica ion p o ides a e y basic ske ch o how he ne wo k ep esen a ion can be
modi ied and adap ed o add ess di e en p oblems. Ha ing in oduced al eady he basic
no ions o complex sys ems and how he language o ne wo ks may o e a sui able amewo k
o s udy hem, we will now u n ou eyes in o ecological sys ems, in o de o unde s and
how he concep s discussed up o now apply o na u al ecosys ems.
1.3 Complexi y in he en angled bank
The no ion o complexi y we ha e discussed so a may be ce ainly ecognized in a
bundle o examples, including na u al ecosys ems. Indeed, Le in a gues ha ecosys ems
in pa icula and he biosphe e in gene al con o m pa adigma ic cases o complex adap i e
sys ems (Le in, 1998), p ima y cha ac e ized by i s signi ican biodi e si y and by being
subjec o e olu iona y change. As such complex sys ems, hey exhibi pa e ns o agg ega ion
and hie a chical o ganiza ion. One c ucial ques ion, as Le in ema ks, is whe he such complex
s uc u es a e d i en by sel -o ganiza ion p ocesses, con ingen en i onmen al ac o s, o a
pa h-dependen e olu iona y his o y.
6
1.3. Complexi y in he en angled bank
In any case, he idea ha complexi y pe ades na u e appea ed conside ably ea lie , and
we may ace i back o Da win’s amous pic u e o he en angled bank. Indeed, in he las
pa ag aph o his seminal wo k On he o igin o he species, Da win w o e:
I is in e es ing o con empla e an en angled bank, clo hed wi h many plan s
o many kinds, wi h bi ds singing on he bushes, wi h a ious insec s li ing
abou , and wi h wo ms c awling h ough he damp ea h, and o e lec ha hese
elabo a ely cons uc ed o ms, so di e en om each o he , and dependen on
each o he in so complex a manne , ha e all been p oduced by laws ac ing a ound
us.
Despi e i s poe ic simplici y, his desc ip ion poin s ou se e al key aspec s o ecosys ems:
he p esence o a in insic he e ogenei y o species and unc ions, in e connec ed among
hem as well as wi h he en i onmen . Mo eo e , such o ganized a ie y is no he p oduc
o a global design by a chie con olle , bu he eme gen esul o wha Da win called he
‘laws’. Al hough he e Da win su ely e e s o e olu ion, we may ex end his iew o include,
in gene al, any se o ules which leads o complex beha io .
This in insically complex cha ac e o na u al sys ems has led ecologis like Odum (Odum,
1977) o ad oca e o a holis ic app oach o s udy hem, which would complemen he
educ ionis esea ching agenda on he componen s o ecosys ems. Wi h his pe spec i e
in mind, he aim o his subsec ion is o o e a sho in oduc ion o he de elopmen o
ecology as a science and, specially, o e iew he ecen incu sion o complex ne wo ks as a
language o modeling, explaining and p edic ing he complex s uc u e and unc ioning o
ecosys ems.
A young science
The ea ly his o ical de elopmen o ecology as a science is in ac a dispu ed subjec , and,
like in he bes o oyal in igues, he e is no ag eed consensus on who is he ‘ a he ’ o he
discipline (McIn osh, 1986). Al hough Da win (1809-1882) is o en c edi ed o unwillingly
se ing he basis o ecological heo y, some his o ians o science conside i s p edecesso
Humbold (1769-1859) as he eal ounde , due o his pionee ing wo ks on biogeog aphy.
O he s ack he o igin s ill ea lie in ime o Whi e (1720-1793) and his obse a ions o he
na u al li e in Selbo ne, a esh iew ha has o en been ca ego ized as ‘A cadian’.
Cu iously enough, he e m ‘ecology’ was no coined by any o hese au ho s bu by he
zoologis and na u alis Haeckel (1834-1919), who in oduced i o Ge man om he ancien
G eek oo ‘oikos’ which means ‘house’, ‘habi a ’. In his sense, he e m was in oduced
o name he s udy o he ela ion o animals –including humans– wi h he en i onmen .
Indeed, as anach onis ic and incomple e as i migh be, he de ini ion gi en by Haeckel o
he e m ecology is s ill ci ed in may ex books and o e s a nice pic u e o wha ela ions
y o cap u e, in pa icula , ecological ne wo ks:
By ecology, we mean he whole science o he ela ions o he o ganism o he
en i onmen including, in he b oad sense, all he “condi ions o exis ence.” These
a e pa ly o ganic, pa ly ino ganic in na u e; bo h, as we ha e shown, a e o
he g ea es signi icance o he o m o o ganisms, o hey o ce hem o become
adap ed. (...)
As o ganic condi ions o exis ence we conside he en i e ela ions o he o ganism
o all o he o ganisms wi h which i comes in o con ac , and o which mos
con ibu e ei he o i s ad an age o i s ha m. Each o ganism has among he
o he o ganisms i s iends and i s enemies, hose which a o i s exis ence and
hose which ha m i . (...)
On he whole, as he his o ian o ecology McIn osh a gues, his di icul y on e-cons uc ing
he o igin o ecological heo y is agg a a ed by he in insic p oblema ics in de ining he
scope and limi s o he e y own discipline (McIn osh, 1986). To his, should be added
a long-s anding human in e es on he composi ion and unc ioning o na u al sys ems,
7
1. In oduc ion
Figu e 1.5:
a)
On he le , illus a ion o Ang aecum sesquipedale, also known as Da win’s
o chid, and i s hypo he ical mo h pollina o . The o chid has a ex emely long nec a spu ,
which lead Da win o hypo hesize he exis ence o an unknown mo h, endowed wi h an
equally long p oboscis ( he " ongue" o he mo hs) which would allow i o pollina e he
lowe (Da win, 1877). The mo h was disco e ed in 1903, 21 yea s a e Da win’s dea h,
and named Hawk Mo h o Xan hopan mo ganii p aedic a which in la in means ‘p edic ed
mo h’ (A di i e al., 2012). The d awing is ex ac ed by he essay C ea ion by Law by
Wallace (1867), who suppo ed Da win’s idea and imagined he o m o he hypo he ical
pollina o .
b)
On he igh , ac ual pho og aphy o he Xan hopan mo gani p aedic a kep a
he Na u al His o y Museum o London.
specialized in e ac ions (Wase e al., 1996), in bo h ecological and e olu iona y e ms. He e
again, we ind an example o how a educ i e app oach, consis ing in disen angling he web
o in e ac ions in o i s basic componen s - he pai wise in e ac ions-, is insu icien o accoun
o he complexi y o he whole ecosys em. The in oduc ion o mu ualis ic ne wo ks as
models o mu ualis ic web in he 90’s aimed a ackling he challenge o conside ing he
en i e communi y, by exploi ing his no el amewo k and i s powe ul se o analy ical and
nume ical ools.
The s uc u e o mu ualis ic ne wo ks
As a o emen ioned, mu ualis ic ne wo ks a e commonly ep esen ed in a bipa i e embedding
whe e each guild depic s a di e en mu ualis ic pa ne o g oup o species, like in he
example in Fig. 1.6 o a plan -pollina o communi y. Despi e he pa icula cha ac e is ics
ha each mu ualis ic sys em may ha e, some gene al ea u es ha e been iden i ied in e ms
o s uc u e. He e ogenei y, modula i y, nes edness o asymme ic in e ac ions a e some o
hese opological egula i ies, o which we will now pay some a en ion.
Al hough we will desc ibe each o his s uc u al ea u es sepa a ely, i is wo h no ing
ha mos o hem co-occu simul aneously in he ne wo k, and wha is mo e, hey a e
some imes no independen o one ano he . Indeed, a g ea bulk o wo k has been de o ed
o explo e he in e ela ions be ween s uc u al p ope ies, and in his hesis we will as well
add ess his ques ion o he pa icula case o nes ed pa e ns.
14

1.4. A ne wo k pe spec i e on iends will be iends
Figu e 1.6: Example o a mu ualis ic ne wo ks composed by plan s and hei pollina o s,
ep esen ed in a bipa i e se ing. The links among he wo guilds can be weigh ed, which
we ep esen he e by a ying he hickness o he line.
Deg ee he e ogenei y
Along wi h he ecogni ion ha mu ualism can be highly gene alized (Wase e al., 1996)
came he acknowledgemen ha mu ualis ic ne wo ks a e signi ican ly he e ogeneous, in he
sense ha in he same communi y coexis species ha hold many con ac s – he gene alis s,
which consequen ly ha e a la ge deg ee– wi h o he s ha in e ac wi h jus a small subse
o he possible mu ualis ic pa ne s - he so-called specialis s, ha ha e a small deg ee-.
Acco dingly, special e o s ha e been made o cha ac e ize he unc ional o m o he deg ee
dis ibu ions o bo h guilds. Indeed, i has been claimed ha o an impo an majo i y
o eal sys ems, hei deg ee dis ibu ion can be i ed by a unca ed powe -law (Jo dano
e al., 2003). Ne e heless, he signi icance o such i is g ea ly hinde ed by he ac ha ,
since mu ualis ic ne wo ks a e in gene al ela i ely small, hei empi ical deg ee dis ibu ions
spans jus a ew o de s o magni ude.
Independen ly o wha he bes i ing unc ion migh be, wha seems clea is ha he
pa e n o connec i i y is signi ican ly he e ogeneous, simila ly o wha happens in many
eal sys ems (see Sec ion 1.2). None heless, he e he ail o he deg ee dis ibu ions depa s
om ha o a scale- ee. This implies ha se e al species ha e a ew in e ac ions, while
jus a ew species hold many in e ac ions bu no as many as could be, a p io i, possible.
This limi a ion in he deg ee o he mos connec ed species has been linked o he exis ence
o o bidden o links due o ecological ba ie s de e mined, o ins ance, by mo phological,
phenological o pheno ypical ai s (Bascomp e and Jo dano, 2013).
Besides deg ee dis ibu ions, i he ne wo k is weigh ed one can s udy he dis ibu ion o
s eng hs. In he con ex o mu ualis ic ne wo ks, he weigh is ypically associa ed o he
equency o in ensi y o he ecological in e ac ion, al hough his quan i y may be di icul o
measu e and some imes i is ins ead es ima ed om he ela i e species’ abundances. In any
case, he s eng h is he weigh ed analogue o he deg ee, which p o ides in o ma ion abou
he connec i i y a he species le el. The s eng h’s dis ibu ions o mu ualis ic ne wo ks ha e
been claimed o be e en mo e he e ogeneous han he deg ee dis ibu ion (Bascomp e and
Jo dano, 2013), sugges ing ha he he e ogenei y in he connec i i y eme ges no only in a
quali a i e ep esen a ion o who in e ac s wi h whom, bu also when we include quan i a i e
in o ma ion on how much hey in e ac .
Asymme ic in e ac ions
The s udy o quan i a i e mu ualis ic ne wo ks e ealed ha he dis ibu ion o in e ac ion
weigh s is signi ican ly igh -skewed, e lec ing a majo i y o weak mu ualis ic connec ions
agains jus a ew s ong in e ac ions (Bascomp e e al., 2006). Mo eo e , he in ensi y o
15
1. In oduc ion
a mu ualis ic in e ac ion is no necessa ily symme ic. This means ha , o ins ance, in a
plan -pollina o communi y he dependency o a lowe on a plan is no , by o ce, he same
as ha o he plan on he lowe .
Indeed, he s udy o empi ical sys ems has shown ha his s eng h he e ogenei y is
pa e ned in a pa icula way: he ew in ense in e ac ions p esen in he ne wo k a e,
ypically, highly asymme ical. In a plan -pollina o sys em, his ansla es in o he ac ha
i a plan elies signi ican ly on he pollina ion se ices o a gi en animal, his pollina o
depends weakly on he esou ces o he plan (Bascomp e e al., 2006). Such asymme y may
play an impo an ole in sus aining biodi e si y by p e en ing posi i e eedback loops, ha
can ac as ampli ie s o possible dis u bances in he popula ion.
Small wo ld
Mo ing om a species-le el desc ip ion o he connec i i y o a communi y-wide pe spec i e,
mu ualis ic ne wo ks exhibi se e al egula i ies. One o hem is a widesp ead phenomenon
ac oss eal sys ems, he a o e-men ioned ‘small wo ld’ p ope y (see sec ion 1.2 o a mo e
de ailed discussion o i s ubiqui y in empi ical complex ne wo ks).
Olesen e al. (2006) s udied i s p e alence in mu ualis ic communi ies by con e ing
he bipa i e ne wo k o mu ualis ic in e ac ions in o wo one-mode ne wo ks o sha ed
neighbo s, one pe guild, such ha species o he same guild a e connec ed whene e hey
sha e a common mu ualis ic pa ne . The analysis o hese p ojec ed ne wo ks e ealed ha
he majo i y o species a e placed jus a ew links away om each o he , wi h an a e age
pa h leng h ypically smalle han wo oge he wi h a ela i ely la ge clus e ing coe icien .
Such combina ion implies ha pe u ba ions may sp ead quickly along he communi y,
o as Mon oya e al. (2006) pu i : “E e y species is closely linked o e e y o he , so –
me apho ically– when a ee alls in a ain o es , e e y species in ha species- ich, complex
sys em would seem o ‘hea ’ ha e en quickly”. This has con ibu ed o he pa adoxical
no ion ha na u al mu ualis ic communi ies may be agile, yielding o he p oposal o he
p esence o al e na i e s uc u al p ope ies o mechanisms ha would enhance he sys em’s
esilience, like he a o e-men ioned in e ac ion asymme y.
Modula i y
Ano he way o inspec ing he closeness be ween species is looking a he p esence o modules,
ha is, g oups o clus e s o species cha ac e ized by being igh ly connec ed, in he sense
ha connec ions among he membe s o he same module a e signi ican ly mo e equen
han links among di e en modules. The exis ence o such modules has been ela ed o
he ac ion o en i onmen al, ecological and e olu iona y o ces, in pa icula since modules
end o g oup oge he species wi h con e gen mo phological ai s (Olesen e al., 2007;
Dupon and Olesen, 2009). Acco dingly, modules ha e been ad oca ed o be he building
blocks o mu ualis ic ne wo ks, e ealing habi a compa men aliza ion a he big scale and
coe olu iona y p essu es a a a close look.
Nes edness
Besides he gene al ea u es men ioned so a , he mos no o ious and widesp ead p ope y o
mu ualis ic is p obably nes edness. This communi y-wide pa e n en ails ha he in e ac ions
o a gi en species esul o be a subse o he in e ac ions o mo e gene alized –highe
deg ee– species. This special o ganiza ion o he in e ac ions, which has been ound o be
ubiqui ous ac oss na u al ecosys ems ega dless o hei di e ences in habi a , clima e o
species composi ion (Bascomp e e al., 2003), has ueled in he ecen yea s an impo an
numbe o wo ks ocusing on i s causes and consequences. In pa icula , special e o s ha e
been made o explain i s implica ions o he assembly and s abili y o he ecosys em.
Anyway, gi en ha he s udy o nes ed pa e ns is he opic o he i s pa o his hesis,
we will no elabo a e u he on i now. Ins ead, we will de o e he en i e i s chap e o his
hesis o discuss, in g ea e de ail, i s de ini ion, o igin, dynamical implica ions and se e al
ways o measu ing i .
16
1.5. Whe e we a e now, and whe e we aim o go
Figu e 1.7: Example o a pe ec ly modula ne wo k o plan s and pollina o s, di ided in o
ou modules.
Mu ualism ou side ecology
Al hough he s udy o mu ualism has mainly conce ned ecologis s, i has also played a ole
in he s udy o human o ganiza ions e e since he publica ion o he book ‘Mu ual aid’
by K opo kin e al. (1902), which explo ed he p e alence o mu ualis ic ela ions in socie y
as well as he na u al wo ld. By challenging he idea ha compe i ion domina es na u e,
K opo kin e alued he impo ance o coope a ion and mu ualis ic in e ac ions as c ucial
ac o s o he su i al o human and animal communi ies. Cu iously enough, i has been
a gued ha i s associa ion wi h ana chis communism may ha e con ibu ed o he gene al
dismissal o mu ualis ic o ces, conside ed almos as inciden al un il he 70’s (Bascomp e
and Jo dano, 2013).
Ou side ecology, mu ualism akes he o m o win-win ela ionships among wo
en i ies o agen s, whose iden i ies span di e se con ex s and le els o o ganiza ion,
om indi iduals o en i e coun ies. In he economic con ex , some examples o
mu ualis ic webs a e manu ac u e -con ac o ne wo ks (Saa ed a e al., 2009) o selle -buye s
ne wo ks (He nández e al., 2018). In sociology, communica ion ne wo ks made o indi idual
use s and memes (e.g. hash ags in Twi e ) ha e been as well s udied h ough he lens o
mu ualism (Bo ge-Hol hoe e e al., 2017).
In he ecen yea s, he e has been an inc easing in e es in explo ing whe he he
s uc u al ea u es and dynamical p ope ies o ecological mu ualis ic communi ies hold o
mu ualis ic ne wo ks ou side ecology (Bu gos e al., 2008; He nández e al., 2018; S aka
e al., 2018; Ma iani e al., 2019). In he ligh o his, while he ocus on his hesis will
be mainly on ecological sys ems, he conclusions we may a ain a e expec ed o o en ha e
an implica ion, as well, o mu ualis ic ne wo ks o economical, sociological o echnological
sys ems.
1.5 Whe e we a e now, and whe e we aim o go
In spi e o i s you h, he ield o ecological ne wo ks –and in pa icula ha o mu ualis ic
ne wo ks-. has wi nessed an imp essi e g ow h du ing he las decades (Heleno e al., 2014),
in pa allel wi h he expansion o ne wo k science and, ce ainly, spu ed by an accompanying
inc ease o da a ga he ing and sha ing (as illus a es o ins ance, he Web o Li e p ojec
1
).
Ne e heless, se e al ques ions emain s ill un esol ed (Ings e al., 2009; Heleno e al., 2014;
Piloso e al., 2017) calling o a combina ion o concep ual, me hodological and expe imen al
endea o s. Such challenges can be amed along he lines o he a o emen ioned ade-o
be ween explana o y powe and simpli ica ion, a ension which appea s in insically a ached
o he cons uc ion o a ne wo k.
1Da abase o ecological ne wo ks, specially mu ualis ic, a ailable a : h p://www.web-o -li e.es/
17
1. In oduc ion
On he one hand, ecologis s ha e been ad oca ing o amelio a ing he ealism o
mu ualis ic ne wo ks by adding u he in o ma ion abou hei empo al and spa ial a ia ion,
as well as conside ing pai wise in e ac ions’ speci ici ies o he exis ence o di e en ial ai s
a he le el o he species o e en he indi iduals (Ings e al., 2009; Heleno e al., 2014).
This could yield o a mo e de ailed ep esen a ion o he na u al complexi y, a he cos o
inc easing he he e ogenei y o bo h nodes and links. All in all, explo ing he implica ions
o changing he le el o o ganiza ion is a necessa y s ep in o de o dismiss he p esence
o a obse e ’s bias, which could esul in a i ac s like hose ega ding he measu e o he
allome ic ela ion in oodwebs (Ings e al., 2009). The a o emen ioned ans o ma ion o
bina y, quali a i e ne wo ks in o weigh ed, spa ially ex ended, o empo al ne wo ks, a e
all di e en ways o e ining he ealism o he ne wo k, by paying he p ice, howe e , o
po en ially complica ing he models.
On he o he hand, he obse a ion and s udy o pa e ns e en in he mos simpli ied
ne wo ks has posed se e al ques ions, some o which a e s ill open. As Le in ecalled, he
de ec ion o pa e ns is in e es ing o he ex en o which i e lec s hidden mechanisms (Le in,
1992), and indeed he implica ions o s uc u al p ope ies o mu ualis ic ne wo ks, like
nes edness, a e con o e sial up o da e. Fu he mo e, a c ucial aspec o he in e play
be ween pa e n and scale is explo ing whe he a gi en ea u e o phenomena is in a ian
ac oss scales. O he wise, he na u al ques ion ha a ises is how in o ma ion is ans e ed
and ans o med om one le el o desc ip ion o ano he , in o de o lead o he eme gence
–o disappea ance– o he pa e n. In his sense, de ining he s uc u al de e minan s o
mu ualis ic ne wo ks is a long-s anding p oblem in ecology, ha conce ns communi y-wide
p ope ies –e.g. nes ed pa e ns– as much as species-cen e ed ea u es –e.g., he deg ee.
The challenges we ha e men ioned so a call o bo h heo e ical and expe imen al
ad ances. Theo e ical, since app op ia e concep ual and me hodological amewo ks a e
o be de eloped in o de o e ine he ne wo k ep esen a ion o ecosys em, as well as o
cha ac e ize pa e ns ac oss scales and in es iga e he dependency be ween s uc u al ea u es
a di e en le els. Expe imen al, on he o he side, because o be able o model de ails one
needs ich and eliable da a, ha should mo eo e ake in o accoun spa ial and empo al
a iabili y. This equi es an inc ease o sampling e o in quali y and quan i y e ms alike,
oge he wi h he elabo a ion o p o ocols o measu e –and possibly enhance– he deg ee o
comple eness o empi ical ne wo ks (Ings e al., 2009). All his implies ha u u e ad ances
in ecological ne wo ks a e no only an in e disciplina y endea o , conce ning oge he pu e
ecologis s, ma hema icians o physicis s, bu also a mul i-app oach ask, since heo e ical
ad ances a e in ima ely condi ioned by ield wo k –and ice e sa.
In his hesis, we will s udy mu ualis ic ne wo ks om a heo e ical iewpoin , conside ing
in de ail wo main opics: (i) he cha ac e iza ion and eme gence o nes edness and (ii) he
in oduc ion o empo al in o ma ion on plan -pollina o ne wo ks. This pa i ioned esea ch
e lec s he pe ennial ension be ween simpli ica ion and ealism in models ha we ha e jus
discussed he e. Indeed, while he in he i s pa we will ocus on explaining nes ed pa e ns
ha appea in he mos basic ep esen a ion o mu ualis ic ne wo ks -bina y, agg ega ed
and monolaye -, in he second pa we will wo k owa ds a mo e ealis ic cha ac e iza ion o
communi ies, by modeling hei empo al dimension h ough empi ical da a on he pe iods
o ac i i y o plan s and pollina o s, he so-called phenology. Be o e de ini ely beginning
wi h he main body o his hesis, le us discuss a bi u he how hese wo subjec s will be
add essed.
Nes ed pa e ns and he h ee wi ches
Why is i ele an o look o pa e ns in ecology? Beyond he sea ch o na u al o de ,
egula i ies can e lec dynamical p ocess o di e en kind, conce ning ei he he pas , he
p esen o he u u e o ecosys ems. In a symbolic way, his ecalls he igu e o he h ee
wi ches. In he i s pa o Skakespea e’s amous play Macbe h, h ee wi ches encoun e
he main cha ac e Macbe h and, when g ee ing him, each hails him di e en ly: one o
hem by his pas ank (‘ hane o Glamis’), he second by his p esen posi ion (‘ hane o
Cawdo ’) and he hi d wi ch by his u u e i le (‘ he u u e king’). This has yield o he
in e p e a ion ha , while he h ee wi ches held Macbe h’s a e in hei hands, each o hem
18
1.5. Whe e we a e now, and whe e we aim o go
ep esen s a di e en empo al momen , espec i ely he p esen , he u u e and he pas .
In a me apho ical sense, an analogue iad can be de ined in he s udy o he implica ions
o ecological pa e ns and wha in o ma ion hey con ey: abou he ecosys em’s pas , in
pa icula i s assembly and eco-e olu iona y his o y; abou he p esen s a e, o ins ance
how species coexis and unc ion oge he , leading o he obse ed biodi e si y; and o cou se
abou he possible u u e, in he sense o how he ecosys em would eac agains ex e nal
pe u ba ions. All h ee aspec s o he iad a e equally impo an o he unde s anding o
he a e o mu ualism.
Figu e 1.8: The h ee wi ches o he Shakespea e’s play Macbe h, also called he h ee wei d
sis e s. Eng a ing by Lo say, ha appea ed in ‘Magasin Pi o esque’ in 1863, a e a 1782
d awing by Fuseli.
Ne e heless, no e e y obse ed pa e n is equally in o ma i e, as nei he e e y obse ed
egula i y is o cibly signi ican . In he i s pa o his hesis, we will add ess he ques ion o
he signi icance o nes ed pa e ns in mu ualis ic ne wo ks, o in o he wo ds, how lowe -o de
s uc u al p ope ies o he ne wo k can de e mine he global nes edness. In pa icula , we
will s a wi h an in oduc o y chap e e iewing he de ini ion and implica ions o nes edness,
as well as he main me ics ha ha e been de eloped o quan i y i . In he subsequen
chap e we will use null models o y o esol e he a o emen ioned ques ion o he o igin
and signi icance o nes edness. Finally, in he hi d chap e o his i s pa , we will use he
de eloped me hods o in es iga e he pe o mance o a a ied se o nes edness me ics.
The phenos o mu ualism
Phenology is he science ha s udies he iming o biological cycles along he li e o o ganisms,
om seasonal pa e ns o ci cadian hy hms. Re u ning, as we s a ed his in oduc o y
chap e , o e ymology, he wo ld phenology is composed by he p e ix phenos, ha comes
om he Ancien G eek ‘phaino’ which means ‘ o appea ’, and he con en ional su ix ology,
om he G eek ‘logia’ ha means ‘ he s udy o ’. Phenology, he e o e, would ansla e as
‘ he s udy o appea ance’.
Gi en ha an impo an pa o ecological mu ualisms akes he o m o plan -animal
in e ac ions, hey can be g ea ly condi ioned by he seasonal cycles o he pa ne s. In
pa icula , plan s end o exhibi ma ked seasonal li e e en s, like lea -p oducing, ui ing o
lowe ing. Al hough such e ec s may change conside ably among habi a s and clima es, in
gene al hey a e no negligible. In he la e yea s, he in e es in modeling he phenology
19

1. In oduc ion
o mu ualism has been spu ed by he ac ha clima e change may se e ely a ec species’
phenology, leading o a po en ial dis up o plan -pollina o o plan -dispe se sys ems.
In his hesis, we will app oach his opic in he second pa o he manusc ip . The i s
chap e will be de o ed o he cha ac e iza ion o wo empi ical da ase s and he p oposal o
some models o gene a e syn he ic con igu a ions o phenology unde di e en cons ain s.
Subsequen ly, we will add ess he ques ion o he dynamical consequences o in oducing
phenology by examining a model ha inco po a es bo h mu ualism and compe i ion o
esou ces.
20
PART I
Re isi ing Nes ed Pa e ns
CHAPTER 2
The h ee W’s and one H o nes ed
pa e ns
Almos wo decades ago, a seminal wo k by Bascomp e e al. (2003) e ealed ha nes ed
pa e ns a e widesp ead in mu ualis ic communi ies. Indeed, mu ualis ic ne wo ks ga he ed
ac oss he globe appea o be egula ly nes ed, despi e di e ences in ecosys em’s geog aphical
loca ion, clima e, o species composi ion. The disco e y o such ubiqui y awoke conside able
in e es wi hin and beyond he ecological communi y and, as a esul , many wo ks ha e been
de o ed since hen o unde s anding he ex ension, impac and o igin o nes ed pa e ns.
In his chap e , we summa ize he main conclusions o hese e o s. We s a p o iding
a exac de ini ion o nes edness ( he ‘wha ’), and con inue wi h he eal sys ems whe e i
has been de ec ed ( he ‘whe e’). Nex , we b ie ly e iew some o he hypo hesis ha ha e
been handled as possible explana ions o i s o igin, as well as i s dynamical consequences
( he ‘why’). Finally, we in oduce he main me ics ha ha e been p oposed o measu e and
quan i y nes ed pa e ns ( he ‘how’).
2.1 The wha
Nes edness is a global p ope y o ne wo ks ha add esses he o e lap be ween in e ac ions,
in pa icula o wha ex en he mu ualis ic pa ne s o a gi en species a e sha ed by i s
mo e gene alized coun e pa s. S ic ly speaking, a pe ec ly nes ed s uc u e is de ined by
he ac ha he in e ac ions o a gi en node a e in a iably a subse o he in e ac ions o all
nodes wi h la ge deg ee (see Fig. 2.1). Tha is, i
B
is he biadjacency ma ix o a
NR×NC
bipa i e ne wo k, he sys em will be pe ec ly nes ed only i he ollowing condi ions a e
bo h ue:
Bi,j = 1 and Bi,k = 1 ⇐⇒ gi en a pai jand ksuch ha
NR
X
i
Bi,j ≤
NR
X
i
Bi,k ,∀i;
(2.1)
Bi,j = 1 and Bl,j = 1 ⇐⇒ gi en a pai iand lsuch ha
NC
X
j
Bi,j ≤
NC
X
j
Bl,j ,∀j;
(2.2)
whe e
NR
and
NC
a e, espec i ely, he numbe o ows and columns o he biadjacency
ma ix B.
The condi ions abo e ansla e in o he ac ha specialis species, ha is, species wi h
ew in e ac ions and hus a small deg ee, a e seldom in e ac ing wi h o he specialis s.
Ins ead, hey end o exhibi deg ee disasso a i i y, appea ing a ached o gene alis species.
Gene alis s, in u n, ha e a la ge deg ee and hence a e connec ed o a a ie y o neighbo s,
including o he gene alis s. This con e s o nes ed biadjacency ma ices i s dis inc i e
iangula shape, composed by a obus co e o connec ions among gene alis s o which
specialis cling (see Fig. 2.1).
23
2. The h ee W’s and one H o nes ed pa e ns
he ma ix whe e he nes ed co e is expec ed (see Fig. 2.3 o an example). Dis ances a e
measu ed in e ms o he Manha an dis ance, which means ha he dis ance be ween a
escaled elemen
bi,j
o he ma ix and he o igin is
di,j
=
xi
+
yi
. This nes edness index is
gi en by:
τ=d−dnes
d and −dnes
,(2.5)
whe e
d
is he sum o e all he elemen s’ dis ances
d
=
Pdi,j
o he eal ma ix (maximally
packed) and
dnes
ep esen s an analogous sum bu o e he co esponding pe ec ly nes ed
ma ix wi h he same size and ill as he empi ical one. Thei di e ence is hen no malized
by he maximum di e ence in a e age dis ances be ween a null model and he pe ec ly
nes ed ma ix. The e a e a ious op ions o he null model used o calcula e
d and
, bu a
common choice is o keep cons an size and ill. In his way 0
≤τ≤
1, and he smalle
τ
he
mo e nes ed he sys em is.
The nes edness me ics based on o e lap and dec easing ill
This index (he ea e NODF), in oduced by Almeida-Ne o e al. (2008), in ol es wo
con ibu ing ac o s o nes edness: dec easing ill, ha quan i ies o wha ex en , a e
o de ing he ows and columns o he ma ix, he deg ee sequences s ic ly dec ease; and
pai ed o e lap, ha accoun s o he numbe o sha ed pa ne s be ween all pai s o columns
( ows), no malized by he smalle deg ee. By ga he ing oge he he ope a ional de ini ion
indica ed by Almeida-Ne o e al., we p oposed he ollowing compac exp ession o calcula e
NODF (Pay a ó-Bo às e al., 2019):
NODF(B)=1
K
NP
X
i<j







[1 −θ( j− i)] ·
NA
P
a=1
biabja
j







+
1
K
NA
X
k<l









[1 −θ(hl−hk)] ·
NP
P
p=1
bpkbpl
hl









,
(2.6)
whe e K=NP(NP−1) + NA(NA−1)
200 .(2.7)
He e we ha e used he ollowing no a ion:
p
is he deg ee o plan
p
and
ha
he deg ee
o animal
a
. The double sums un o e wo indexes and we conside ha he bipa i e
adjacency ma ix is labeled such ha ow
i
is placed abo e ow
j
and column
k
a he le o
column
l
. The
K
ac o con ains he no maliza ion o e he numbe o all possible pai s,
and he ac ha NODF is de ined o ake alues be ween 0 and 100. Finally, he
θ
s ands
o he Hea iside s ep unc ion, which is ze o when i s a gumen is nega i e, and one i i s
a gumen is posi i e o ze o. As a esul , he 1
−θ
(
j− i
) e m encapsula es he dec easing
ill condi ion.
As can be seen, he NODF me ics sepa a ely in o ms on he con ibu ion o ows and o
columns o he obse ed nes edness. I is impo an o emphasize ha he o e laps be ween
all he possible pai s o ows (columns) a e only aken in o accoun i he conside ed pai is
o de ed in dec easing deg ee, o he wise i assigns a null alue o he o e lap. Mo eo e , he
highe he index, he mo e nes ed he sys em is.
The NODF me ics co ec ly assigns a e y low nes edness alue o modula ne wo ks
because, in gene al, elemen s wi hin he same block ha e simila deg ee. Howe e , as
ema ked by S aniczenko e al. (2013), i may gi e a alse nega i e in he case o a nes ed
ne wo k wi h mul iple ows o columns wi h he same deg ee. This is due o he dec easing
ill ac o , which hea ily penalizes deg ee degene acy. Un o una ely, his si ua ion is qui e
30

2.4. The how
common o mu ualis ic ecosys ems which a e in gene al e y spa se and o en eccen ic,
wi h ypically much mo e animal species han plan species, which al oge he leads o a non
negligible deg ee degene acy. Fo his eason, a a ian o his me ics called
s able-NODF
has ecen ly been p oposed by Ma iani e al. (2019). The de ini ion o his index (also named
s-NODF) is analogous o he classic me ic NODF excep o he dec easing ill e m. In
pa icula , keeping he same no a ion, i eads:
s-NODF(B)=1
K
NP
X
i<j
NA
P
a=1
biabja
j
+1
K
NA
X
k<l
NP
P
p=1
bpkbpl
hl
,(2.8)
whe e K=NP(NP−1) + NA(NA−1)
200 .(2.9)
No e ha he de ini ion abo e equi es he ne wo k
B
o be o de ed by dec easing deg ee
in bo h guilds. This a ian does no inco po a e he dec easing ill e m and hence does
no penalize he deg ee epe i ion, he e o e solely measu ing he numbe o sha ed pa ne s
among pai s o ows and columns. This esul s in his me ic being mo e obus agains
sligh a ia ions in he deg ee sequence and, impo an ly, in sol ing he d awbacks ou lined
by S aniczenko e al. (2013).
The B ualdi and Sande son disc epancy
This me ics is, a e he empe a u e, one o i s ly p oposed indexes. Ins ead o ocusing on
he dis ances, hough, i is based on he gap-coun ing app oach men ioned abo e. Indeed,
s a ing om he eal ma ix in i s maximally packed s a e, his me ics coun s he numbe
o misplaced absences o p esences o con ac s, called disc epancies, ha should be ‘co ec ed’
in o de o p oduce a pe ec ly nes ed ma ix wi h equal size and ill (B ualdi and Sande son,
1999).
Since such measu e is based on he compa ison o he eal ma ix wi h a pe ec ly nes ed
one o he same pa ame e s (numbe o ows, numbe o columns and numbe o links), i is
independen o any pa icula null model. Howe e , gi en ha he e may be some ambigui y
on he maximally packed con igu a ion, he esul depends on he chosen one. The e o e,
an op imal calcula ion would in ol e a e aging o e he di e en ini ial maximally packed
con igu a ions, which can be howe e a qui e demanding p ocess in compu a ional e ms.
Fu he mo e, gi en ha he numbe o possible disc epan links is di ec ly p opo ional
o he o al numbe o links, he esul g ea ly depends on he ne wo k’s ill. To p e en
such dependency, i has been p oposed o no malize he disc epancy by he o al numbe o
links (G e e and L. Chown, 2006). As i was he case o he empe a u e
TAP
, he de ini ion
o his me ics implies ha he lowe he alue o he index, he mo e o de ed he sys em is.
The nes ing index based on ne wo k’s obus ness
This me ics (sho ened as NIR he ea e ) is based on he no ion o he obus ness o a
ne wo k, ha is, he capaci y o he sys em o emain connec ed when subjec o node
emo al (Bu gos e al., 2007; Memmo e al., 2004). This me ics uses wo ex eme node
emo al p ocedu es, o a ack s a egies, whose ou comes e eal he amoun o nes edness o
he ne wo k. On he one hand he nodes o one guild a e emo ed in dec easing deg ee o de
(DDR s a egy), and o he o he in inc easing deg ee o de (IDR s a egy). The ac ion o
species o he o he guild ha s ill keeps con ac s (su i e) as he coun e pa s a e emo ed
leads o he A ack ole ance cu e (ATC).
Once he a ack s a egy is ixed, he ATC depends on he deg ee o nes edness. Fig 2.4
illus a es h ee di e en ypical beha io s o he ATC o each s a egy, when he p ocedu e
is applied on a pe ec ly nes ed ne wo k, on a eal ne wo k and on a null model wi h he
same size and ill. The DDR s a egy be e e eals he di e ences o s uc u e o he h ee
ne wo ks.
31
2. The h ee W’s and one H o nes ed pa e ns
0.0 0.5 1.0
0.0
0.5
1.0
F ac ion o a acked columns
F ac ion o su i ing ows
Figu e 2.4:
A ack Tole ance Cu es o h ee di e en ne wo ks ha ing he same
pa ame e s.
T iangles co espond o he eal mu ualis ic ecosys em o Clemen s and Long
(1923), squa es o a andomiza ion o his sys em and ci cles o an a i icial pe ec ly nes ed
ne wo k wi h he same pa ame e s (size and numbe o links). Open and ull symbols
co espond o he IDR and DDR a ack s a egies, espec i ely.
I can be easily shown ha , o he pe ec ly nes ed ne wo k, he a ea unde he ATC
is
RIDR
= 1 o IDR s a egy, while i is
RDDR
=
φ
o he DDR (Bu gos e al., 2009).
Thus, his index is no malized by he a ea be ween wo ex eme cu es, which is maximum
o a nes ed ne wo k. Mo eo e , he a ea is minimum o a andom ne wo k, while o he
eal ne wo ks he a ea lies be ween hese wo ex emes. Al oge he , he con ibu ion o he
nes edness coe icien o ows o columns is de ined as:
NIR =RIDR −RDDR
1−φ,(2.10)
which measu es, like NODF, he con ibu ion o nes edness o ows and columns,
sepa a ely. NIR looses sensi i i y as he densi y o links inc eases, which is no a p oblem
o ecosys ems ha a e, in gene al, e y spa se. Finally, his index may in p inciple sligh ly
depend on he chosen ma ix o de ing wi h espec o he deg ees o he guild being supp essed.
As such o de is no unique due o deg ee degene acy, a e aging o e a se o equi alen ly
o de ed ma ices would p eclude any possible biases.
The spec al adius
The spec al adius was ecen ly p oposed by S aniczenko e al. (2013) as an al e na i e
me ic o nes edness ha di ec ly elies on he spec al p ope ies o he adjacency ma ix.
Le us call
I
he iden i y ma ix and
A
he adjacency ma ix o a bipa i e ma ix
B
, such
ha :
A=0B
B|0,(2.11)
which is a squa e, symme ic and non-nega i e ma ix, gi en ha
ai,j ≥
0. The spec al
adius o he ma ix
A
(also called dominan eigen alue o la ges eigen alue) is de ined as
ollows:
ρ(A) = max{|λi|}.(2.12)
32
2.4. The how
Whe e
λii∈ {
1
, ..., n}
a e he eigen alues o
A
, hus he oo s o he equa ion:
de (Iλ−A)=0. Since Ais a symme ic ma ix, λi∈Re ∀i.
The capabili y o he spec al adius o quan i ying he deg ee o nes edness o a ne wo k
is oo ed in a heo em by Bell e al. (2008), ha s a es ha wi hin he se o ne wo ks ha ing
he same numbe o links and nodes, he one yielding he maximum spec al adius will be
pe ec ly nes ed. In ac , S aniczenko e al. (2013), showed ha mo e nes ed ne wo ks end
o ha e la ge spec al adius. Howe e , impo an ly, his ela ion is only ue in s a is ical
e ms. Indeed, hei esul s e eal ha , i we ake wo sligh ly di e en ne wo ks, he one
wi h a la ges spec al adius is no , necessa ily, he mos nes ed (see, o ins ance, Fig. 1
om S aniczenko e al. (2013)). The e o e, he sensibili y o he spec al adius a a ine scale
(i.e. o dis inguish be ween small di e ences in he deg ee o nes edness o wo ne wo ks) is
a he limi ed. This impo an law, howe e , is sca cely -i a all- epo ed in he li e a u e,
and o ou knowledge we we e he i s one o d aw a en ion o i in Pay a ó-Bo às e al.
(2019).
Ano he ca ea o he spec al adius is, u he mo e, ha i is no no malized. This
implies ha nes edness measu es a e a ec ed by ne wo k p ope ies like he densi y o
links o he size, hus hinde ing he compa ison among ne wo ks which do no sha e hose
cha ac e is ics. We will explo e wi h mo e cau ion his cha ac e is ic in Chap e 4, and
p opose -and s udy- a possible no maliza ion.
33
CHAPTER 3
Nes edness and chance
Non omne quod ni e au um es
(All ha gli e s is no gold)
Ancien La in P o e b
Hope ully, he p e ious chap e would ha e se ed o illus a e, a leas pa ly, how he
disco e y o nes ed pa e ns s i ed an a id in e es no only among ecologis s, bu among
ne wo k scien is s in gene al. Admi edly, his esul ed in a g owing numbe o e idences o
hei ubiqui y, e lec ed as well in he quan i y o hypo hesis and obse a ions b ough ou
ega ding i s causes and consequences. Along he p esen chap e , pe haps pa adoxically,
we will a emp o undo his cons uc ed no ion o nes edness. Indeed, we will y o show
ha nes ed pa e ns a e no independen , signi ican pa e ns in hei own, bu jus he
mac oscopic esul o imposing ce ain cons ain s in he le el o gene aliza ion o each
species. Tha is, nes edness eme ges om he combina ion o lowe -o de p ope ies o he
ne wo k wi h chance, a esul ha calls o e isi ing i s o e all ele ance.
F om he poin o iew o a busy eade , i may seem a poin less oundabou o elabo a e
so much upon he impo ance o nes edness, in o de o subsequen ly a gue ha i is no , in
ac , a ele an pa e n. Howe e , only by doing so i is possible o accoun o he ex en
o he implica ions o such esul , since, as he popula saying ypically a ibu ed o Ma k
Twain says: ‘I is easie o ool people, han o con ince hem ha hey ha e been ooled’.
In pa icula , we will show how nes ed pa e ns can eme ge in eal sys ems as an en opic
consequence o he deg ee sequences alone, i.e., o he numbe o con ac s held by each species.
The discussion o his esul , which has been published in Pay a ó-Bo às e al. (2019), will be
he cen al pa o his chap e (sec ion 3.3). Howe e , be o e add essing his co e p oblem,
we will in oduce in sec ion 3.1 he e ec o chance in he s uc u e o ecological ne wo ks,
which will equi e a gene al -ye sho - accoun o he main p inciples o e olu iona y heo y.
Secondly, in sec ion 3.2, we will summa ize he di e en ypes o null models ha ha e been
p oposed o deal wi h -and possibly ule ou - he e ec o chance in de e mining he s uc u e
o a ne wo k. Finally, his chap e will be closed explo ing he ecological in e p e a ion, as
well as he po en ial applica ions, o a meaning ul se o pa ame e s o he null model ha
a e empi ically de e mined - he so-called Lag ange Mul iplie s-.
3.1 Na u e does play dice
In 1926, Eins ein w o e a le e o Max Bo n whe e he amously said: ‘The heo y yields
much, bu i ha dly b ings us close o he Old One’s sec e s. I, in any case, am con inced
ha He does no play dice’. The ‘ heo y’ men ioned by Eins ein is Quan um Mechanics, abou
which, despi e ha ing con ibu ed o i himsel , he held se ious doub s. In his celeb a ed
quo e, Eins ein c i icizes he inhe en p obabilis ic na u e o he quan um heo y, a guing
ha i s emb acing o chance and he esul ing challenge o adi ional de e minism was
me ely epis emological -bu no on ological. In o he wo ds, he a gued ha he p obabilis ic
basis o he quan um desc ip ion o eali y was due o he incomple eness o he heo y, and
35

3. Nes edness and chance
no a p ope y o he physical wo ld
1
. Despi e he undeniable commo ion ha Quan um
Mechanics supposed o he scien i ic concep ion o eali y, u h o be hold he impo ance
o he ole o chance in shaping na u e had been made e iden some yea s be o e, wi h he
in oduc ion o he heo y o e olu ion. Da win, who cas s a long shadow ac oss his hesis
-o o in e se he nega i e meaning o he ph ase, a long-las ing glow-, is a he hea o he
scien i ic e olu ion which ans o med he concep ion o he o igin o li e, including ha o
humans. Indeed, his heo y has been some imes conside ed as a complemen a y s age o he
Cope nican Re olu ion, which se he basis o he mode n scien i ic iew o he wo ld (Ayala,
2007).
While he heo y was o iginally concei ed o explain, p ima y, he e olu ion o li ing
o ganisms, i has been la e applied as well o he selec ion o ela ionships among species,
pa icula ly coope a ion (Nowak, 2006), and e en mo e, o he s udy o non-biological sys ems
like cul u e, echnology o language (A hu e al., 1993; Solé and Val e de, 2020) –all o
which ha e been o en de ined as complex adap i e sys ems, as discusses in he in oduc o y
chap e . O cou se, he la e o ms o e olu ion can no be exac ly ansla ed om he
o iginal heo y, bu may s ill in ol e, a leas in pa , analogous p inciples o ules. In wha
ollows we will b ie ly summa ize he gene al no ion o e olu iona y heo y in o de o be e
ame he impo ance o chance in ecology, and pa icula ly he eme gence o he so-called
spand els.
God as a gamble
The pa played by chance in biological e olu ion may seem i ling when compa ed o ha
o quan um physics, ye Da win had o ace an es ablished p econcep ion o his ime: ha
he e could no be ‘design wi hou designe ’ (Ayala, 2007). The exis ence o o gans ha
pe o med complex asks, mo eo e seemingly app op ia e o hei en i onmen , yield he
p edecesso s o Da win, like William Paley (1743-1805), o hink ha , in wo ds o Ayala:
‘The pu pose ul unc ion e eal, in each case, an in elligen designe , and he di e si y,
ichness, and pe asi eness o he designs show ha only he omnipo en C ea o could be
his In elligen Designe ’. Da win challenged his iew by a guing, no only ha species
e ol e o e ime -which was al eady a sound idea a his ime-, bu also ha he e en ual
complexi y o o gans and o ganisms is he esul o an unsu eilled and unplanned p ocess
called na u al selec ion, o wha Dawkins amously called ‘ he blind wa chmake ’. In his
p ocess, chance is p esen in he appea ance o andom mu a ions which may inc ease, o
dec ease, he i ness o a species. In ac , i Paley’s God was o be seen as he be o in his
mu a ion game, he would be a a he a spend h i , since i is es ima ed ha mo e han 99%
o he species ha ha e e e li ed ha e become ex inc (Ayala, 2007).
In p inciple, his selec ion-o - he- i es mechanism may shape as well he obse ed
ne wo ks o in e ac ions, pa icula ly in mu ualis ic communi ies. As discussed in he
p e ious chap e , na u al selec ion can mani es i sel a a sys em-wide le el by il e ing ou
less esilien con igu a ion (Suweis e al., 2013). Besides, i may also ac upon indi idual
ai s ha egula e in e -species in e ac ions - o ins ance, hose ha de e mine exploi a ion
ba ie s p hose ha de e mine he e en ual bene i and cos o a mu ualis ic ela ionship-,
ein o ce ce ain pai wise in e ac ions h ough he e ec o coe olu ion o e en sculp he
s uc u e o he en i e ne wo k (Nuisme e al., 2013).
Selec ion on he eye o he beholde
Na u al selec ion o e ed a scien i ic answe o he p oblem o design. Ne e heless, jus as
he obse a ion o a pa icula complex elemen does no en ail he exis ence o a c ea o ,
such complexi y does no necessa ily imply, likewise, he p esence o selec i e p essu es. In
a seminal pape , Gould and Lewon in (1979) c i icized wha hey called ‘panselec ionism’,
ha is, he belie in ‘ he nea omnipo ence o na u al selec ion in o ging o ganic design and
ashioning he bes among possible wo lds’. Gould and Lewon in p e en ed agains he allacy
1
This lead Eins ein o p opose, oge he wi h Podolsky and Rosen, his amous EPR pa adox, which
hypo hesized he exis ence o some local hidden a iables.
36
3.1. Na u e does play dice
o assuming ha , since na u al selec ion is able o p oduce e ol ed complex s uc u es, e e y
e olu iona y p ocess o complex s uc u e is o cibly he p oduc o na u al selec ion. In
sho , he whole c i icism can be summed up by he popula saying ha goes: ‘i all you
ha e is a hamme , e e y hing looks like a nail’
In pa icula , Gould and Lewon in p oposed he exis ence o he a o emen ioned spand els,
de ined as s uc u es ha could seem o be e olu iona y selec ed –bu a e no . The e m
spand el o iginally e e s, in ac , o an a chi ec onic elemen , namely he oid space be ween
an a ch and a ans e sal ame o be ween wo (o mo e) adjacen a ches (see Fig. 3.1).
Such emp y space has no p ima y use and is me ely a byp oduc o he con igu a ion o he
suppo ing s uc u al elemen s. Ye , a chi ec onic spand els a e usually illed wi h o namen s
such as pain ings o sculp ed elie s, which may ick he obse e in o hinking ha such
space is in en ionally designed and ca ies ou a key pu pose.
Figu e 3.1: Example o an a chi ec u al spand el in he doo way o he Sain Geo ges Chu ch
in G ea B omley (Essex, UK). The spand el is deco a ed wi h a D agon, as pa o he
ep esen a ion o Sain Ge oge’s legend. Pic u e adap ed om an o iginal pho og aphy by
Michael Ga lick (CC BY-SA 2.0)
In he analogy p oposed by Gould and Lewon in (1979), he a chi ec onic cons ain s
ela ed o he cons uc ion o he building a e ansla ed as biological cons ain s ha may
d i e and es ic he e olu ion o an o gan o an o ganism h ough non-selec i e mechanisms.
In wha conce ns us he e, he no ion o spand els was in oduced om e olu iona y heo y
in o he ne wo k language by Solé and Val e de (2006). In hei a icle, he au ho s p oposed
ha mo i s ound in cellula ne wo ks a e in ac ne wo k spand els. Indeed, while ce ain
ne wo k mo i s a e signi ican ly abundan in compa ison o andom models (Milo e al.,
2002), Solé and Val e de e iewed se e al e idences poin ing a he ac ha no pa icula
selec i e p essu e seem o wo k upon such mo i s, and mo eo e simple models o andom
duplica ion and mu a ion can ep oduce hei obse ed abundances. Mo e ecen ly, Val e de
e al. (2018) in he i s place, ollowed by Mayna d e al. (2018), showed ha he s uc u e
o ecological ne wo ks, including mu ualis ic sys ems, can eme ge as a by-p oduc o he
assembly p ocess. The e o e, non- i ial opological p ope ies o such ne wo ks, like deg ee
he e ogenei y o nes edness, could be ne wo k spand els ha a e no speci ically op imized
h ough na u al selec ion.
In he nex sec ions we will y o show ha , om a pu ely s uc u al poin o iew,
empi ical nes ed pa e ns o mu ualis ic sys ems u n ou o be non-signi ican when aking
in o accoun in o ma ion abou he deg ee sequences. This iew is complemen a y o he
one exposed up o now, whe e he no ion o spand el is jus i ied by explo ing he dynamics
o assembly and e olu ion o mu ualis ic sys ems. Ins ead, in wha ollows we will ocus on
s udying he p ope ies o s a ic, agg ega ed ne wo ks, in o de o show ha nes edness is an
en opic e ec , in he sense ha i appea s as he mos p obable mac oscopic con igu a ion
o he in e ac ions gi en ce ain cons ain s on he numbe o con ac s o each species.
37
3. Nes edness and chance
3.2 Null models o how o il e chance
The concep o null model is closely ela ed o he no ion o null hypo hesis, a e m ha was
coined by Fishe (1935). In his seminal book, Fishe illus a ed he idea wi h a simple, ye
well-known expe imen : he lady as ing ea. In such hough expe imen , a woman decla es
o be able o di e en ia e whe he an English cup o ea has been p epa ed by pou ing i s
he ea and secondly he milk, o ice- e sa. Fishe p oposed o challenge he claim by
ca ying ou a blind as ing es , in ol ing se e al cups o ea di e en ly mixed. Such es is
based on a simple null hypo hesis, namely ha he lady’s s a emen is alse and, in ac , she
is guessing he o de o he mix haphaza dly. In o de o ejec his hypo hesis -i.e. o show
ha he abili y o dis inguish he ea as e is no simply a ma e o luck- we would need
o know how many cups she could co ec ly iden i y when playing andomly. Then, i in
compa ison o ha quan i y she ac ually does a su icien ly la ge numbe o co ec guesses,
we can dismiss he op ion ha she is jus ossing a coin o decide.
Following his line o hough , he idea o a null model o igina es in he con ex o ne wo k
heo y, whe e we need o es a pa icula null hypo hesis on a g aph – ypically whe he a
gi en p ope y can be p oduced by he e ec o chance alone. In his sense, he null model is
used o p oduce a se -o ensemble- o andomiza ions, adi ionally syn he ic ne wo ks whe e
a ce ain numbe o pa ame e s ha e been cons ained, agains which o compa e he o iginal
ne wo k. This pe mi s assessing whe he a pa icula obse a ion, o ins ance communi y
s uc u e (Ba be , 2007) o deg ee asso a i i y (Newman, 2002; Pa k and Newman, 2003;
Johnson e al., 2010), is a ele an p ope y gene a ed by non- i ial mechanisms o ne wo k
o ma ion, o , on he con a y, i can be me ely explained by s a is ical co ela ions na u ally
eme ging om o he es ic ions on he g aph, e.g. he ne wo k’s ini e size, a pa icula ly
high o low densi y o links, o he p esence o deg ee he e ogenei y.
In wha ollows, we will use a null model in o de o de e mine whe he , as a o emen ioned,
nes edness can eme ge as an en opic consequence o he deg ee sequences. In ac , he
applica ion o null models o he analysis o ecological ne wo ks is no new. Fa om i , he e
is a long his o y o deba e abou which null model is mo e sui able o assess he signi icance
o pa e ns –including nes edness– da ing back o he i s s udies in biogeog aphy (Conno
and Simbe lo , 1979; Go elli and G a es, 1996) ollowed by a la e ex ension o in e ac ion
ne wo ks– (Bascomp e e al., 2003), which has emained a li ely opic un il oday (see o
ins ance S ona e al. (2018) o a ecen con ibu ion and he e iew o he opic by Ma iani
e al. (2019)).
The majo i y o he ea ly null models ely on algo i hmic p ocedu es ha nume ically
andomize he ne wo k by s ic ly keeping some cons ain s, wha we will call he ha d
cons ain s app oach. On he o he hand, i is also possible o elax such condi ions, wha
esul s in a gene al amily o null models based on so cons ain s. Wi hin his g oup, a
pa icula class o null model has been ecen ly p oposed (Squa ini and Ga laschelli, 2011),
ha he au ho s named canonical ensemble ollowing s a is ical mechanics e minology. I
consis s in cons uc ing a s a is ical ensemble whe e cons ain s a e kep only on a e age, by
imposing some key condi ions on he en opy and likelihood o he dis ibu ion o p obabili y
o exis ence o each g aph in he ensemble. Such no el amewo k o e s se e al echnical
and concep ual ad an ages wi h espec o o me me hods, which is why we will exploi i
o es whe he nes edness is a signi ican pa e n. Be o e doing so, we summa ize below
he main amilies o null models wi h hei cha ac e is ics and laws, keeping in mind ha
we a e specially in e es ed on ecological ne wo ks and in keeping he deg ee sequences as
cons ain s.
Ha d cons ain s
As a o emen ioned, his amily o null models is cha ac e ized by en o cing a se o cons ain s
s ic ly. Gi en ha s a is ical mechanics e minology is o en used o desc ibe di e en ypes
o ne wo k ensembles –a p ac ice ha was inaugu a ed, likely, by (Pa k and Newman, 2004)–
we could also iden i y his class o null models wi h he mic onanical ensembles (Squa ini
and Ga laschelli, 2011). Indeed, jus like some cons ain s on he ne wo k pa ame e s a e
p ese ed exac ly, in physics he mic onanical ensemble includes he se o all mic os a es
38
3.2. Null models o how o il e chance
ha sha e a p ecise, comple ely speci ied o al ene gy. In heo y, he numbe o condi ions
ha one could impose o he null model is i ually as la ge as he numbe o obse able
quan i ies, anging om he mos basic p ope ies o elabo a ed and sys em-wide pa e ns.
None heless, in mos cases we a e in e es ed in unde s anding whe he undamen al ne wo k
p ope ies – ha we may also call lowe -o de ea u es– a e able o de e mine global –o
highe -o de – ones. Acco dingly, mos null models a e cons uc ed by con olling one o
a ious o he ollowing pa ame e s: (i) ne wo k size, (ii) numbe o links and (iii) deg ee
sequences.
In he pa icula case o bipa i e ne wo ks, i is impo an o no e ha bo h (i) and
(iii) can be disen angled pe guild, ha is, no only he o al numbe o nodes is gi en bu
also how many o hem belong o each kind. Likewise, we do no ha e one bu wo deg ee
sequences. This bipa i e cha ac e o mu ualis ic ne wo ks mul iplies as well he spec a
o possible combina ions o cons ains, because a pa icula condi ion may be applied o
jus one guild, i s coun e pa , o bo h. Following Ul ich and Go elli (2007), he subsequen
classi ica ion can be made:
•Equip obable model.
In his null model only he size o he ne wo k and he numbe
o links is p ese ed. Consequen ly, he ensemble is cons uc ed by imposing ha ,
wi hin a guild, each node has in heo y he same p obabili y o ha ing a link (hence
he name equip obable). Since bo h guilds a e andomized ul illing his condi ion, his
null model is adi ionally named in sho as EE (equip obable-equip obable).
•Fixed models.
In his case, no only he size and densi y o links a e kep , bu also
he deg ee sequences a e conse ed s ic ly ( ha is, he deg ees a e ixed). The mos
es ic i e null model, he so-called ixed- ixed (FF), cons ain s he deg ee sequences
o bo h guilds o mee s ic ly he o iginal ones. None heless, i is also possible o elax
his condi ion upon one o he guilds, esul ing on he ixed-equip obable model (FE)
o i s complemen a y equip obable- ixed (EF), depending on which o he wo deg ee
sequences ( ows o columns) is kep cons an .
This classi ica ion was o iginally de eloped in he con ex o biogeog aphy (Conno and
Simbe lo , 1979; Go elli, 2000), bu was apidly adap ed and adop ed in he con ex o
mu ualis ic ne wo ks (Bascomp e e al., 2003; Ul ich e al., 2009). In he bigge pic u e,
simila null models a e used in he analysis o ne wo k s uc u e. Indeed, he EE model
p oduces E dős-Rényi andom g aphs wi h a ixed numbe o edges
m
and nodes
n
, a
model ha in g aph heo y is o en named as G(n, m)(Newman, 2010). A s aigh o wa d
implemen a ion o his me hods is o ake he o iginal ne wo k and andomly ewi e he
in e ac ions, picking and eshu ling hem ollowing a uni o m dis ibu ion. Al e na i ely, i
is possible o ‘ ill’ he ne wo k, assigning in e ac ions andomly among no -p e iously linked
nodes, un il he numbe o links ma ches he empi ical one (Go elli, 2000). Howe e , as
discussed in he in oduc ion, his so o null models p oduce syn he ic g aphs ha a e in
gene al e y di e en om eal examples, gi en ha hey show e y low clus e ing – ha
ends o ze o in he limi o la ge ne wo ks– and i ually no communi y s uc u e o deg ee
co ela ions. The e o e, he EE null model is adi ionally ega ded as a loose benchma k
agains which o challenge empi ical obse a ions.
In wha ega ds he FF null model and i s a ian s, he e is a a he p oli ic li e a u e
on wha is he op imal –o in o he wo ds less s a is ically biased– implemen a ion. He e
again we ind ha his ques ion is no exclusi e o ecology, since ne wo k scien is ha e been
o long in e es ed in gene a ing mo e ealis ic deg ee sequences, pa icula ly esembling
he he e ogeneous, igh -skewed deg ee dis ibu ions ha a e ypically ound in empi ical
sys ems (Newman, 2010). Despi e his mul iplici y o me hods, we can ind, undamen ally
speaking, wo p incipal ways o a acking he p oblem o cons uc ing a ne wo k ma ching a
speci ic deg ee sequence:
•Swapping algo i hms
. In a sense, his app oach pu sues ai ly closely he no ion o
andomiza ion p oposed by Fishe , concei ed as diso de ing a gi en sys em in o de
39
3. Nes edness and chance
Gene al analy ical exp essions o he i s wo momen s o he dis ibu ion
o a gi en p ope y o he ne wo k.
The andomizing scheme discussed up o now may be exploi ed o achie e he inal goal
o his whole p ocedu e: o measu e a gi en ne wo k p ope y ac oss he s a is ical ensemble.
As a o emen ioned, in gene al he e a e wo possible ways o pe o ming his calcula ion. On
he one hand, as long as he p ope y ha we aim o e alua e can be analy ically o mula ed
in e ms o he elemen s o he bipa i e adjacency ma ix, Squa ini and Ga laschelli (2011)
showed ha i is possible o ob ain, a i s o de , he analy ical exp ession o he i s and
second momen s o he co esponding dis ibu ion. These exp essions depend only on he
link p obabili ies. In o he wo ds, i is no necessa y o sample he ensemble, ins ead he
mean and he s anda d de ia ion o he nes edness index can be analy ically calcula ed. On
he o he hand, one can always sample he ensemble in o de o s udy he s a is ics o he
a ge p ope y on a gene a ed, unbiased sampling (Squa ini e al., 2015)
Le us s a e iewing he analy ical p ocedu e. We will call a ne wo k p ope y by
X
and i s a e age ac oss he s a is ical ensemble by
hXi∗
. The as e isk supe sc ip indica es
ha he s a is ical ensemble is buil o a gi en eal bipa i e ma ix
B∗
. I his p ope y
X
can be calcula ed h ough an analy ical exp ession – ha is, as a unc ion o he ma ix
elemen s o
B
–, hen Squa ini and Ga laschelli (2011) showed ha i is possible o pe o m
an app oxima e bu accu a e measu e o he i s and second momen o
X
,di ec ly on
hB∗i
. In pa icula , o he bipa i e case, he exp essions o he a e age and he s anda d
de ia ion o X ead:
hXi∗≃X(hB∗i),(3.18)
σX≃
u
u
NP
X
p=1
NA
X
a=1 ∂X(B)
∂bpa 2
σ2
bpa ,(3.19)
whe e
σbpa
is he s anda d de ia ion o he bipa i e ma ix elemen
bpa
. These wo
exp essions can be ega ded as he esul o linea ly app oxima ing he dependence o
X
on
he ma ix elemen s, whe e second-o de e ms a e neglec ed. In pa icula , Squa ini and
Ga laschelli (2011) showed ha his es ima ion is accu a e as long as ha he p ope y
X
is
gaussian-dis ibu ed in he andom ensemble, which wa an s ha highe o de co ec ions
will be compa a i ely small.
Secondly, any p ope y –ha ing an analy ical exp ession o no – can always be measu ed
in a sample o he s a is ical ensemble. The sampling consis s in gene a ing ne wo ks by
placing links among species acco ding o he p obabili ies p o ided by he ma ix elemen s
o
hB∗i
(Eq. 3.17). The ac ha he p obabili ies
ppa
a e independen om one ano he
o e s a g ea ad an age wi h espec o o he andomizing me hods, like he swapping
algo i hm desc ibed abo e. Mo eo e , gi en ha hese p obabili ies a e ob ained imposing
he maximum-en opy and maximum-likelihood condi ions, he esul ing sample will be
signi ican ly less biased han o he choices (Squa ini e al., 2015).
We ha e desc ibed in g ea de ail his andomizing amewo k since i will be he basis o
calcula ions in he es o his chap e , as well as pa o chap e 4. Indeed, in wha ollows
we will apply hese esul s o he s udy o nes ed pa e ns, in o de o elucida e whe he
nes edness is –o is no – an independen , signi ican p ope y. Besides, he desc ip ion o
al e na i e null models will ha e hope ully p o ided a glimpse o he bigge pic u e, and
ce ainly will help us explain how ou wo k ela es o p io in es iga ions on he ele ance o
nes ed pa e ns.
3.3 B eaking he spell o nes edness
As i has been discussed in Chap e 2, ecen ly he pe inence o nes edness as a sui able
indica o o cha ac e ize he dynamics o mu ualis ic communi ies has been challenged
by a ious wo ks. In ac , i has been a gued ha ei he nes edness has no signi ican
impac on he coexis ence o mu ualis ic communi ies (James e al., 2012; G illi e al., 2017),
46

3.3. B eaking he spell o nes edness
ei he i de imen ally a ec s local s abili y (Allesina and Tang, 2012; S aniczenko e al.,
2013). Mo eo e , o he p ope ies o he obse ed ne wo ks ha e been claimed o be key
d i e s o he communi y dynamics (James e al., 2012; Feng and Takemo o, 2014). In
pa icula , he ne wo ks’ deg ee asso a i i y o he deg ee he e ogenei y ha e been iden i ied
as de e minan s o biodi e si y pe sis ence. This leads o he a o emen ioned key ques ion o
whe he nes edness, concei ed as a global ai o he eme ging a chi ec u e, is ac ually a
genuine and independen p ope y, o con a ily, i jus de i es om lowe o de ea u es
o he in e ac ion ne wo k. In wha ollows we will a emp o answe his ques ion om
a s uc u al poin o iew. In pa icula , his sec ion s a s wi h a b ie discussion on
p e ious wo ks ha add essed a simila p oblem; nex , we will desc ibe how we implemen ed
he maximum-en opy and maximum-likelihood amewo k o cons uc a null model o
ecological ne wo ks; and, inally, we will apply his scheme o e alua e he signi icance o he
nes edness o a la ge da ase composed by 167 empi ical ne wo ks ep esen ing di e se ypes
o plan -animal mu ualis ic communi ies ac oss he globe.
P edecesso s
Ea lie in es iga ions on he s uc u al de e minan s o nes edness we e concu en on he
ele ance o he deg ee sequences. Fi s ly, Medan e al. (2007) heo e ically showed ha he
isocline o pe ec nes edness can be ul ima ely ela ed, in he con inuous limi app oxima ion,
o he deg ee dis ibu ions o bo h guilds. On he o he hand, Joppa e al. (2010) iden i ied
he deg ee sequences as a ea u e ha conside ably explains empi ical nes edness, al hough
hey s ill claimed o ind ‘a s a is ically signi ican excess o ne wo ks wi h unusual nes edness
pa e ns’. Las bu no leas , Jonhson e al. (2013) explo ed he eme gence o co ela ions in
a ini e size con igu a ion model, and no ably a gued ha deg ee he e ogenei y oge he wi h
dissaso a i i y a e wo c ucial de e minan s o nes edness, o in hei wo ds: ‘... mos o he
empi ically ound nes edness s ems om he e ogenei y in he deg ee dis ibu ion. Once such
an in luence has been discoun ed – as a second ac o – we ind ha nes edness is s ongly
co ela ed wi h disasso a i i y and hence – as andom ne wo ks ha e been ecen ly ound o
be na u ally disasso a i e – hey also end o be na u ally nes ed jus as he esul o chance’.
Indeed, he esul s de i ed in his sec ion ha e much in common wi h hei conclusions,
al hough we ake a di e en me hodological pa h in o de o demons a e such dependencies.
Fu he mo e, in he con ex o building null models o measu e he signi icance o empi ical
nes ed pa e ns, o he wo ks had been con on ed o he ela ionship be ween nes edness
and he e ogenei y o deg ees, ye wi hou undamen ally add essing i . Ins ead, he majo i y
o such s udies placed he ocus on he echnical capabili y o each amily o null model o
‘de ec ’ nes edness – esul ing in o he well-known classi ica ion be ween conse a i e and
non-conse a i e es s, and he a ousal o Type I and Type II e o s– and lea ing aside
any u he , concep ual implica ions o he ob ained nes edness signi icance. Mo eo e , he
ac ual exis ence o andomizing issues o en monopolized he discussion. A good example o
his si ua ion is he deba ed abou he a o emen ioned ixed- ixed model, which has been
ex ensi ely used o assess nes ed pa e ns. A common ca ea o his model is ha he numbe
o null ne wo ks compa ible wi h he cons ained deg ee sequences migh be highly limi ed.
Fo ins ance, Ul ich and Go elli (2007) and Ul ich e al. (2009) obse ed poo ly signi ican
nes ed pa e ns when using his null model. Howbei , hey explained such esul a guing ha
he FF null model induces a bias in he sampling, due o he ac ha he gene a ed null
ma ices closely esemble he eal ne wo k, con a ily o he case whe e a andomiza ion ha
elaxes he deg ee sequences – o ins ance he EE model– is applied. In he wo ds o Ul ich
and Go elli (2007): ‘This simila i y makes i mo e di icul o he FF algo i hm o de ec
nes edness’. Likewise, S aniczenko e al. (2013) poin ed ou he limi a ions o he FF model,
gi en ha he numbe o possible null ne wo ks dec eases as he nes edness o densi y o
links o he eal ne wo k inc eases.
In spi e o all hese di e en hin s poin ing a he c ucial ques ion o he s uc u al
ele ance o nes edness, he deba e s ill emains open (Allesina, 2012). On he whole, he
di icul y in ob aining a de ini i e answe o his p oblem is ela ed o he na u e o he
me hodology ollowed in p e ious s udies. As i has been explained, such me hodology mainly
elayed on s a is ically co ela ing he obse ed deg ee o nes edness in eal sys ems wi h he
47
3. Nes edness and chance
expec a ion p o ided by he null model(s) o choice. Gi en ha , as discussed abo e, p e ious
null models ypically andomize algo i hmically he obse ed ne wo k unde some cons ain ,
hey a e mo e o en han no lawed by he eme gence o undesi ed bias —enhanced, in
u n, by he ypical small size o ecological ne wo ks. In wha ollows, we ake a di e en
me hodological pa h ha allows us o es , in a non-biased andomizing amewo k, he
signi icance o nes ed pa e ns.
Cons uc ion o he s a is ical ensemble
The maximum-en opy and maximum-likelihood ensemble in oduced abo e sol es many
o he d awbacks o al e na i e null models ha , ei he so ly o ha dly, cons ain he
deg ee sequences. In he pa icula case o ecological da a, his canonical app oach has
he ad an age ha possible missing links o o e a ed in e ac ions, which migh lead o
impo e ished ecological da a (Blü hgen e al., 2008; Olesen e al., 2010), a e deal wi h in a
p ope way. In o he wo ds, om a heo e ical pe spec i e elaxing such cons ain s e lec s
he ac ha he obse ed deg ee sequences may p o ide impe ec in o ma ion, i.e., he
epo ed ne wo k may be incomple e o con ain noisy da a like mislead in e ac ions. Indeed,
en o cing he andomized deg ee sequences o be equi alen o he empi ical ones only on
a e age limi s he possible impac o hese sho comings, while assu ing ha esul s a e no
dependen on speci ic de ails. Mo eo e , om a me hodological iewpoin , he ac ha his
null model p o ides an analy ic exp ession o he p obabili y o in e ac ion be ween species,
esul s in he compu a ional gene a ion o null ne wo ks being as , e icien and demanding
ew nume ical esou ces (Squa ini and Ga laschelli, 2011).
In o de o apply he amewo k desc ibed abo e o he s udy o mu ualis ic ne wo ks,
i s we need o cons uc a maximum en opy s a is ical ensemble o each one o he
empi ical ne wo ks we aim o examine, unde he cons ain ha he deg ee sequences in
he ensemble ma ch on a e age he empi ical ones – his being ue o he wo guilds o he
co esponding bipa i e g aph. E en ually, his p o ides a se o coupled equa ions o sol e o
he Lag ange mul iplie s (Eqs. 3.12-3.13 abo e). Hence, de e mining he s a is ical andom
ensemble o each eal mu ualis ic ne wo k en ails sol ing he co esponding op imiza ion
p oblem. In pa icula , o each ne wo k in he empi ical da ase we nume ically ound
he Lag ange mul iplie s ha maximize he likelihood using wo di e en , independen
algo i hms: (i) simula ed annealing, which is a global, pseudo- andom nume ical me hod
o op imizing he likelihood, and (ii) a de e minis ic, g adien -based algo i hm o sol ing
non-linea sys ems o equa ions. Fu he de ails on how hese me hods a e implemen ed a e
gi en in he Appendix B. Mo eo e , he codes o p oduce his ensemble o any ne wo k a e
published as an open eposi o y named nullnes in gi hub (see Appendix G).
As i has al eady been discussed, a p ima y ad an age o cons uc ing a maximum
likelihood and maximum en opy ensemble is ha , in he case o local cons ain s, he
p obabili y o exis ence o a g aph in he ensemble can be exac ly ac o ized in o he
p obabili ies o exis ence o a link be ween any wo species (see Eq. 3.15). The e o e, a e
nume ically de e mining each op imal se o Lag ange mul iplie s, we buil he ma ix
con aining he a e age p obabili y o in e ac ion co esponding o each empi ical ne wo k, as
illus a ed in Fig. 3.4 o he empi ical ne wo k epo ed by Inoue e al. (1990). The esul ing
da ase , comp ising 167 p obabilis ic ne wo ks, is as well made public as pa o he nullnes
eposi o y.
S a is ical measu es o nes edness
We pe o med he s a is ical measu es on he ensemble applying bo h he analy ical and he
nume ical app oach desc ibed in he sec ion 3.2. In wha ollows, hough, we ocus on de i ing
he analy ical exp essions o he mean and he s anda d de ia ion o he dis ibu ion o
nes edness ac oss his ensemble. In pa icula , we do so o he wo indices o nes edness o
which his is possible: he well-known nes edness me ic based on o e lap and dec easing ill
(NODF) (Almeida-Ne o e al., 2008), and he ecen ly p oposed spec al adius (S aniczenko
e al., 2013). The measu es pe o med by nume ically sampling he ensemble a e de ailed in
Appendix E.
48
3.3. B eaking he spell o nes edness
L
T
P
O
L
I
N
A
O
RS
PLANTS
Figu e 3.4:
Compa ison be ween empi ical mu ualis ic in e ac ions and p obabil-
i y o in e ac ing in he ensemble.
The p obabili y o in e ac ion be ween species in
he s a is ical ensemble gi en by
hB∗i
is shown as a colo hea map, o he plan -pollina o
ne wo k eco ded by (Inoue e al., 1990). The empi ical co esponding bipa i e ma ix o
in e ac ions
B∗
is supe imposed in black. Bo h plan s and pollina o s species ha e been
o de ed in dec easing o de o hei deg ees ( om op o bo om and om le o igh ).
As i can be seen a a glance, he ob ained p obabili ies a e consis en wi h he obse ed
in e ac ions, wi h he da k egions delimi ing an uppe le iangle, as in an ideally nes ed
s uc u e. No e ha he colo legend is in loga i hmic scale.
De i a ion o he analy ical exp essions o NODF
In his subsec ion, we de i e he analy ical exp essions o he a e age and he s anda d
de ia ion o nes edness measu ed by he index known as nes edness me ic based on o e lap
and dec easing ill (NODF). We chose his me ic among he as ange o indices in he
li e a u e due o a a ie y o easons: i s , i can be calcula ed h ough an analy ical and
compac exp ession in e ms o he ma ix elemen s; second, and con a ily o o he me ics,
i s de ini ion is based on a clea and explici quan i ica ion o condi ions in Eqs. 2.1-2.2,
p ecluding any ype o geome ic o algo i hmic app oach; and inally, i is widely used no
only in ecology bu also in ne wo k applica ions o economics (Sa acco e al., 2015; He nández
e al., 2018) o sociology (Bo ge-Hol hoe e e al., 2017).
De ini ion and dis ibu ion o NODF
As we explained in Chap e 2, he NODF index quan i ies wo aspec s o nes edness: on
he one hand, he dec easing ill, ha quan i ies he a ia ion in he deg ee sequence; and on
he o he hand he pai ed o e lap, ha weigh s he o e lap o in e ac ing pa ne s among
he nodes o a guild. To encapsula e his calcula ion, we will use he analy ical de ini ion
p oposed in he p e ious chap e (see Eq. 2.6), which, o he sake o cla i y –and a he
expense o some edundancy–, we emind:
NODF(B)=1
K
NP
X
i<j







[1 −θ( j− i)] ·
NA
P
a=1
biabja
j







+1
K
NA
X
k<l









[1 −θ(hl−hk)] ·
NP
P
p=1
bpkbpl
hl









,
(3.20)
49
3. Nes edness and chance
12345678
1
2
3
4
5
6
7
8
i
j
kl
Figu e 3.5: Example o an o de ed ma ix o in e ac ions, no pe ec ly nes ed. Species o
bo h guilds ha e been o de ed in dec easing deg ee, and he numbe ed labels indica e hei
ank ( he la ge he deg ee, he smalle he ank). The indexes
i
,
j
,
k
and
l
illus a e ou
no a ion o ows and columns.
whe e K=NP(NP−1) + NA(NA−1)
200 .(3.21)
He e we conside ha he bipa i e adjacency ma ix is labeled as shown in Fig. 3.5,
such ha ow
i
is placed abo e ow
j
and column
k
a he le o column
l
. The
K
ac o
in oduces he no maliza ion and he θs ands o he Hea iside s ep unc ion.
In ac , om now on we will use he ollowing abb e ia ions o he dec easing ill e m:
DFij = 1 −θ( j− i)
such ha , i j≥ i hen DFij = 0,
and i j< i hen DFij = 1
(3.22)
DFkl = 1 −θ(hl−hk)
such ha , i hl≥hk hen DFkl = 0,
and i hl< hk hen DFkl = 1
(3.23)
Despi e being a popula me ics, some au ho s ha e aised conce ns abou he use o
NODF o measu e nes edness. In pa icula , as discussed in Chap e 2, S aniczenko e al.
(2013) a gued ha , due o he DF ac o , he NODF is unable o de ec nes ed pa e ns
when he p opo ion o epea ed deg ees in he ne wo k is la ge. In o de o ensu e ha
ou esul s a e no a ec ed by his limi a ion in he sensibili y o NODF, we ob ained as
well he analy ical exp essions o he i s wo momen s o he al e na i e e sion o he
me ic called s able-NODF, p oposed by (Ma iani e al., 2019). As discussed in Chap e 2,
his a ian elaxes he dec easing ill condi ion and hence sol es he d awbacks ou lined
by S aniczenko e al. (2013). The analy ical exp essions o he i s wo momen s o i s
dis ibu ion in he ensemble can be ound in Appendix C. S ill, we keep ou main ocus
in NODF gi en i s widesp ead use o measu ing nes edness, oge he wi h he ac ha
he men ioned lack o sensibili y is no pa icula ly ele an bu o small o highly dense
ne wo ks (see Appendix C).
We nex e i y ha NODF is Gaussian-dis ibu ed in he ensemble, as equi ed i we
aim o apply Eqs. 3.18-3.19, by pe o ming a check on a subse o he empi ical ne wo ks.
To his end, o each o he co esponding s a is ical ensembles we gene a ed a sample o
10
4
ne wo ks obeying he p obabili y o link exis ence gi en by
hBi∗
. We hen compu ed
he nes edness o each sampled ne wo k, using NODF, in o de o gene a e he nes edness
50
3.3. B eaking he spell o nes edness
0
0.005
0.01
0.015
0.02
0.025
0.03
30 35 40 45 50 55 60 65
a io o occu ences
NODF
eal ne wo k
analy ical a e age
Figu e 3.6: Nes edness dis ibu ion, measu ed by NODF, o a sampling o he s a is ical
ensemble co esponding o he empi ical ne wo ks by (Small, 1976). In blue, i o a gaussian
unc ion using he mean and s anda d de ia ion ex ac ed om he dis ibu ion (mean
µ
= 45
.
8and s anda d de ia ion
σ
= 4
.
2). In g ey, alues o he nes edness o he eal
ne wo k and o he analy ical a e age, which co esponds o he a e age compu ed using he
analy ical exp ession in Eq. 3.24.
dis ibu ion. In all cases we could success ully i a Gaussian unc ion (see Fig. 3.6 o an
example).
Analy ical exp ession o he i s momen o NODF
The analy ical and packed exp ession o NODF ha appea s in Eq. 3.20 can hen be
plugged in o Eq. 3.18. Acco dingly, we ob ain ha he i s momen o he andomized
NODF o a gi en eal bipa i e ma ix B∗ eads:
hNODF(B)i∗=1
K
NP
X
i<j







DFij ·
NA
P
a=1 hbiaihbjai
NA
P
a=1 hbjai







+1
K
NA
X
k<l









DFkl ·
NP
P
p=1 hbpkihbpli
NP
P
p=1 hbpli









.
(3.24)
No e ha
PNA
a=1 hbpai
=
p
and
PNP
p=1 hbpai
=
ha
, gi en ha he andomized ma ix
necessa ily ul ills he en o ced cons ains. Addi ionally, his wa an s ha he o de ing o
he ma ix is equal o he o iginal one, which is impo an since NODF is o de ing-dependen
h ough he dec easing ill e ms. I is also o in e es o ema k ha he p e ious exp ession
can be unde s ood in p obabilis ic e ms. Indeed, gi en ha
hbpai
=
ppa
, whe e
ppa
a e
independen link p obabili ies, he o e lap e m migh be seen as a join p obabili y o
wo independen e en s, di ided by a no malizing ac o which is he union o independen
p obabili ies. Fo example, o he pai o animal species
k
and
l
, he o e lap e m esul s in:
NP
P
p=1 hbpkihbpli
NP
P
p=1 hbpli
=
NP
P
p=1
ppk ppl
NP
P
p=1
ppl
.(3.25)
51

3. Nes edness and chance
Analy ical exp ession o he second momen o NODF
The s anda d de ia ion is gi en by Eq. 3.19, which o NODF eads:
σNODF =
u
u
NP
X
p=1
NA
X
a=1 ∂NODF(B)
∂bpa 
2
B=<B>∗
σ2
bpa
wi h σ2
bpa =ppa (1 −ppa),
(3.26)
whe e we ha e used he ac ha he exis ence o a link in he ne wo k is a Be noulli p ocess.
Fu he mo e, he de i a i e wi h espec o a gene al ma ix elemen
b c
( he index s ands
o ows and cs ands o columns) can be spli in o he con ibu ions o plan s and o
animals:
∂NODF(B)
∂b c
=∂NODF(B)plan s
∂b c
+∂NODF(B)animals
∂b c
.(3.27)
A e de i ing, we ob ained ha :
K∂NODF(B)plan s
∂b c
=
NP
X
j= +1
DF j
bjc
j
+
−1
X
i=1
DFi
bic
−
−1
X
i=1
NA
X
a=1
DFi
bia b a
2(3.28)
K∂NODF(B)animals
∂b c
=
NA
X
l=c+1
DFcl
b l
hl
+
c−1
X
k=1
DFkc
b k
hc−
c−1
X
k=1
NP
X
p=1
DFkc
bpk bpc
hc2,(3.29)
which a e being plugged in o Eq. 3.26 p o ides an analy ical exp ession o he s anda d
de ia ion o he dis ibu ion o NODF in he ensemble.
De i a ion o he heo e ical exp essions o he spec al adius
In his subsec ion, we de i e he heo e ical exp essions o calcula ing he a e age and
s anda d de ia ion o nes edness using he so-called spec al adius (S aniczenko e al., 2013).
We pe o med he s a is ical measu es using his me ic due o i s inc easing popula i y
among nes edness indices, and, mo eo e , o i s compu a ional ad an ages: i s ly, i is a
ma hema ical p ope y o he g aph which does no depend on he o de ing o he ma ix
and, secondly, i s nume ical calcula ion is as . None heless, i is wo hy o emind well
ha he measu es ob ained wi h he spec al adius should be handled wi h ca e, since as
discussed in Chap e 2 his me ic does no eliably quan i y nes edness a a ine scale and,
mo eo e , i is no no malized.
De ini ion and dis ibu ion o he spec al adius
The spec al adius was ecen ly p oposed by (S aniczenko e al., 2013) as an al e na i e
me ic o nes edness ha di ec ly elies on he spec al p ope ies o he adjacency ma ix.
Al hough we al eady in oduced his me ics in Chap e 2, o imp o e he eadabili y o
his hesis we will emind i s de ini ion he e. Le us call
I
he iden i y ma ix and
A
he
adjacency ma ix o a bipa i e ma ix B, such ha :
A=0B
B|0,(3.30)
which is a squa e, symme ic and non-nega i e ma ix, gi en ha
ai,j ≥
0. The spec al
adius o he ma ix
A
(also called dominan eigen alue o la ges eigen alue) is de ined as
ollows:
52
3.3. B eaking he spell o nes edness
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
7 7.5 8 8.5 9 9.5 10 10.5
a io o occu ences
spec al adius
eal ne wo k
heo e ical a e age
Figu e 3.7: Dis ibu ion o he spec al adius o e he ensemble calcula ed o he eal
ne wo k collec ed by Small (1976), o a sampling made o 10
4
ne wo ks. In blue, i o a
gaussian unc ion using he mean and s anda d de ia ion ex ac ed om he dis ibu ion
(mean
µ
= 8
.
8and s anda d de ia ion
σ
= 0
.
4). In g ey, alues o he nes edness o he eal
ne wo k and o he heo e ical a e age, which co esponds o he a e age compu ed using
he heo e ical exp ession in Eq. 3.33.
ρ(A) = max{|λi|}.(3.31)
Whe e
λii∈ {
1
, ..., n}
a e he eigen alues o
A
, hus he oo s o he equa ion:
de (Iλ−A)=0. Since Ais a symme ic ma ix, λi∈Re ∀i.
In o de o apply he analy ical exp essions o he i s wo momen s o i s dis ibu ion,
i s we checked whe he he spec al adius is Gaussian-dis ibu ed o e he ensemble. In
pa icula , o each ne wo k in ou da ase , we gene a ed a sample o 10
4
ne wo ks obeying
he p obabili y o link exis ence p o ided by
hBi∗
. Then, we calcula ed he spec al adius
o each sampled ne wo k algo i hmically using he
R
package ARPACK (Qiu and Mei, Qiu
and Mei). Finally, we e i ied ha he esul ing dis ibu ion is indeed no mal, as can be
seen in Fig. 3.7.
Theo e ical exp ession o he i s momen o he spec al adius
Gi en ha
ρ
(
A
)is a unc ion o he ma ix en ies o
A
, we can apply he linea
app oxima ion p oposed by Squa ini and Ga laschelli in Eq. 3.18, in o de o es ima e he
a e age o e he ensemble compu ed o a eal bipa i e ma ix B∗:
hρ(A)i∗≈ρ(hAi∗),(3.32)
whe e:
hAi∗=0hBi∗
hBi∗|0(3.33)
This means ha he a e age spec al adius can be ound as:
ρ(hAi∗) = max{|hλii|},(3.34)
whe e hλiia e he oo s o he equa ion:
53
3. Nes edness and chance
de (Ihλi−hAi∗)=0.(3.35)
In p ac ice, Eq. 3.35 has o be sol ed nume ically, which implies ha no analy ical
exp ession o he a e age o he spec al adius exis s. In pa icula , we nume ically
implemen ed he calcula ion o he spec al adius o each ma ix
hAi∗
using again he
R
package ARPACK (Qiu and Mei, Qiu and Mei).
Analy ical exp ession o he second momen o he spec al adius
Using Eq. 3.19, he s anda d de ia ion o he spec al adius o e he ensemble can be
es ima ed by:
σρ≃
u
u
NP
X
p=1
NA
X
a=1 ∂ρ(A)
∂Apa 
2
A=<A>∗
σ2
Apa .(3.36)
He e, he calcula ion o he de i a i e o he spec al adius,
∂ρ(A)
∂Apa |A=<A>∗
, is non- i ial,
gi en ha he e is no gene al analy ical exp ession o he spec al adius because Eq. 3.35
needs o be sol ed nume ically. None heless, we will now show how is i possible o ob ain
such de i a i e by applying he esul s by Deu sch and Neumann (1984).
Le us s a by assuming ha
M
is a squa e, non-nega i e and i educible ma ix. Then,
i has been shown ha i s spec al adius
ρ
(
M
)is a simple eigen alue and i is equal o i s
Pe on oo . Since
ρ
(
M
)is a simple eigen alue, i has mul iplici y one and i is possible o
ob ain i s i s de i a i es wi h espec o
Mij
. Indeed, i we deno e by
D
he ma ix whose
ma ix elemen s a e:
Dij =∂ρ(M)
∂Mij
,(3.37)
hen, ollowing (Deu sch and Neumann, 1984),
D
can be compu ed using he exp ession:
D=I−QQ#|
.(3.38)
He e,
Q
is a special ype o ma ix known as M-ma ix (Ki kland and Neumann, 2012)
and de ined as:
Q=ρ(M)I−M,(3.39)
while
Q#
is he g oup in e sion o
Q
(Ben-Is ael and G e ille, 2003). The g oup in e sion
is a mo e gene al ype o in e se ha can be applied as well o singula ma ices. Fo a ce ain
ype o ma ices, he g oup in e se is equi alen o ano he class o in e sion known as Moo e-
Pen ose in e se. This is ue i and only i he ma ix o s udy is ange-He mi ian (Ben-Is ael
and G e ille, 2003). One o he condi ions ha wa an s ha a ma ix is ange He mi ian is
he ollowing:
ange (Q) = ange (QH)(3.40)
whe e
QH
is he conjuga e anspose (also called He mi ian conjuga e) o
Q
. I we now
assume ha Qis ange-He mi ian, exp ession 3.38 can be ew i en in o:
D=I−QQ†|
,(3.41)
whe e Q† ep esen s he Moo e-Pen ose in e se o Q.
Le us show now ha Eq. 3.41 can be used o calcula e he de i a i e wi h espec o ou
ma ix o in e es
A
, in pa icula in he case whe e
A
=
hAi∗
. Fi s ly, we will show ha
hAi∗
ul ills he condi ions ha allow us o apply equa ion 3.38. Nex , we will p o e ha a
ma ix QAde ined as:
QA=ρ(hAi∗)I− hAi∗,(3.42)
54
3.3. B eaking he spell o nes edness
is a ange-He mi ian ma ix and, consequen ly, Q#
A=Q†
A.
Fi s , we know al eady ha
hAi∗
is a squa e and non-nega i e ma ix, ye i emains
o be shown whe he i is i educible. A ma ix is said o be i educible i and only i
i s co esponding g aph is s ongly connec ed, ha is, i i is possible o ind a pa h ha
connec s any pai o nodes o he ne wo k. Because
hBi∗
is a comple e bipa i e g aph
(
hbiji∗>
0
∀i, j
), hen i is clea ha i is s ongly connec ed and he e o e
hAi∗
is an
i educible ma ix.
Second, he condi ion o he ma ix
QA
o be ange-He mi ian is p o ided by Eq. 3.40.
In ou case,
ρ
(
A
)
∈Re
and
Aij ∈Re ∀i, j
, he e o e
QA,ij ∈Re ∀i, j
. F om his ollows
ha :
ange (QH
A) = ange (Q|
A)(3.43)
Whe e we ha e used ha he conjuga e anspose o a eal ma ix is simply i s anspose.
We s ill need o p o e ha :
ange (Q|
A) = ange (QA).(3.44)
This condi ion is equi alen o:
ow space o QA=column space o QA.(3.45)
No e ha his is no ue in gene al. In ou case, gi en ha
QA
is a squa e and symme ic
ma ix, i s ow and column spaces a e equal and he e o e Eqs. 3.44 and 3.45 a e e i ied.
This p o es ha
QA
is ange-He mi ian and consequen ly i s g oup in e se is equi alen o
i s Moo e-Pen ose in e se. Wi h his we ha e shown ha a ma ix DAde ined by:
DA,ij =∂ρ(A)
∂Aij A=<A>∗
,(3.46)
can be compu ed using he ollowing exp ession:
DA=I−QAQ†
A|
,(3.47)
which comple es Eq. 3.36 and hus p o ides an analy ical exp ession o he calcula ion
o he s anda d de ia ion o he spec al adius. We implemen ed Eq. 3.47 using he
R
MASS package (Venables and Ripley, 2002), in pa icula he unc ion gin o calcula e he
Moo e-Pen ose in e se.
Signi icance o empi ical nes ed pa e ns
Wi h hese analy ical exp essions cha ac e izing he nes edness’ dis ibu ion a hand, one can
now a emp o examine how he nes edness o eal ne wo ks ela es o i s null expec a ion.
Indeed, we do so o each o he 167 empi ical ne wo ks in ou da ase . This empi ical se
includes h ee di e en kinds o mu ualis ic communi ies –plan -pollina o , seed-dispe se
and plan -an –, co e ing a wide a ie y o geog aphical loca ions, clima e condi ions and
species composi ion (see Appendix A o u he de ails).
Using exp essions 3.24 and 3.26 i is possible o compu e he expec a ion alue o
nes edness measu ed by NODF,
hNODF(B)i∗
, and i s s anda d de ia ion, o each empi ical
ne wo k in ou da ase . A compa ison be ween he expec ed alue o nes edness calcula ed
o e he s a is ical ensemble co esponding o each eal ne wo k and he ac ual nes edness
o he eal ne wo k shows a s iking ag eemen , see Fig. 3.8. As epo ed in Table 3.1, he
absolu e di e ence be ween hese wo quan i ies is less han one s anda d de ia ion o 100
ou o 167 ne wo ks (59.9%), aising o 158 ou o 167 ne wo ks (94.6%), i we accoun o
wo s anda d de ia ions. A e pe o ming a mul iple es ing co ec ion (see Appendix D),
we ind ha only 3 ou o he 167 empi ical ne wo ks show signi ican nes edness (co ec ed
p
- alue
<
0
.
01). The h ee o hem, which a e o a ela i ely small size (
≤
55 species), we e
ound o be less nes ed han p edic ed by he s a is ical ensemble.
Mo eo e , we e i ied ha hese esul s a e no a ec ed by he sho comings ela ed o
he dec easing ill ac o , by epea ing he abo e calcula ions using he index s able-NODF,
55
3. Nes edness and chance
heo e ical dis ibu ion, assigning hem acco ding o he dis ibu ion o a eal ea u e. A
success ul example ha ca ies ou his idea is he wo k by Ga laschelli and Lo edo (2004),
whe e he au ho s ind ha he g oss domes ic p oduc (GDP) o coun ies can be iden i ied
as he i ness a iables de e mining he opology o he ne wo k o comme cial in e ac ions
among hem, i.e. he Wo ld T ade Web (WTW). An e en mo e in e es ing s udy, ha
mo eo e inally ela es o he cen al ques ion o his sec ion, is a con inua ion wo k in
which Ga laschelli and Lo edo (2008) showed ha such i ness alues, and hus in u n he
GDP, coincide wi h he Lag ange mul iplie s ob ained when cons uc ing a canonical null
model o he WTW – ha is, by maximizing he p obabili y o occu ence o he eal deg ee
sequence o he WTW in a maximum-en opy ensemble wi h cons ained deg ee sequences.
Al oge he , his sugges s ha in he p esen applica ion o he ERG o malism o ecological
mu ualis ic ne wo ks, he LMs a e no only a heo e ical ool bu could con ey some ele an
in o ma ion abou he s uc u al cons uc ion o he web o in e ac ions. The e o e, ollowing
he spi i o he ela ionship ound by Ga laschelli and Lo edo (2004, 2008), we examined
whe he he species’ abundances could be playing he ole o he i ness a iables. As
a o emen ioned, he dis ibu ion o abundance has been ecu en ly iden i ied as one o he
signi ican ac o s shaping he s uc u e o ecological communi ies (Vázquez e al., 2009),
pa icula ly modula ing he deg ee o mu ualis ic gene aliza ion (see sec ion 2.3, specially
he discussion on he passi e sampling hypo hesis).
a) b)
c)
d)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 1 2 3 4 5 6 7
Rela i e abundance o plan s
Lag ange mul iplie s
0
0.05
0.1
0.15
0.2
0.25
0 1 2 3 4 5 6 7
Rela i e abundance o lowe s
Lag ange mul iplie s
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 5 10 15 20 25
Rela i e abundance o pollina o s
Lag ange mul iplie s
0
0.005
0.01
0.015
0.02
0.025
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Rela i e abundance o pollina o s
Lag ange mul iplie s
Figu e 3.12: Rela ion among he ela i e abundance and he lag ange mul iplie s o :
a)
plan s,
b)
lowe s and
c)
pollina o s. The
d)
panel shows he de ail o plo c). The Lag ange
mul iplie s a e eescaled o he mean. The da a co espond o a plan -pollina o ne wo k,
collec ed in he es o ed si e by Kaise -Bunbu y e al. (2009).
In o de o add ess his ques ion, a leas p elimina y, we analyze wo plan -pollina o
ne wo ks epo ed by Kaise -Bunbu y e al. (2009, 2010). Bo h ne wo ks we e obse ed in
Mau i ian ecosys ems, bu one o hem co esponds o a ecen ly es o ed habi a , while
he second one was eco ded in a un es o ed si e. Kaise e al. also p o ide he abundances
o species, bu while he measu es o ela i e abundance o plan s we e di ec ly es ima ed
h ough a sampling me hod based on coun ing indi iduals – and he lowe ing densi y was
es ima ed simila ly–, o pollina o s he au ho s p o ide an in e ed measu e, aking as he
o al abundance pe specie i s o al numbe o obse ed isi s.
Ou esul s (Figs. 3.12-3.13) clea ly indica e a posi i e co ela ion be ween LMs and
ela i e abundance o pollina o s, bu ins ead we ind no co ela ion o plan s species in any
o he possible measu es o abundance –i.e. lowe s abundance o whole plan s abundance.
62

3.5. Conclusions and pe spec i es
a) b)
c)
d)
0
0.05
0.1
0.15
0.2
0 2 4 6 8 10 12 14
Rela i e abundance o lowe s
Lag ange mul iplie s
0
0.2
0.4
0.6
0.8
0 2 4 6 8 10 12 14
Rela i e abundance o plan s
Lag ange mul iplie s
0
0.05
0.1
0.15
0.2
0.25
0 2 4 6 8 10 12 14 16 18 20
Rela i e abundance o pollina o s
Lag ange mul iplie s
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Rela i e abundance o pollina o s
Lag ange mul iplie s
Figu e 3.13: Rela ion among he ela i e abundance and he lag ange mul iplie s o :
a)
plan s,
b)
lowe s and
c)
pollina o s. The
d)
panel shows he de ail o plo c). The Lag ange
mul iplie s a e eescaled o he mean. The da a co espond o a plan -pollina o ne wo k,
collec ed in he un es o ed si e –also named con ol si e– by Kaise -Bunbu y e al. (2009).
As can be seen, in compa ison o he es o ed si e, he e he plan popula ion is domina ed
by a supe abundan species.
As a ma e o ac , he la e esul is in acco dance wi h o he s udies which claim ha ,
while abundan species end o be gene alis s, no all gene alis s a e abundan Fo e al.
(2016). Mo eo e , conside ing he obse ed co ela ion o pollina o s, we can no dismiss
he possibili y o i being an a i ac o he measu ing echniques. 3
On he whole, his sugges s ha we may be acing a mo e complex ela ionship han he
one encoun e ed by (Ga laschelli and Lo edo, 2008) in he con ex o in e na ional ading
and calls o u he esea ch. P ospec i ely, o he ac o s such as phenology o ai s could
be aken as well in o accoun in o de o be e alua ed as candida es o i ness a iables ha ,
hope ully, explain he ecological meaning o he LMs.
3.5 Conclusions and pe spec i es
Along his chap e we ha e examined he igh ela ion be ween nes edness and chance,
by looking i s a how e olu ion migh –o in ac migh no – accoun o he complex
a chi ec u e o in e ac ions wi hin ecological communi ies, and secondly ocusing on he
eme gence o s uc u al pa e ns in mu ualis ic ne wo ks.
In pa icula , by exploi ing a powe ul andomizing amewo k based on a maximum
en opy and maximum likelihood ensemble, we ha e demons a ed ha he deg ee sequences
ul ima ely de e mine he obse ed nes edness and deg ee disasso a i i y o eal mu ualis ic
sys ems. The eme gence o hese non- i ial ea u es is jus i ied by i s highly en opic
cha ac e , ha is, by he ac ha hey a e he mos p obable con igu a ion o links gi en
he condi ion o so ly cons aining he deg ee sequences, in he absence o o he o ces. As i
has been ex ensi ely discussed, such inding has b oad implica ions o he unde s anding o
he o igin and he ole o nes ed pa e ns in ecological communi ies. Indeed, i alludes ha ,
3
Indeed, Kaise -Bunbu y e al. sampling me hod was based on an equal epa i ion o obse a ional ime
among plan s, in o de o minimize he possible unde sampling o a e species. Howe e , his me hodology
could easily lead o a biased quan i ica ion o pollina o s’ abundance, gi en ha gene alis animal species
could ha e a la ge p obabili y o being obse ed.
63
3. Nes edness and chance
despi e nes ed pa e ns en ail a ce ainly non- i ial o ganiza ion o mu ualis ic in e ac ions,
hei complexi y is no he p oduc o na u al selec ion –as he so-called panselec ionis s
would assume– bu a he he mac oscopic consequence o o he pe inen s uc u al ea u es,
pa icula ly he deg ee sequences.
Mo eo e , he adequacy o his null model o accu a ely ep oduce he obse ed s uc u e
o eal ne wo ks sugges s he possibili y o de eloping u he applica ions beyond he p esen
analysis o he signi icance o nes ed pa e ns. Fo ins ance, he encoun e ed Lag ange
mul iplie s can be ega ded as i ness a iables ha no only s and as a me hodological ool
o de e mine he null ensemble, bu also may ca y some ele an con en abou he empi ical
d i e s o he ne wo k’s a chi ec u e. As a o emen ioned, hough, ou esul s conce ning his
opic a e s ill inconclusi e and equi e u u e in es iga ions. Ano he , mo e di ec applica ion
o he me hods de eloped along his chap e is he e alua ion o o he nes ed me ics, in
o de o assess i s pe o mance and un eil i s dependencies on o he ne wo k pa ame e s.
This opic is, indeed, he endea o o he nex Chap e .
64
CHAPTER 4
Many ule s o one leng h: how o
quan i y nes ed pa e ns
Admi edly, as a esul o he e o o quan i y nes edness, a a ie y o me ics wi h hei
co esponding nes edness indices coexis in he ecological li e a u e. Howe e , since hey a e
based on di e se bu no necessa ily independen p ope ies o he nes ed ne wo ks, how o
compa e he deg ee o nes edness o di e en ecosys ems emains unclea . The si ua ion ecalls
he well-known his o y o he de ini ion o empe a u e in The modynamics. Ini ially de ined
ope a ionally, i.e., by lis ing he p o ocol o measu e i , he ob ained empe a u e alues
su e ed om he law ha hey depended on he he mome e used. This p oblem was sol ed
by he heo e ical de ini ion o he empe a u e based on he Second P inciple o Clausius,
and inally he no ion o empe a u e was comple ely unde s ood by he mic oscopic app oach
o S a is ical Physics in oduced by Bol zmann and Gibbs. In e es ingly, he i s me ics
o nes edness de ined by A ma and Pa e son (1993) was called empe a u e. This ini ial
p oposal was ollowed by a long s uggle o ind he bes index o measu e nes edness, wi h
he de elopmen o a ious app oaches anging om algo i hmic p ocedu es o analy ical
me hods. In his chap e I add ess he ques ion o how o quan i y nes ed pa e ns by
ocusing on he compa ison be ween he mos ele an o such me ics and hei philosophy,
pe o mance and de ails.
As a ma e o ac , he me ics de ined o quan i y nes edness su e om a c i ical
d awback: as hey a e s ongly dependen on di e en ne wo k pa ame e s like size, ill, e c,
he compa ison among ecosys ems is di icul , e en in he case whe e he same me ics ( he
same he mome e ) is used o measu e all he sys ems. These p oblems ha e been epo ed by
se e al au ho s, no ably on he occasion o he in oduc ion o each new index and/o package
de o ed o co ec some o he sho comings o p e iously exis ing ones (Rod íguez-Gi onés
and San ama ía, 2006; Almeida-Ne o e al., 2008; Do mann e al., 2009; Bu gos e al., 2009;
Galeano e al., 2009). S ill, hese wo ks mainly ocus on he dependence on he size and he
densi y o links o he ne wo k o a ew me ics, lea ing aside o he impo an nes edness
indices as well as he in e dependencies among ne wo k pa ame e s.
In o de o o e come he a o emen ioned di icul ies when measu ing and compa ing he
nes edness o di e en ne wo ks, he s anda d p ocedu e is o con as he nes edness alue
o a gi en eal ne wo k wi h ha o a null model (see sec ion 3.2), bo h calcula ed using
he same me ics. Howe e , while he majo i y o nes edness me ics ha e been es ed o
algo i hmically-based null models (Ul ich and Go elli, 2007; Almeida-Ne o e al., 2008), hei
beha io in maximum-en opy ensembles is s ill la gely unexplo ed.
In his chap e , we ocus on he p oblem o measu ing nes edness by p esen ing a
compa a i e s udy o he beha io o he six nes edness me ics e iewed in sec ion 2.4, mos
o which a e commonly included in popula packages and ci ed in he li e a u e. Ou pu pose
is wo- old: i s , we aim o es he pe o mance o hese me ics unde he maximum-en opy
null model explained in Chap e 3 and ecen ly used in Pay a ó-Bo às e al. (2019), and
secondly, we in end o c i ically assess he unc ioning o each me ics by analyzing i s
dependencies wi h ne wo k pa ame e s. By doing his we mean o, i s , ill a gap in he
li e a u e conce ning null models, and second, o p o ide a p ac ical guide o he ad an ages
and disad an ages o each nes edness me ics.
65
4. Many ule s o one leng h: how o quan i y nes ed pa e ns
This chap e is s uc u ed as ollows. The i s sec ion is de o ed o in oducing a
no malized e sion o he spec al adius. Secondly, I measu e he nes edness in he ensemble
buil o each o he eal ne wo ks acco ding o each o he me ics and I compa e he esul s
wi h he co esponding nes edness alue o he obse ed ne wo k. In he subsequen sec ion,
I pe o m a ious s a is ical analyses o de e mine he ela ion o each me ics wi h ne wo k
p ope ies, like size, ill, deg ee degene acy, e c. Nex , I discuss he implica ions o hese
indings o he pe o mance o each me ics; and I inish he chap e wi h some gene al
ema ks and conclusions on how o choose a nes edness me ics.
4.1 No malizing he spec al adius
As discussed in sec ion 2.4, he spec al adius is a ecen ly p oposed nes edness me ics ha
exploi s he spec al p ope ies o nes ed g aphs (S aniczenko e al., 2013), cha ac e ized
by being independen o he o de ing o he g aph and as o calcula e. None heless, an
impo an d awback o he spec al adius is ha i is no no malized. Rema kably, he
heo em on which i is based (Bell e al., 2008) equi es he size o he ma ix and he numbe
o links o be ixed, a condi ion ha should be conside ed in o de o se a benchma k
agains which o escale he inal measu e. Al hough he FF null model espec s he equi ed
hypo hesis, we al eady discussed ha i leads o a poo s a is ics due o he ela i ely
ew numbe o ne wo ks ma ching hese cons ain s in ini e and small sys ems. Fo hese
easons, we p opose o no malize he spec al adius ob ained o each sys em wi h ha
o he pe ec ly nes edness ma ix ha ing he same size and ill, as was al eady sugges ed
by S aniczenko e al. (2013). Tha is, i
ρ
ep esen s he spec al adius o a eal ne wo k
and
ρmax
he spec al adius o a pe ec ly nes ed g aph wi h he same size and ill, he
no malized index ρno m is gi en by he exp ession:
ρno m = 100 ρ
ρmax
(4.1)
In o de o calcula e he no malized e sion o his me ics, we need an es ima ion o he
la ges spec al adius o a pe ec ly nes ed ne wo k o he same size and ill. To es ima e
each o hese alues, o each eal ne wo k in ou da ase we p oduced 100 new ne wo ks,
cha ac e ized by being pe ec ly nes ed. These ne wo ks we e gene a ed using he SNM
algo i hm in oduced by Bu gos e al. (2007), which p ese es he numbe o connec ed nodes
and links, bu modi ies he pa e n o connec ions and he deg ee sequences. This algo i hm
is di ided in o wo p ocedu es. Fi s , he eal ne wo k is andomized p ese ing only he ill
and he size – ha his, ensu ing ha e e y node has a leas one connec ion. Second, he
SNM algo i hm is pe o med, which consis o i e a ing he ollowing ules:
•
We a emp o modi y a link by p oposing a new pa ne , andomly selec ed bu
di e en o he o iginal node. The ewi ing is suscep ible o being accep ed only i he
new pa ne has a la ge deg ee han he p e ious one. This s ep pe o ms a s a ic
e sion o p e e en ial-a achmen .
•
I he p oposed econnec ion lea es one o he nodes wi h ze o deg ee, he mo e is
disca ded. This ensu es ha he numbe o connec ed nodes does no change, hus
p ese ing he ne wo k size.
By i e a ing o e hese s eps iand ii, one gene a es a new ma ix which is mo e nes ed as
well as mo e he e ogeneous in i s deg ee sequences han he o iginal one (see Fig. 3.10). The
i e a ion s ops when no mo e mo es a e allowed. Howe e , gi en condi ion ii, his p ocess is
no unique and migh end up in mul iple pe ec ly nes ed con igu a ions. To handle his, we
gene a ed se e al op imal con igu a ions pe each eal ne wo k. Speci ically, we gene a ed
100 new ne wo ks o each empi ical ne wo k, and excep ionally, o compu a ional easons,
50 ne wo ks o he e y la ge Robe son ne wo k. This means ha he no malized spec al
adius is ac ually calcula ed as:
ρno m = 100 ρ
PNpe
i
ρpe ec ,i
Npe
,(4.2)
66
4.2. Me ics’ beha io in he canonical ensemble
whe e
ρ
is he spec al adius o he eal ne wo k and
ρpe ec ,i
ep esen s an op imal
con igu a ion wi h he same size and ill o he eal ne wo k, p oduced by he SNM algo i hm.
Npe
co esponds o he numbe o pe ec ly nes ed gene a ed, ha in gene al we se o 100.
When sampling he ensemble, we gene a ed 10 pe ec ly nes ed ne wo ks pe each null
ne wo k (
Npe
= 10), and in o de o keep he calcula ions compu a ionally easible we
educed he sampling size o 500 null ne wo ks (Nsamp = 500).
Acco dingly, he a e age no malized spec al adius is calcula ed as:
hρno mi= 100
Nsamp
X
j
ρnull,j
Nsamp PNpe
i
ρpe ec ,i,j
Npe
,(4.3)
whe e
ρnull,j
ep esen s a null ne wo k sampled om he s a is ical ensemble and
ρpe ec ,i,j
ep esen s a pe ec con igu a ion p oduced wi h he SNM algo i hm, ha ing he same size
and ill as he co esponding null ne wo k.
In he nex sec ions, we s udy bo h e sions o his me ics: he unno malized o iginal
one along wi h he no malized modi ica ion gi en by Eq. 4.1.
4.2 Me ics’ beha io in he canonical ensemble
To s a wi h, we ha e measu ed he nes edness o 191 empi ical ecological ne wo ks ex ac ed
om he Web o Li e as well as 8 economic ne wo ks which ep esen he ading in e ac ions
be ween he buye s and he selle s o wo di e en ish ma ke s s udied by He nández e al.
(2018) (see Appendix A o a de ailed accoun o he da ase ). In o de o compa e he
a e age nes edness o e he ensemble wi h ha co esponding o empi ical ne wo ks, we ha e
used he six me ics desc ibed in he sec ion 2.4 plus wo a ia ions, namely: he Tempe a u e,
he nes edness me ics based on he Manha an dis ance (NMD), he NODF and i s ecen
a ian he s able NODF, he disc epancy, he nes edness index based on obus ness (NIR),
and inally he spec al adius and he abo e in oduced no malized modi ica ion.
As i has been shown analy ically and nume ically in he p e ious chap e , he nes ed
s uc u e o mu ualis ic ne wo ks is a consequence o he double he e ogenei y in he deg ee
sequence which esul s om en opic e ec s. In o de o in es iga e i he o he nes edness
indices a e able o e eal his dependence, we buil a null model o each eal ne wo k –as
explained in sec ion 3.2 and Appendix B– and we compa ed he nes edness o eal ne wo ks
wi h hei co esponding a e age o e he ensemble. In pa icula , o each eal ne wo k in
ou da ase , we sampled 10
4
null ne wo ks wi h he ob ained p obabili y in e ac ion ma ix
o Eq. 3.17. Ac oss he same sample, each o hese null ma ices may a y in i s size (numbe
o connec ed nodes), densi y o links, deg ee sequence, edundancy o deg ees o bipa i e
ma ix eccen ici y. Ne e heless, he deg ee sequences a e main ained, on a e age, equal o
he empi ical ones.
Nex , o each o he s udied me ics, he a e age alue o nes edness o e he andomized
ensemble has been ob ained by nume ically calcula ing each me ics o e he sampled ne wo ks
and hen cons uc ing he o al es ima ed dis ibu ion. The de ails o he implemen a ion o
hese measu es a e gi en in he Appendix F. Fo he sake o cla i y and o homogenize he
eading o he di e se igu es, I ha e ans o med he de ini ion o he empe a u e, he NMD
and he disc epancy indices so ha he la ge he index he mo e nes ed he sys em is. We
ha e also escaled hese indices so ha hey a y be ween 0 and 100. These modi ica ions
ead as ollows:
T= 100 −TAP ,(4.4)
NMD = 100 (1 −τ),(4.5)
∆0= 100 1−∆
E.(4.6)
(4.7)
whe e
T
is he empe a u e,
NMD
he nes edness me ics based on he Manha an
dis ance, ∆is he disc epancy index, and E he o al numbe o edges in he ne wo k.
67

4. Many ule s o one leng h: how o quan i y nes ed pa e ns
The Fig. 4.1 shows he nes edness measu ed o e he ensemble e sus he nes edness o he
co esponding eal ne wo k. Consis en ly wi h he esul s ob ained in he p e ious chap e
using NODF and he spec al adius, NIR and NMD also show ha he nes edness alues
o he empi ical ne wo ks a e s a is ically equi alen o he a e age o he co esponding
andomized ensemble. This leads o he conclusion ha he obse ed nes edness measu ed
by hese indices is no signi ican . On he con a y, he disc epancy and empe a u e indices
show a clea bias, wi h an impo an ac ion o he eal ne wo ks being less nes ed han he
andom a e age.
4.3 In luence o ne wo k ea u es
The esul s p esen ed in Fig. 4.1 e eal ha he me ics s udied beha e in di e en ways
unde he same null model, showing dis inc le els o luc ua ions and some imes a sys ema ic
bias, as i is he case o he disc epancy and empe a u e indices. This inding sugges s
ha he di e en algo i hms implemen ed by each me ics may e en ually ansla e in o
non-equi alen nes edness measu es. We explo e u he his si ua ion in Fig. 4.2, whe e
we compa e he alues o nes edness ob ained o a g oup o mu ualis ic ne wo ks when
measu ed using each o he me ics. As i can be obse ed, o he same da ase no only he
alue o nes edness i sel bu also he anking o he ne wo ks acco ding o hei deg ee o
nes edness is s ongly me ics’ dependen .
Ideally, as i has been ecalled by se e al au ho s (S aniczenko e al., 2013; Ul ich e al.,
2009; Almeida-Ne o e al., 2008), a well-beha ed nes edness me ics ough o be independen
o he pa icula ne wo k pa ame e s and, u he mo e, ank he deg ee o nes edness o
a gi en se o ne wo ks uni e sally. The esul s discussed abo e pu in e idence ha he
second condi ion is no always ue. Rega ding he i s equi emen , we nex explo e mo e
ca e ully how he nes edness alues gi en by each me ics depend on he ne wo k pa ame e s.
In pa icula , since he ne wo ks o he da ase co e a wide ange o pa ame e alues (see
Fig. 4.2 o an example), we analyze he e ec s o h ee cha ac e is ic ne wo k p ope ies:
size, densi y o links and eccen ici y. These quan i ies a e de ined as ollows:
size ≡s=n+m , (4.8)
densi y o links ≡φ=E
n+m,(4.9)
eccen ici y ≡=
n−m
n+m
,(4.10)
(4.11)
whe e, as be o e,
n
and
m
a e, espec i ely, he numbe o ows and columns o he
bi-adjacency ma ix, while
E
is he o al numbe o links. The eccen ici y quan i ies he
di e ence be ween he numbe o nodes o he wo guilds, o in o he wo ds, he de ia ion
om a squa e-shaped bi-adjacency ma ix. Indeed,

= 0 o a squa e ma ix and
→
1when
one o he guilds is much la ge han he o he . In e es ingly, mos o he la ge ecological
ne wo ks obse ed show mo e columns (animal species) han ows (plan species), wi h a
equen a io o 1 o 3. This obse a ion, hough, canno be gene alized o all mu ualis ic
ne wo ks, specially o small ne wo ks (which can be much mo e eccen ic) o o non ecological
sys ems.
Addi ionally, we s udy he dependence o nes edness on a ou h pa ame e , he deg ee
degene acy. In pa icula , a pe ec nes ed ma ix wi h an a bi a y
φ
migh ha e se e al
species o each guild wi h he same deg ee. We measu e his quan i y as:
degene acy in deg ees ≡g=numbe o species wi h he same deg ee
n+m.(4.12)
The s udy o his pa ame e emains a special case, since he known connec ion be ween
he nes ed pa e ns and he deg ee sequences en ails ha a ce ain dependency wi h he
68
4.3. In luence o ne wo k ea u es
50
60
70
80
90
100
110
50 55 60 65 70 75 80 85 90 95 100
a e age empe a u e in he ensemble
empe a u e eal ne wo k
20
30
40
50
60
70
80
90
100
20 30 40 50 60 70 80 90 100
a e age NMD in he ensemble
NMD eal ne wo k
0
10
20
30
40
50
60
70
80
90
100
0 10 20 30 40 50 60 70 80 90
a e age NODF in he ensemble
NODF eal ne wo k
0
10
20
30
40
50
60
70
80
90
100
0 10 20 30 40 50 60 70 80 90 100
a e age s-NODF in he ensemble
s-NODF eal ne wo k
10
20
30
40
50
60
70
80
90
100
110
10 20 30 40 50 60 70 80 90 100
a e age disc epancy in he ensemble
disc epancy eal ne wo k
0
10
20
30
40
50
60
70
80
90
100
110
10 20 30 40 50 60 70 80 90 100
a e age NIR in he ensemble
NIR eal ne wo k
0
5
10
15
20
25
30
35
40
0 5 10 15 20 25 30 35 40
(
ρ(λ) eal ne wo k
ρ(λ) andomized
40
50
60
70
80
90
100
110
120
40 50 60 70 80 90 100
no malized
(
ρ(λ) andomized
no malized ρ(λ) eal ne wo k
a) b)
c) d)
)
h)
g)
e)
Figu e 4.1:
Signi icance o he nes edness o eal ne wo ks.
The igu e shows he
empi ical measu e o nes edness agains he a e age alue o nes edness in he gene a ed
s a is ical ensemble o he 199 empi ical ne wo ks in ou da ase . The di e en panels
co espond o di e en me ics: (a) empe a u e, (b) NMD, (c) NODF, (d) s able-NODF, (e)
disc epancy, ( ) NIR, (g) spec al adius and (h) no malized spec al adius. The shadowed
a eas ep esen one (salmon colo ) and wo (ligh g ay) s anda d de ia ions o he mean.
The black line depic s he iden i y cu e. T iangle symbols s and o small ne wo ks (less
han 50 nodes), ci cles o medium size ne wo ks (mo e han 50 nodes and less han 410)
and squa es o la ge ne wo ks (mo e han 410 nodes). Ecological ne wo ks a e colo ed in
blue, economic ne wo ks in ed.
deg ee degene acy is in ac expec ed. All in all, we analyze i s in luence gi en ha each
69
4. Many ule s o one leng h: how o quan i y nes ed pa e ns
Me ics colo code
Ne wo k
III III
IV VVI VII VIII
IX
X
0
10
20
30
40
50
60
70
80
90
100
Nes edness alue
Tempe a u e
NMD
NODF
S able NODF
Disc epancy
NIR
Spec al adius
No malized spec al adius
I) Olesen A (27,0.29) II) Olesen B (84, 0.09) III) Olle on (65, 0.20) IV) Hocking (110, 0.08) V) Pe anidou (797, 0.03)
VI) He e a (205, 0.09) VII) Memmo (104, 0.15) VIII) Olesen C (144, 0.09) IX) Inouye (125, 0.08) X) Ke an (111, 0.10)
Figu e 4.2:
Compa ison among nes edness indices.
The his og am on he op o he
igu e shows how eigh di e en me ics measu e he nes edness o se e al di e en ne wo ks.
Each ne wo k, indexed I o X, is ep esen ed in he bo om o he igu e by i s bi-adjacency
ma ix o de ed by dec easing deg ee, wi h he in e ac ions among species ep esen ed by
black pixels. All ne wo ks ep esen plan -pollina o mu ualis ic communi ies ex ac ed om
he Web o li e da ase Bascomp e Lab (Bascomp e Lab). Each ne wo k is labeled wi h he
name o he i s au ho o he co esponding e e ence, ollowed wi hin b acke s by, i s , i s
o al numbe o species (numbe o plan s plus numbe o animals), and second, i s densi y
o links.
me ics deals wi h deg ee degene acy in a di e en way.
In o de o quan i y he dependencies discussed abo e we ha e pe o med a wo- old
analysis. Fi s , we ha e calcula ed he Spea man’s ank co ela ion be ween he nes edness
index gi en by each me ics and he di e en ne wo k pa ame e s. This coe icien allows
o assess he ela ion be ween bo h a iables wi hou assuming a linea beha io . Fig. 4.3a
summa izes he esul o he analysis, showing he Spea man coe icien along wi h i s
s a is ical signi icance o all pai s o nes edness alues and ne wo k pa ame e s (see he
Appendix F o he de ails on he nume ical calcula ion). Secondly, I ha e pe o med a
mul i-linea eg ession. In pa icula , I ha e aken he nes edness alues ob ained by each
me ics as he dependen a iable while he ne wo k pa ame e s beha e as he explana o y
a iables. Impo an ly, in his second analysis we do no conside he e ec o he deg ee
degene acy, since we a e mainly in e es ed on he dependence on pa ame e s ha should
no , in p inciple, de e mine nes edness. The linea unc ion we ha e i ed has he ollowing
s anda d o m:
νj=β0,j +β1,js+β2,jφ+β3,j+ε , (4.13)
whe e
νj
,
j
= 1
, ...,
8 ep esen s he nes edness me ics indexed by
j
,
β0,j
is he in e cep
and
βi,j
,
i
=
1, .., 3
a e he pa ial eg ession coe icien s. The
ε
ep esen s an e o e m.
This so o eg ession in o ms on he e ec o a single ne wo k pa ame e when he es o
70
4.3. In luence o ne wo k ea u es
pa ame e s a e kep ixed. Such conside a ion is specially impo an gi en ha , in na u al
sys ems, ne wo ks’ p ope ies a e o en co ela ed ( o ins ance, la ge ne wo ks end o be
less dense) and he e o e bi- a ia e eg essions may misleadingly quan i y he in luence o a
ce ain p ope y due o he uncon olled coupled in luence o ano he one. On he o he hand,
ou model assumes a linea ela ion among he a iables which migh no always be accu a e.
Fig. 4.3b shows he esul s o he eg ession o each nes edness me ics, in pa icula , he
signi icance o he pa ial coe icien s co esponding o he di e en ne wo k pa ame e s as
well as he alue o he adjus ed coe icien o mul iple de e mina ion (see he Appendix F
o mo e de ails).
No malized
spec al adius
●
●●
●
●
●
●
●
●
●
●
●
●
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Size
Densi y o links
Eccen ici y
Tempe a u e
NMD
NODF
S able NODF
Disc epancy
NIR
Spec al adius
Deg ee
degene acy
0.62
−0.09
−0.44
−0.48
−0.5
0.09
0.72
−0.56
−0.63
0.52
0.86
0.84
0.89
−0.1
−0.1
0.85
−0.02
−0.03
−0.07
0.03
−0.09
0.13
0.25
−0.11
0.49
−0.44
−0.71
−0.63
−0.74
0.17
0.15
−0.69
b) Mul ilinea eg ession
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−25
−20
−15
−10
−5
0
5
10
15
20
25
Size
Densi y o links
Eccen ici y
Adjus ed R2
Tempe a u e
NMD
NODF
S able NODF
Disc epancy
NIR
4.37
0.79
−0.59
−2.2
−3.18
−0.63
16.07
−8.69
−5.74
8.4
21.06
19.23
21.2
−0.73
4.24
14.45
−1.28
−0.47
−2.09
2.05
−0.66
2.44
−0.33
−0.36
0.31
0.28
0.74
0.72
0.76
0.02
0.59
0.72
No malized
spec al adius
Spec al adius
a) Spea man coefficien s
Figu e 4.3:
Dependency o nes edness me ics on ne wo k pa ame e s.
The le
panel,
a)
, shows he Spea man co ela ion ac o be ween he ne wo ks pa ame e s (columns)
and he eigh nes edness me ics unde s udy ( ows). The numbe s ep esen he alue o
he Spea man ank coe icien o each co esponding pai o nes edness alue and ne wo k
pa ame e . Only hose coe icien s ha a e s a is ically signi ican (
p
- alue
<
0
.
01) a e
highligh ed by a colo ed ci cle, being he size and he colo o he ci cle p opo ional o he
coe icien . The igh panel,
b)
, summa izes he esul s o he mul i-linea i de ailed in
Eq. 4.13. Each ow co esponds o a di e en nes edness me ics. The i s column om
he igh shows he adjus ed coe icien o mul iple de e mina ion (adjus ed
R2
). The o he
h ee columns show he - a io o he eg ession coe icien co esponding o each explana o y
a iable (as labelled by he column name). Only hose coe icien s ha a e s a is ically
signi ican (
p
- alue
<
0
.
01) a e highligh ed by a colo ed ci cle, being he size and he colo
o he ci cle p opo ional o i s - a io.
Once we ha e quan i ied he dependencies o he a ious nes edness me ics on di e en
ne wo k pa ame e s, we nex explo e whe he we can explain he de ia ions wi h espec o
he null model obse ed in Fig. 4.1. In pa icula , we pe o m a mul i-linea i o he ype
de ailed in Eq. 4.13, whe e we eplace he nes edness alues by he z-sco es ob ained o each
me ics when applying he null model discussed in sec ions 3.2 and 4.2. Such z-sco es a e
calcula ed as ollows:
z-sco ej=νj−hνji
σj
,(4.14)
whe e
νj
ep esen s, as be o e, he eal alues ob ained wi h a nes edness me ics indexed
by
j
,
hνji
ep esen s he a e age nes edness alue calcula ed wi h me ics
j
o e he null
71

CHAPTER 5
Beyond he agg ega ed pa adigm
This chap e inaugu a es he second pa o he hesis, along which we will con inue looking
a mu ualis ic ne wo ks ye om a ai ly di e en pe spec i e. Indeed, while in he p e ious
pa we we e occupied disen angling he exis ence o edundan pa e ns in empi ical sys ems,
he e we will y o b ing ou a en ion o a e y di e se, almos opposi e -in e ms o he
gene al spi i - ask. In pa icula , we will a emp o unde s and how by neglec ing he
empo al dimension o eal mu ualis ic communi ies, ha is, by wo king wi h he agg ega ed
e sion o he in e ac ion ne wo ks, we can loose ele an insigh in o he o ganiza ion and
unc ioning o na u al ecosys ems. In his sense, we will y o inco po a e in o he ne wo k
o malism he in o ma ion abou empi ical phenology, namely he biological ac i i y cycles o
species ha , o a ce ain ex en , cons ain and a icula e how ecological ela ionships occu
among indi iduals.
This endea o is ce ainly no en i ely new (Olesen e al., 2008; Encinas-Viso e al., 2012;
Sajjad e al., 2017; Ramos-Jilibe o e al., 2018; Chaco e al., 2018) and, in ac , du ing he
ecen yea s he need o mo ing owa ds a mo e ealis ic depic ion o ecological communi ies
has been s essed mo e han once (Ings e al., 2009; Heleno e al., 2014). All in all, agg ega ed
ne wo ks a e s ill pa adigma ically used in he cha ac e iza ion o he s uc u e and dynamics
o mu ualis ic sys ems. Along he ollowing sec ions we will add ess his issue by examining
he e ec s on he pe cei ed ne wo k a chi ec u e o aking in o accoun -a leas pa ly-
he empo al dimension o mu ualism, while he nex chap e will be de o ed ins ead o
explo ing i s implica ions o species pe sis ence by using dynamical popula ion models.
Acco dingly, he p esen chap e is o ganized as ollows: we will s a in oducing some
basic no ions on phenology; nex , we will cha ac e ize wo eal da ase s and how hei
ne wo ks o in e ac ions a y along he season; and, in he hi d place, we will p opose a
se o syn he ic models o phenology ha pe mi assessing and con as ing he e ec o
empo al a iabili y beyond he sca ce numbe o open da ase s.
5.1 A sho ale on phenology
F om che y blossom o digi al came as
The bloom o lowe s in sp ing, he a i al o he i s mig a o y bi ds o he shedding o
lea es a all a e all simple ye beau i ul examples o phenological e en s ha undeniably
shape, since ancien ages, ou pe cep ion and na a ion o ime, specially he passage o
he seasons. Ou language and cul u e is ull o e e ences o his so o phenomenon and
hei iming, om wo ds like ‘la e bloome ’ o A is o le’s amous ph ase ‘one swallow does
no make a summe ’. Na u ally, hence, we can ind as well ema kably old ins ances o
documen ed phenological pa e ns, no only ela ed o ag icul u al needs –as was he case o
he i s empi ical ecological ne wo ks desc ibed in Chap e 1– bu also in cases in which hei
iming ma ked he da e o eligious o adi ional es i i ies. A ascina ing illus a ion is he
iewing o che y blossom in Japan, known as hanami, a celeb a ion ha s a ed in he eigh
cen u y as an eli is ce emony and g adually became a gene al es i i y. Nowadays, i is so
popula ha he che y blossom is nigh ly o ecas in Japan and he ad ance o he on o
blossom ac oss he coun y - he so-called saku a zensen- is keenly ollowed (see Fig. 5.1). As
79
5. Beyond he agg ega ed pa adigm
a esul , i is possible o ind ex ensi e da ase s on he phenology o a ious species o che y
ee along he yea s –mainly he P unus se ula a-, comp ising mo e han se en cen u ies o
blossoming in o ma ion (Aono and Kazui, 2008). In ac , his kind o sequen ial da a ha e
been used o econs uc sp ing ime empe a u es o undocumen ed old pe iods. Simila ly,
analogous s udies ha e been ca ied using al e na i e species’ phenology a ound he globe,
such as he g ape plan o p oduce wine in F ance (Chuine e al., 2004).
Figu e 5.1: In
a)
, pic u e called Asukayama hanami no zu (Che y-blossom iewing a Asuka
hill) by U agawa Hi oshige, da ed abou 1831. In
b)
, o ecas ing o he da es o blossoming
ac oss Japan in 2007. The numbe s ep esen he mon h and day o he blossoming, being
he da ke a eas he ones whe e blossoming is expec ed ea lie . Sou ce: Wikimedia commons.
While hese da ase s we e cons uc ed by elying, basically, on isual and small-scale
obse a ions, du ing he las decades echnological ad ances ha e pe mi ed a mul iplica ion
and di e si ica ion o he means o measu e phenology o bo h plan s and animals. Such
me hodology anges om ci izen science p ojec s whe e pa icipan s o ma and sha e
indi idual obse a ions o phenology in hei local a ea (Ha ens and Hende son, 2013), o
he use o sophis ica ed emo e sensing ools based on sa elli e da a (Zhang e al., 2003).
A pa icula ly ingenious case among hese no el app oaches is he wo k by G aham e al.
(2010), in which hey p ocess he images eco ded by public came as connec ed o he In e ne
-p ima y ela ed o a ic su eillance o na ional pa ks- o ga he in o ma ion on ege a ion
phenology ac oss No h-Ame ica, exploi ing a ee sou ce o glances o ees and plan s
unin endedly caugh by he came as. Nu u ed by hese me hodological ad ances as well
as spu ed by he in es iga ions on how clima e change in luences phenology (Memmo
e al., 2007; Hegland e al., 2009) –a opic ha we will discuss in some mo e in de ail in he
nex chap e –, nowadays we ace an undeniable inc ease in he quan i y and he quali y o
documen ed phenology.
Pa adoxically, hough, his copious amoun o phenological da a comes a li le help, a
leas di ec ly, when ying o be e unde s and mu ualism along ime. This is due o he
ac ha public empi ical da a on bo h he ne wo k o in e ac ions and he iming o hei
mu ualis ic ac i i y is, o pu i mildly, sca ce. Consequen ly, p e ious s udies examining he
e ec o phenology on mu ualis ic communi ies ei he ocus on a e y educed numbe o
highly- esol ed ne wo ks (Olesen e al., 2008; Ca aDonna e al., 2017; Ramos-Jilibe o e al.,
2018), ei he hey succeed in keeping he big numbe s o phenological da a a he expense
o oughly app oxima ing he pa e ns o mu ualis ic ela ionships – ha is, by neglec ing
he eal complex ne wo ks o in e ac ions (Duchenne e al., 2020). A hi d app oach, ye , o
add ess he lack o da a, is o explo e ins ead syn he ic models ei he o phenology o o he
ne wo k o in e ac ions (Kallimanis e al., 2009; Encinas-Viso e al., 2012). In wha ollows,
hese limi a ions will condi ion us as well. Bu be o e jumping in o ha , le us summa ize
he main cha ac e is ics o he phenology o plan -pollina o communi ies.
80
5.1. A sho ale on phenology
Ecological and e olu iona y de e minan s o phenology
Al hough phenology ce ainly plays a pa in di e en ypes o ecological mu ualism, in his
hesis we will ocus on i s implica ions o plan -pollina o s communi ies, bo h because i is
a pa adigma ic example in which seasonali y is s ongly ma ked and also because, despi e
he a o emen ioned da a limi a ions, he phenology o plan s and pollina o s is in gene al
be e documen ed han ha o o he mu ualis ic species (Ra e y e al., 2015).
The s udy o he phenology in ol es all empo al aspec s o species’ li e cycle, ega ding
on he one hand he iming o hei a ious s adiums o de elopmen , i.e. egg/seed, la ae,
adul , e c; and on he o he hand he onse and du a ion o di e en biological p ocesses
such as lowe ing, ge mina ion, pollina ion, lea e alling, e c. Such iming is de e mined by
a my iad o ac o s, simila ly o wha occu s wi h mu ualis ic in e ac ions as desc ibed in
sec ion 2.3. Indeed, bo h bio ic and abio ic o ces shape he phenology o species, which, in
addi ion, is hough o be subjec o e olu iona y change (Ra hcke and Lacey, 1985; Fo es
and Mille -Rushing, 2010). To complica e hings u he , se e al sou ces o in a-species
phenological a iabili y occu simul aneously: in e -annual (Olesen e al., 2008; Ci will
e al., 2018) – ha is, among di e en seasons–, geog aphical (Pos e al., 2018) – o he
same species bu on di e se si es– and las bu no leas , indi idual, i.e. among di e en
indi iduals o he same popula ion, e en hose coexis ing on he same si e and a he same
season (Fo es and Mille -Rushing, 2010).
All in all, om a s a is ical iewpoin a ew gene al pa e ns ha e been iden i ied ha ,
hope ully, will pe mi us gain u he insigh in o he gene al ules go e ning phenology.
In pa icula , we will ocus on wo undamen al phenological quan i ies o plan -pollina o
sys ems: i s , we will look a he s a ing da es, ha is, he ime a which he lowe ing –in
he case o plan s– o he pollina ion – o animals– begins, and secondly we will conside he
so-called pe iods, ha is, he leng h o du a ion o his ac i e s a e, du ing which mu ualis ic
in e ac ions a e, on pape a leas , possible. O cou se, his selec ion o phenological indica o s
is a om being exhaus i e, and o he wo ks ha e modeled he di e en s ages o plan s and
pollina o s in mo e ealis ic de ail (Ramos-Jilibe o e al., 2018). Ins ead, in ou app oach we
will app oxima e his complex landscape o empo al a iabili y by ocusing only on he adul
phase o he species, and mo eo e , educing he whole possible se o biological p ocesses
and s adiums o a couple o s a es: ac i e, namely when he species could po en ially hold a
mu ualis ic ela ionship –i.e. pollina e o be pollina ed–, o inac i e. In Fig. 5.2 we depic
his schema ic ep esen a ion o an hypo he ical plan and i s pollina o . Indeed, his so
o simpli ica ion is no no el and o he wo ks ha e adop ed analogous app oaches (Memmo
e al., 2007; Encinas-Viso e al., 2012; Bu kle e al., 2013).
Figu e 5.2: Schema o he phenology o a plan and a pollina o , ep esen ed by hei ac i e
s a es. Each species is de ined by i s pe iod and i s s a ing da e, called he e
s,
o he
lowe ing plan and
s,p
o he pollina o . The sec ion o ime du ing which he wo species
o e lap and can ac ually in e ac is highligh ed in g ey.
Rega ding he pe iods o ac i i y, in gene al bo h plan s’ and pollina o s’ pe iods end
o ollow igh skewed dis ibu ions, wi h a small numbe o species exhibi ing a long
phenophase while a la ge numbe o species a e ac i e only du ing a sho ime (Bawa e al.,
2003; Kallimanis e al., 2009). Pa icula ly, Kallimanis e al. (2009) examined he s a is ical
81
5. Beyond he agg ega ed pa adigm
dis ibu ion o pe iod’s leng hs in a Medi e anean sc ub communi y obse ed along ou
yea s by Pe anidou e al. (1995), and concluded ha he dis ibu ion o pollina o ’s pe iods
could be i ed by a dec easing exponen ial, while plan s’ ollowed a logno mal dis ibu ion.
On he o he hand, in a plan -pollina o sys em in he A ic, Olesen e al. (2008) iden i ied
a logno mal shape o bo h plan s and pollina o s’ pe iods, and a no mal dis ibu ion o
plan s in one o he yea s o hei obse a ion.
In wha conce ns he onse o lowe ing and pollina ing ac i i y, a complemen a y measu e
ha is o en used a e he middle da es. In his sense, e idence sugges s ha hey end
o be ela i ely synch onized (Ra hcke and Lacey, 1985; Téba e al., 2004). Fo example,
bo h Olesen e al. (2008) and Bawa e al. (2003) obse ed ha mos ac i e pe iods empo ally
coexis a a peak, p obably due o a simila eac ion o a common se o physicochemical
s imuli such as empe a u e, pho o-pe iod, humidi y, e c (Ra hcke and Lacey, 1985). In
s a is ical e ms, his would co espond o a scena io whe e he middle da es a e ela i ely
clus e ed, as modeled o ins ance by Kallimanis e al. (2009) using a no mal dis ibu ion. A
he same ime, gene ic ac o s seem o play as well an impo an ole in de e mining he iming
o lowe ing o pollina ing ac i i y, which hence con ibu es o explain he he e ogenei y o
s a ing da es. On he o he hand, i has also been hypo hesized ha species may sp ead
along he season in o de o minimize compe i ion. Al hough he e idence suppo ing his
hypo hesis is con o e sial (Ra hcke and Lacey, 1985), some au ho s ha e p oposed ha
such minimiza ion could s ill occu wi hin a empo al ange de e mined by gene ic and
en i onmen al cons ain s (Kochme and Handel, 1986).
In conclusion, his sho summa y illus a es he complexi y o he empo al dimension
o plan -pollina o s communi ies. Indeed, empo al a iabili y no only appea s a di e en
scales, om days o decades and om indi iduals o species, bu i is u he mo e egula ed by
a mul iplici y o ac o s. In he nex sec ion, we analyze wo empi ical da ase s o phenology
in o de o examine whe he he s a is ical cha ac e is ics a o emen ioned hold, and hen
we use hem o e alua e how ou unde s anding o mu ualis ic ne wo ks can change when
mo ing beyond he agg ega ed pa adigm.
5.2 Phenology in a ne wo k: cha ac e iza ion o wo da ase s
In his sec ion we will ocus on s udying wo empi ical examples o plan -pollina o sys ems,
ha con ain, a he same ime, de ailed in o ma ion abou hei phenology and hei web
o mu ualis ic in e ac ions. In pa icula , we will analyze he plan -pollina o communi y
eco ded by Bu kle e al. (2013) oge he wi h he communi y measu ed by Kan sa e al.
(2018), which includes wo consecu i e yea s o obse a ions and hence wo ne wo ks. The
speci ici ies ega ding each da ase and how we p ocessed hem be o e he analysis can be
ound in Appendix A. In wha ollows, o he sake o con enience we will e e o hem
simply as he Bu kle da ase and he Kan sa da ase .
Wi h he aim o unde s anding how he ne wo k desc ip ion changes when we in oduce
empo al a iabili y, his sec ion is di ided in o wo pa s: i s , we will examine he main
cha ac e is ics o he empi ical phenology unde s udy; secondly, we will moni o how
he s uc u e o he ne wo k a ies on a daily basis beyond he s a ic desc ip ion we a e
al eady amilia ized wi h. In de ail, we will cha ac e ize how i a ec s bo h mu ualis ic and
compe i i e in e ac ions wi hin he communi y.
S a is ical p ope ies o phenology
We begin by explo ing he s a is ical cha ac e is ics o he wo phenological da ase s. We
will i s look in o he dis ibu ion o pe iods leng hs and compa e hem wi h he li e a u e
discussed abo e. Secondly, we will add ess he e ec o coupling his phenological in o ma ion
wi h he ne wo k o in e ac ions, which esul s in a dis ibu ion o mu ualis ic and compe i i e
o e laps. These analysis will pa e he way o , e en ually, in oducing some syn he ic models
o ec ea e he e ec s o phenology on he ne wo k’s s uc u e a he end o his chap e .
82
5.2. Phenology in a ne wo k: cha ac e iza ion o wo da ase s
Dis ibu ion o pe iods
We s a by examining he dis ibu ion o he du a ion o he ac i i y o plan s and pollina o s.
In pa icula , we i ed a a ie y o unc ional o ms on he cumula i e dis ibu ions using
a maximum likelihood es ima ion. Then, we es ed he quali y o he i by pe o ming a
Kolmogo o -Smi no es by boo s apping, as explained in de ail in Appendix H.1.
Figu e 5.3: Empi ical dis ibu ion o he pe iods leng h o plan s ( igh pannel) and
pollina o s (le pannel), co esponding o h ee di e en eal da ase s. We show he i ed
be a dis ibu ion o he Bu kle’s da ase , o bo h plan s and animals. Fo he Kan sa
da ase , we could i a logno mal o plan s and we i ed a exponen ial o pollina o s. In
Table 5.1 we summa ize he signi icance o each i . Al hough we show he e he binned
his og ams, he pa ame e s o each i ing unc ion we e es ima ed using he unnbined,
disc e e empi ical dis ibu ions.
83

5. Beyond he agg ega ed pa adigm
In mo e de ail, we es ed h ee di e en unc ional o ms: logno mal, exponen ial and
he be a unc ion. As a o emen ioned, he i s wo dis ibu ion ypes ha e been claimed o
co ec ly desc ibe some empi ical obse a ions, while we in oduce he la e as a gene ic
i ing. The esul s o he p- alue o he KS- es a e epo ed in Table 5.1, whe e we
highligh ed in bold he good-quali y i s. As can be seen in Fig. 5.3, he wo da ase s exhibi
e y di e en shapes: he Bu kle da ase is be e i ed by he be a unc ion, while in
he Kan sa da ase we could i a logno mal o he dis ibu ion o plan s’ pe iods bu we
ound no unique i o he pollina o s. In gene al e ms, he dis ibu ion o pe iods o he
Bu kle da ase is less he e ogeneous, while he Kan sa dis ibu ions a e clea ly igh e -skewed.
This is specially no o ious o he pe iods o he pollina o s, among which we ind a la ge
p opo ion o species wi h e y sho pe iod -one o wo days- as can be seen in Fig. 5.3.
This pa icula i y is wha makes his case specially di icul o i , al hough i s plo sugges s
a exponen ial shape.
Da ase Logno mal i Exponen ial i Be a unc ion i
Plan s Pollina o s Plan s Pollina o s Plan s Pollina o s
Bu kle 0.021 0.018 0.001 0.001 0.63 0.18
Kan sa 1s yea 0.67 0.014 0.029 0.001 0.41 0.001
Kan sa 2nd yea 0.50 0.001 0.018 0.001 0.02 0.001
Table 5.1: Resul s o i ing di e en unc ional o ms, disen angled in o plan s and pollina o s.
We show he p- alue o he i , and highligh in bold he good quali y i s, ha is, hose which
do no signi ican ly di e om he es ima ed dis ibu ion (p- alue
>
0
.
05). The p- alues
a e ob ained by pe o ming a Kolmogo o -Smi no es be ween he i ed dis ibu ion and
he empi ical sample, hen compa ing i o he co esponding K-S dis ibu ion sampled by
boo s ap as explained in Appendix H.1.
Dis ibu ion o o e laps
As a ma e o ac , he ele ance o phenology goes beyond he indi idual cha ac e iza ion o
species’ ai s. Indeed, i egula es as well he ecological in e ac ions wi hin he communi y,
a ec ing bo h hei occu ence and in ensi y. Such ela ionships in ol e o cou se he
di e en kinds o mu ualism we ha e been conside ing so a , bu also indi ec in e ac ions
like compe i ion o sha ed mu ualis ic esou ces (Jones e al., 2012), which na u ally eme ges
among species o he same kind. The nega i e e ec s o hese an agonis ic in e ac ions
a e known o coexis wi h mu ualism, yielding o a ade-o be ween cos s and bene i s
as explained in Chap e 1 (B ons ein, 2001). This means ha add essing how phenology
impac s an agonis ic in e ac ions migh be jus as impo an as unde s anding i s e ec on
mu ualism.
In o de o be able o calcula e he consequence o empo al a iabili y no only among
mu ualis ic pa ne s bu also among compe i o s, we need o compu e he ne wo k o
compe i i e in e ac ions based on sha ed esou ces. We do so ollowing he p oposal by G acia-
Láza o e al. (2018), namely, p ojec ing he empi ical biadjacency ma ix o mu ualis ic
in e ac ions Bi,k in o he subspace o in a-guild in e ac ions such ha :
i Bi,kBj,k = 1 hen plan s iand jcompe e o he pollina ing se ices o animal k,
(5.1)
i Bi,kBj,k = 0 hen iand jdo no compe e o he esou ces o pollina o k,
(5.2)
which leads o a compe i i e ne wo k be ween all pai s o plan species
i
and
j
.
Analogously, we can de ine he compe i i e in e ac ions among wo pollina o species
k
and las ollows:
84
5.2. Phenology in a ne wo k: cha ac e iza ion o wo da ase s
i Bi,kBi,l = 1 hen pollina o species kand lcompe e o he lowe esou ces o plan species i,
(5.3)
i Bi,kBi,l = 0 hen kand ldo no compe e o he esou ces o plan species i.
(5.4)
This comple es he desc ip ion o he ecological communi y in he sense ha he obse ed
ne wo k o pollina ing con ac s no only media es he explici mu ualis ic in e ac ions, bu
also he implici , compe i i e ela ionships among species o he same guild.
Re u ning o ou main p oblem, he mos s aigh o wa d consequence o in oducing
phenology in o an ecological ne wo k is he modi ica ion o he amoun o empo al coexis ence
– he so-called o e lap– among species. Indeed, we ind a con inuum o possible scena ios
anging om ull concu ence o absence o o e lap, as shown in Fig. 5.4. These o e laps
conce n ei he wo species in he mu ualis ic case, o h ee species – wo species o he same
kind and he sha ed esou ce– when conside ing compe i ion.
Figu e 5.4: Di e si y o phenological con igu a ions gi ing ise o di e en ypes o mu ualis ic
and compe i i e o e lap. The pannels on he le ep esen he mu ualis ic o e lap be ween
he pollina o species
k
and he plan species
i
, om he iewpoin o he pollina o species
k
.
In
a)
he e is ull mu ualis ic o e lap, in
b)
he e is only pa ial o e lap and in
c)
he e is no
o e lap a all. Pannel
d)
depic s he compe i i e o e lap be ween h ee species, pa icula ly
wo plan s
i
and
j
compi ing o a sha ed pollina o species
k
. In his case he o e lap is
e e ed o he i s pollina o species,
j
. In
d)
he compe i i e o e lap is ull, in
e)
he e is
only pa ial o e lap and in ) he e is no o e lap a all be ween he compe i o s.
Wi hin his amewo k, we can in oduce a se o phenological coe icien s
{
Φ
}
and
{
Ω
}
o quan i y he e ec o he phenological o e lap on, espec i ely, he mu ualis ic and he
compe i i e in e ac ions. In he pa icula case o a plan species
i
in e ac ing mu ualis ically
wi h a pollina o
k
and compe i i ely wi h ano he plan
j
, we de ine hese coe icien s as
ollows:
ΦP
ik =τik
τi
,(5.5)
85
5. Beyond he agg ega ed pa adigm
ΩP
ijk =τijk
τi
,(5.6)
whe e,
τi
s ands o he pe iod o lowe ing o he plan species
i
,
τik
ep esen s he empo al
o e lap be ween he plan species
i
and i s pollina o
k
(see Fig. 5.4 a-c o a g aphical
ep esen a ion), and inally
τijk
s ands o he o e lap be ween plan s species
i
and
j
and
hei pollina o
k
(see Fig. 5.4 d-g o an example). An analogous se o coe icien s can be
d awn o he pollina o s.
No e ha , as can be seen in Eqs. 5.5-5.6, he o e laps a e ponde ed by he pe iod o
ac i i y o he species –in he example abo e, he plan
i
–, which means ha each coe icien
is always e e ed o a ce ain plan o animal species. This pa icula no maliza ion implies,
mo eo e , ha he e ec o empo al o e lap is non-symme ic among he in e ac ing pa ne s.
Al hough his migh no seem e y ele an now, we will see i s consequences in he nex
Chap e when we add ess he s udy o he communi y dynamics.
In o de o ob ain some insigh s in o he impac o phenology a a ne wo k le el, we
calcula ed he dis ibu ion o phenological o e laps
{
Φ
}
and
{
Ω
}
o each da ase . The
esul s p esen ed in Fig. 5.5 show ha he wo empi ical cases we ha e a hand exhibi
clea ly di e en shapes. In wha ollows, we will explo e how hese quan i ies can con ibu e
o changing ou pe cep ion o he s uc u e o he communi y along ime.
Figu e 5.5: His og am o he phenological coe icien s
{
Φ
}
and
{
Ω
}
de ined in Eqs. 5.5-5.6
o accoun o he mu ualis ic (in blue) and he compe i i e (in ed) o e lap. Each pannel
co esponds o a di e en da ase , as indica ed on he op o he igu e.
Disen angling he s uc u e
Wi h his summa y abou he gene al cha ac e is ics o each da ase a hand, we can now
u n ou a en ion in o he a o emen ioned ques ion: how does he pe cei ed s uc u e o
he ne wo k change i we ake his empi ical in o ma ion in o accoun ? In o de o a emp
o answe his p oblem and depic he empo al a ia ion o he ne wo k’s s uc u e, we
cons uc a disc e e empo al sequence o ne wo ks, whe e each elemen co esponds o a
di e en ‘snapsho ’ o he sys em, aken a a di e en momen o he season. In pa icula ,
gi en ha he le el o de ail o bo h empi ical phenologies is na owed o days, we cons uc
a se o daily ne wo ks, ep esen ing he mu ualis ic in e ac ions obse ed among plan s and
pollina o s on a gi en da e. To do so, we use he empi ical in o ma ion on he s a ing and
ending da es o ac i i y o plan s and pollina o s, and emo e om he daily ne wo k ce ain
mu ualis ic links among species whene e hey do no coincide in ime – i. e., when hei
phenological o e lap, as de ined in Eq. 5.5, is ze o. Addi ionally, we also emo e inac i e
species as well as ac i e species wi h ze o deg ee. On he whole, his p o ides a coa se-g ained
desc ip ion o he communi y as shown in Fig. 5.6, ha , e en i i is no a pu e empo al
ne wo k, i does accoun o a g ea ex en o he in a-annual empo al a ia ion, mo ing
beyond he s a ic ne wo k o malism.
Using his in o ma ion, we measu e a se o undamen al s uc u al ea u es o he
di e en ne wo ks o he sequence, in pa icula : he numbe o ac i e nodes, he numbe
o ac i e links, and he maximum deg ee. Fig. 5.7 p o ides a summa y o he e olu ion o
hese p ope ies along ime, o each o he empi ical ne wo ks in ou da ase . To begin
86
5.3. Going syn he ic
Figu e 5.6: Schema ic ep esen a ion o he coa se-g ained desc ip ion o empo al a ia ion.
Each ne wo k in g ey-scale co esponds o a di e en day o he season, whe e ce ain species
and in e ac ions migh be absen . Summing up hese in e ac ions p oduces he agg ega ed
ne wo k, depic ed in colo , whe e he in o ma ion on he u no e o species and in e ac ions
is los . Despi e his agg ega ion p ocess is indeed qui e simple, a pe inen ques ion is
whe he he s uc u e and dynamics o he empo al snapsho s is compa able o ha o he
agg ega ed ne wo k.
wi h, in bo h cases he empo al measu es a e signi ican ly smalle han hei agg ega ed
coun e pa s, al hough his was al eady expec ed. Wha is mo e in e es ing is ha he
e olu ion o he wo da ase s exhibi a clea ly di e en beha io o e he season: while in he
Bu kle da ase he h ee quan i ies show a a ma ked peak app oxima ely a he middle o
he season, in he Kan sa da ase hey s ay ela i ely s able a ound a ce ain alue, showing
an almos la shape. Mo eo e , in he Bu kle da ase he maximum o he peak ep esen s
a ound he 75-80%o he agg ega ed coun e pa , whe eas in bo h ins ances o he Kan sa
da ase his quan i y descends o less han 30-40%.
In he bigge pic u e, he e a e e y ew s udies which pe mi assessing he gene ali y
–o a i y– o hese obse a ions, al hough a ecen wo k by Sajjad e al. (2017) examined
he s uc u e o a plan -pollina o sys em in analogous e ms and ob ained simila esul s o
hose desc ibed o he Bu kle da ase . O e all, he ema kable di e ences among empi ical
examples sugges ha in oducing he phenology in o he ne wo k o malisms does no yield
o a unique pa e n, bu ins ead he esul ing pa e ns a e highly sys em-dependen . T u h
be old, his inding aises mo e ques ions han answe s, anging om he easons benea h
he di e gences among he communi ies o he implica ions o he s abili y and obus ness
o ecosys ems. The inhe en di icul y in add essing hese issues is agg a a ed by he lack o
publicly a ailable da ase s beyond he ew examples we s udy he e. As a esul , we de o e
he nex sec ion o explo e a se o syn he ic models, cons uc ed using a minimal numbe
o assump ions, and seeking hose ha be e ep oduce he obse ed cha ac e is ics o
empi ical sys ems.
5.3 Going syn he ic
The idea o cons uc ing syn he ic models ha accoun o unobse ed ea u es o he
communi y, ei he mu ualis ic in e ac ions o phenology, has been pa ly add essed be o e as
we ecalled in sec ion 5.1. Ne e heless, he majo i y o hese p e ious wo ks inco po a e
empi ical phenology as a ixed inpu while ocusing, hence, on modeling he ecological
ne wo k. An illus a ing pai o examples o his app oach a e he wo ks by Kallimanis e al.
(2009) and Vázquez e al. (2009), in which he au ho s p opose a di e se se o null models o
explo e he ole o phenology as a possible de e minan o he in e ac ion p obabili y among
species.
In his chap e we adop an almos in e se pe spec i e and conside , ins ead, he p oblem o
modeling he undocumen ed phenology o a sys em whose ne wo k o mu ualis ic in e ac ions
is known. Tha is, we place he emphasis on modeling he empo al a iabili y o he sys em.
As a o emen ioned, his is a pe sis en p oblem gi en ha , despi e he exis ence o la ge
eposi o ies o eal ecological ne wo ks, such da ase s o dina ily lack he de ails abou he
species’ phenology. Shi ing he ocus on o modeling he phenology o a gi en ne wo k
pe mi s, he e o e, add essing ques ions ha adi ionally e ol ed a ound he ne wo k’s
s uc u e, like he eme gence o pa e ns o hei implica ions o he communi y dynamics.
87
5. Beyond he agg ega ed pa adigm
undamen al s uc u al p ope ies along ime. To quan i y hese co ela ions, we calcula ed
he co esponding Pea son coe icien o di e en alues o ime lag be ween he dis ibu ions.
By doing so, we a e aking in o accoun he possibili y ha he null model ep oduces well
he empi ical scena io bu in a delayed o ad anced ime. Tha is, we a e elaxing he
assump ion o a pe ec ma ching be ween he empi ical and he syn he ic s a ing da es,
ocusing ins ead on he ela i e posi ion among he s a ing da es o di e en species. In
able 5.4 we show he maximum alue o he Pea son co ela ion calcula ed ollowing his
p ocedu e, as i is u he de ailed in Appendix H.4.
Table 5.4 and Fig. 5.8 show ha he analysis o he s uc u e lead o simila esul s o
he a o emen ioned analysis o he phenological o e lap. Indeed, he be e i ing models o
he Bu kle da ase a e he ones based on synch onized phenology and maximum en opy,
while o he Kan sa da ase he maximal co ela ion is ound o he model which minimizes
in a-guild compe i ion and, again, he one maximizing he en opy.
Da ase / P ope y Maximum Pea son coe icien o each null model
Bu kle Synch onized Maximum Va iance Minimum compe i ion Maximum en opy
Maximum deg ee 0.98 0.88 0.93 0.98
Numbe o links 0.98 0.87 0.78 0.98
Size 0.99 0.79 0.92 0.98
A e age 0.99 0.85 0.88 0.98
Kan sa 1s yea Synch onized Maximum Va iance Minimum compe i ion Maximum en opy
Maximum deg ee 0.78 0.76 0.91 0.91
Numbe o links 0.74 0.71 0.89 0.83
Size 0.81 0.58 0.89 0.85
A e age 0.78 0.69 0.90 0.86
Kan sa 2nd yea Synch onized Maximum Va iance Minimum compe i ion Maximum en opy
Maximum deg ee 0.90 0.67 0.87 0.93
Numbe o links 0.85 0.78 0.89 0.85
Size 0.89 0.63 0.90 0.88
A e age 0.88 0.69 0.88 0.88
Table 5.4: Maximum Pea son co ela ion among he empi ical and he null expec a ion o
he s uc u al p ope ies along ime.
O e all, hese mul iple s a is ical es s e eal a leas wo gene al conclusions. On he one
hand, we ind ha each da ase is be e desc ibed by a dis inc mechanis ic assump ion, i.e.
he synch oniza ion o he species’ phenologies in he Bu kle da ase and he minimiza ion
o he compe i ion in he Kan sa da ase . This dispa i y seems o e lec he subs an ial
di e gences be ween he wo da ase s, ha we desc ibed in de ail in sec ion 5.2. On he o he
hand, bo h ne wo ks’ phenology a e accu a ely ep oduced by he s a is ical model ha
maximizes he en opy associa ed o he dis ibu ion o he middle da es o ac i i y, despi e
he conside able di e ences among he wo da ase s. Wha his la e inding sugges s is ha
he obse ed ne wo k o in e ac ions oge he wi h he pe iods o each species p o ide su icien
in o ma ion o ep oduce, ai ly closely, he obse ed empo al pa e ns o ac i i y. This is
pa icula ly in e es ing gi en he men ioned di e ences among da ase s, which p o ides, o a
ce ain ex en , a wa an y o gene ali y despi e he limi ed da a a ou disposal. This inding
does no imply ha , o cedly, he s a ing da es o ac i i y a e se a andom, bu ha he
in o ma ion enclosed in he co esponding ne wo k o in e ac ions and he species’ pe iods
may be su icien o ep oduce well he main cha ac e is ics o he communi y’s phenology.
94

5.4. Conclusions and pe spec i es
5.4 Conclusions and pe spec i es
Along his chap e we ha e add essed he possibili y o mo ing beyond he agg ega ed
pa adigm by inco po a ing –in he o m o pe iods o ac i i y– some o he empo al
a iabili y ha plan -pollina o communi ies exhibi along he yea . In pa icula , we
ha e concen a ed on seasonal ecosys ems, assessing he daily change in bo h in e -guild,
mu ualis ic ela ionships and in a-guild, compe i i e in e ac ions.
The analyses ca ied ou along his chap e ha e e ealed, in he i s place, ha non- i ial
in o ma ion is los when po aying he eal ne wo k o in e ac ions by a s a ic ep esen a ion,
ha is, neglec ing i s empo al dimension. Indeed, we ha e obse ed ha he consequences
o in oducing he empi ical phenology in o he ne wo k o malism a e sys em-dependen ,
and hence no gene al pa e n can be expec ed a p io i. Mo eo e , he p ocess o agg ega ion
no only dis ega ds he ichness o empo al a iabili y, bu i also ends o o e es ima e he
alue o he main undamen al s uc u al ea u es, as had been ema ked as well by Sajjad
e al. (2017). This is specially ele an gi en ha s uc u al ea u es like he deg ee o he
connec i i y a e consis en ly used o cha ac e ize, espec i ely, he ela i e ulne abili y o
species (Dakos and Bascomp e, 2014) o he communi y s abili y (Thébaul and Fon aine,
2010).
In he ligh o he limi a ions o he agg ega ed pa adigm, and d i en by he sca ci y
o a ailable da ase s, we p oposed a g oup o models o p oduce, gi en a ixed mu ualis ic
ne wo k, a compa ible hypo he ical phenology. The compa ison o hese esul s wi h he
empi ical da ase s e ealed ha he soundness o ce ain mechanisms is, a leas in he i s -
o de app oach we ha e adop ed he e, speci ic o he pa icula sys em unde conside a ion.
Ins ead, we ha e ound ha he pu ely s a is ical assump ion o maximizing he en opy
associa ed o he dis ibu ion o middle da es pe o ms gene ally well, as we ha e es ed in
wo dissimila empi ical examples. Impo an ly, he ema kable pe o mance o he maximum
en opy hypo hesis is pa ly explained by he ac ha , ac ually, he ne wo k o in e ac ions
is closely dependen upon he s a ing da es, in he sense ha he mu ualis ic con ac s
obse ed co esponded, o cedly, o concu en species –a condi ion ha , indeed, we impose
o ou models. The e o e, p ese ing he ne wo k o in e ac ions is a s ong cons ain , which
could jus i y he gene al adequacy o his model.
In pe spec i e, hese syn he ic models o e a me hodological se ha migh p o e use ul
in di e en aspec s. On he one hand, hey can be exploi ed me ely as a g oup o ealis ic
models o cons uc syn he ic ensembles in he absence o highly- esol ed empi ical da a, in
hose cases whe e he main d i ing o ces o he phenology a e known. On he o he hand,
hey can also be applied as null models ha pe mi es ing a a ie y o null hypo hesis, om
he mechanis ic o ces shaping he phenology o he exis ence o empo al, s uc u al o
dynamical pa e ns. A his poin , i is wo hy o emind as well ha ou s is jus a i s
app oach o modeling he empo al a iabili y o ne wo ks. In pa icula , we conside ed he
desc ip ion le el a which species a e ac i e o inac i e du ing a ce ain ac ion o he season.
Howe e , i could be possible o e ine he scale o desc ip ion o include he weekly o e en
daily u no e o in e ac ions, a so o hype - ealis ic depic ion o he empo al a iabili y
o he ne wo k ha is gaining a en ion du ing he ecen yea s (Ca aDonna e al., 2017).
This is due, pa ly, o he echnological ad ances ha pe mi moni o ing phenology in g ea
de ail, and he e o e i is p obable ha in he coming yea s we will ind a ising numbe o
his ype o s udies.
In he bigge pic u e, he in e es in mo ing owa ds a mo e ealis ic po ayal o
in e ac ing sys ems is no exclusi e o ecology, and indeed he s udy o empo al complex
ne wo ks has ecei ed g ea a en ion in ecen yea s (Holme and Sa amäki, 2012). How
his change o pa adigm will e en ually challenge ou unde s anding o na u al sys ems is
some hing we a e jus now beginning o explo e.
95
CHAPTER 6
Dynamics one mo e ime
This six h chap e , ha closes bo h he second pa and he main body o he hesis,
cons i u es he na u al con inua ion o he inqui ies abou he empo al dimension o
mu ualism we had ini ia ed abo e. He e, hough, we will lea e aside he emphasis on he
ne wo k’s s uc u e and ackle ins ead he impac o conside ing phenology on he communi y
dynamics. Be o e s a ing, i is wo hy o o ewa n ha some o he esul s p esen ed along
his chap e a e s ill inconclusi e and call o u he in es iga ions, bu we will s ill p esen
hem he e in o de o in oduce some o he p ospec i e wo k ha emana es om his hesis.
This chap e is s uc u ed as ollows. Fi s , we will in oduce he igh ela ion be ween
phenology, ecosys em’s dynamics and clima e change, a opic ha has no ceased o gain
a en ion in ecen yea s. Nex , we will p opose a me hodological app oach o inco po a e
phenology in a dynamical model ha conside s bo h he mu ualis ic and he compe i i e
in e ac ions. Finally, we will es his model on he wo empi ical da ase s we s udied in
Chap e 5 and discuss he esul s.
6.1 S abili y in a changing clima e, o why ime ma e s
P e iously, we ha e in es iga ed how ou ep esen a ion o a mu ualis ic communi y changes
when we ake in o accoun he phenology, concluding ha se e al s uc u al ea u es a e
dis o ed by he lens o he agg ega ed pa adigm. Gi en ha he link be ween s uc u e and
dynamics is, ce ainly, one o he keys ones o he ield o ecological ne wo ks in pa icula
and ha o complex ne wo ks in gene al, a pe inen ques ion ha na u ally ollows is how
ecosys ems’ dynamics a e modi ied by he phenology. In his sense, he e a e, a leas , wo
main que ies. Fi s , how does ou cu en unde s anding o he s abili y o communi ies
ansla e in o a amewo k ha accoun s o empo al a iabili y? And secondly, and maybe
e en mo e impo an ly, wha e ec does phenology ha e on ou p edic ions abou he u u e
obus ness o ecosys ems?
Rega ding he i s ques ion, admi edly he as majo i y o s udies on s abili y a e
based upon agg ega ed ne wo ks, dis ega ding he phenology o species (Bas olla e al., 2009;
Thébaul and Fon aine, 2010; Suweis e al., 2013). A guably, his choice may be jus i ied, o a
ce ain ex en , as long as he popula ion e ec s de i ed om he ecological in e ac ions build
up along he season. None heless, i is clea ha he lack o phenological in o ma ion no
only implies a loss o de ail, bu may also lead o o e o unde es ima ing ce ain ecological
ela ionships. This is specially ele an , as we will discuss below, o indi ec in e ac ions such
as compe i ion, which can be s ikingly modi ied when conside ing he empo al dimension o
he sys em. In his line o hough , some wo ks ha e explo ed he consequences o accoun ing
o phenology in he cha ac e iza ion o he dynamics. Fo ins ance, Encinas-Viso e al. (2012),
p oposed a mic oscopic dynamical model ha in ol ed bo h mu ualism and compe i ion
in o de o analyze, by explo ing a wide ange o heo e ical o ms o he con igu a ion
o phenology, he p ope ies o he esul ing hypo he ical ne wo ks o in e ac ions. Mo e
ecen ly, Ramos-Jilibe o e al. (2018) p oposed a highly- ealis ic popula ion model and
applied i o s udy a se o empi ical ne wo ks wi h i s co esponding phenology, upon
which hey applied di e en ypes o pe u ba ions. In bo h o hese examples, he au ho s
concluded ha he phenology plays a undamen al ole in d i ing he communi y’s s abili y
97
6. Dynamics one mo e ime
and dynamics, by p oducing phenomena ha could no be explained om he agg ega ed
pe spec i e alone.
On he o he hand, he s udy o he impac o phenology on he dynamics is ine i ably
in e wined wi h a global pe u ba i e p ocess ha menaces a g ea a ie y o na u al sys ems,
namely he clima e change. This comes as no su p ise i we conside he mul iplici y o
en i onmen al ac o s ha play a ole in de e mining he s a ing da es o ac i i y, as e iewed
in Chap e 5. In pa icula , bo h he onse o lowe ing and he da e o i s eme gence o
insec s ha e been claimed o ad ance as he en i onmen al empe a u e aises, which can
lead o wha is commonly known as phenological shi s (Hegland e al., 2009). The mos
immedia e consequence o his phenological pe u ba ion is he possibili y ha mu ualis ic
pa ne s desynch onize, leading o a lack o empo al o e lap –also called misma ch– ha
hampe s pollina ion se ices and may e en ually yield o biodi e si y loss. As a esul , du ing
he las decade he e has been a majo explosion in he numbe o wo ks de o ed o quan i y
he possible ex en o such misma ches (Memmo e al., 2007; Hegland e al., 2009; Bu kle
e al., 2013; Duchenne e al., 2020) as well as hei impac on he pe sis ence o species and
he communi y s abili y (Re illa e al., 2015; Ra e y e al., 2015).
T u h be old, in his chap e we will ocus mainly on he i s ques ion, al hough i would
be wo hy o keep in mind he possible applica ions o his kind o dynamical models o he
s udy o phenological shi s and misma ches. In wha ollows, hence, we s a by in oducing
one o hese possible models.
6.2 A model o inco po a e phenology
The aim o his sec ion is o p esen a model ha pe mi s assessing he e ec s o phenology
on he o ganiza ion o ecological mu ualis ic sys ems, pa icula ly on biodi e si y pe sis ence.
To his end, we wo k upon a p e ious popula ion model p oposed by G acia-Láza o e al.
(2018), ha in es iga es he in luence o he ne wo k s uc u e on he pe sis ence o species.
In pa icula , his model exploi s a bilaye amewo k like he one depic ed in Fig. 6.1,
which accoun s, simul aneously, o mu ualis ic links be ween species o di e en kind and
compe i i e in e ac ions among membe s o he same guild. Such compe i ion is d i en, as
de ailed in sec ion 5.2, by he sha ing o common mu ualis ic esou ces (Jones e al., 2012),
he e o e being de i able om he empi ical ne wo k o mu ualis ic in e ac ions. As we had
done in he p e ious chap e , we will concen a e on he pa icula case o plan -pollina o
communi ies, al hough some o he esul s a e in ac gene alizable o o he ecological o
social sys ems ha a e equally based on consume - esou ce ela ions.
Le us ocus he e on plan -pollina o sys ems and conside a communi y consis ing o
NPspecies o plan s and NAspecies o animals, being N=NP+NA he o al numbe o
species. The plan s’ pa ame e s and a iables a e ep esen ed by he supe sc ip
P
, while
A
s ands o animals. As always, he mu ualis ic ela ionships a e gi en by he bipa i e
NP×NA
ma ix,
B
, wi h
Bik
= 1 i animal species
k
pollina es he plan species
i
, and
Bik
= 0 o he wise. Wi hin his amewo k, in he o iginal model by G acia-Láza o e al.
(2018) he e olu ion o he abundance o each species is desc ibed by a di e en ial equa ion,
which akes in o accoun bo h he abundance o o he species and he in e ac ion wi h hem,
esul ing in a sys em o
N
coupled di e en ial equa ions. In de ail, le
sP
i
be he abundance
o he plan species
i
, being
αP
i
i s in insic g ow h a e. Then, he ela i e abundance o a
gi en plan ie ol es acco ding o:
1
sP
i
dsP
i
d =αP
i−βP
isP
i−βP
0PjP,i6=jsP
jPk∈ABikBjksA
k
Pk∈ABiksA
k
+γP
0Pk∈ABiksA
k
1 + hPγP
0Pk∈ABiksA
k
,(6.1)
which akes in o accoun he mean- ield in a-species compe i ion o esou ces egula ed
by pa ame e
βP
i
, he s uc u ed in e -species compe i ion o mu ualis ic esou ces ponde ed
by pa ame e
βP
0
and inally he mu ualis ic bene i . This las e m models he e ec o
mu ualism ollowing a Holling Type II unc ional esponse, as p oposed by Bas olla e al.
(2009), ha in ol es an in e ac ion s eng h pa ame e
γP
0
and he so-called handling ime
e m,
hP
, which egula es he sa u a ion o he mu ualis ic e m. Mo eo e , he exis ence o
98
6.2. A model o inco po a e phenology
Figu e 6.1: Bilaye ep esen a ion o a plan -pollina o ne wo k. Mu ualis ic in e ac ions a e
depic ed in g ey as he in e -laye links, while compe i i e in e ac ions o sha ed esou ces
a e ep esen ed in whi e as in a-laye connec ions.
compe i i e in e ac ions is media ed by he e m
BikBjk
, as p e iously in oduced in Eq. 5.2.
Al hough we ocused he e on he empo al e olu ion o plan s’ popula ion, an analogous
exp ession can be d awn o he pollina o s. O e all, as a o emen ioned he sys em can be
seen as a mul ilaye ne wo k like he one depic ed in Fig. 6.1, whe e in e -laye connec ions
ep esen he mu ualis ic in e ac ions, while in a-laye links accoun o he compe i ion.
In o de o include he empo al dimension in o his o malism, we adop a i s -o de
app oach ha quan i ies he dynamical consequences o phenological o e lap in an e ec i e
manne . This app oxima ion p esumes ha species in e ac homogeneously along ime, and
hence he esul ing amoun o mu ualis ic bene i o compe i ion s ess is p opo ional o he
ac ion o sha ed o e lap in hei pe iods o ac i i y, as de ined in sec ion 5.2 and depic ed
in Fig. 5.4. Subsequen ly, he mu ualis ic and compe i ion g ow h e ms can be modula ed
by a linea unc ion o he empo al coexis ence be ween species. Tha is, he la ge he
amoun o ime du ing which wo species coexis , he la ge he co esponding mu ualis ic o
compe i i e e m will be. As a way o model his mechanism, we can exploi he phenological
coe icien s Ω
P
ijk
and Φ
P
ik
, de ined in Eqs. 5.5-5.6. Indeed, ollowing he e ec i e assump ion
desc ibe he e, we can in oduce hem di ec ly in o equa ion 6.1, which leads o:
1
sP
i
dsP
i
d =αP
i−βP
isP
i−βP
0PjP,i6=jsP
jPk∈ABikBjksA
kΩP
ijk
Pk∈ABiksA
kΦP
ik
+γP
0Pk∈ABiksA
kΦP
ik
1 + hPγP
0Pk∈ABiksA
kΦP
ik
,
(6.2)
whe e he mu ualis ic and compe i i e e ms a e now escaled by he phenological
coe icien s 5.5-5.6.
As a ma e o ac , exp ession 6.2 can be w i en mo e compac ly. Le
MP
i
be he
biomass o he pollina o s o a gi en plan species
i
, hen
MP
i
=
Pk∈ABiksA
k
Φ
P
ik
, and
le
WP
ij
be he biomass o he pollina o s sha ed by wo plan species
i, j
, such ha
WP
ij =Pk∈ABikBjksA
kΩP
ijk. Wi h his no a ion, Eq. 6.2 u ns in o:
1
sP
i
dsP
i
d =αP
i−βP
isP
i−βP
0Pj∈P,i6=jsP
jWP
ij
MP
i
+γP
0
MP
i
1 + hPγP
0MP
i
.(6.3)
In he same way, he ela i e abundance a ia ion o an animal species kis gi en by:
1
sA
k
dsA
k
d =αA
k−βA
ksA
k−βA
0Pl∈A,k6=lsA
lWA
kl
MA
k
+γA
0
MA
k
1 + hAγA
0MA
k
.(6.4)
99

6. Dynamics one mo e ime
I is wo hy o ema k ha he la e equa ions 6.3-6.4 a e o mally iden ical o he
o iginal model in oduced in e e ence G acia-Láza o e al. (2018). This illus a es he ac
ha , ollowing he p ocedu e p esen ed he e, we ha e ansla ed he bina y agg ega ed
ne wo k in o a weigh ed, bu s ill agg ega ed, o malism. Ce ainly, his is s ill a a he
basic, i s -o de app oach o he modeling o empo al a iabili y, since he s uc u e o
he ne wo k does no change o e ime. Howe e , while o he mu ualis ic ne wo k he
bina y s uc u e is no modi ied wi h espec o he agg ega ed ep esen a ion –i.e. ea u es
like he deg ee, connec i i y, e c, a e p ese ed–, in wha conce ns he compe i i e ne wo k
some links may emo ed, pa icula ly in hose cases in which he in e ac ing species do no
coincide along he season. In his sense, he agg ega ed case p esen s a so o wo s -case
scena io in e ms o he compe i ion, ha is empe ed by he in oduc ion o he phenology.
6.3 Resul s in wo empi ical da ase s
Wi h his me hodology a hand, we a e eady now o es he p edic ions o he a o emen ioned
model on a empi ical da ase . In pa icula , we s udy he h ee eal ne wo ks ex ac ed om
he Bu kle and he Kan sa da ase s. The de ails on each da ase a e gi en in Appendix A,
while he main ea u es o each sys em ha e been cha ac e ized in sec ion 5.2. Along he
p esen sec ion we will analyze how hei communi y dynamics is a ec ed by phenology:
i s , by applying he popula ion model in oduced abo e and, secondly, by compa ing hese
esul s wi h he p edic ions o a null model whe e he s a ing da es o ac i i y o bo h guilds
a e shi ed.
To s a wi h, ollowing an analogous p ocedu e o he one p oposed by G acia-Láza o
e al. (2018), we nume ically in eg a e he sys em o Eqs. 6.4-6.3 o a wide ange o di e en
alues o mu ualis ic and compe i i e s eng h (see Appendix J o mo e de ails on he
compu a ional implemen a ion). In Fig. 6.2, we plo he numbe o su i ing species in he
s eady s a e, o bo h he agg ega ed empi ical case as modeled by Eq. 6.1 and he ne wo k
wi h empi ical phenology as gi en by Eq. 6.2.
Mo eo e , in o de o complemen he in o ma ion p o ided by hese wo scena ios, we
cons uc a null model based on he shi ing o he s a ing da es unde he condi ion o
cons aining he bina y mu ualis ic ne wo k and he pe iods’ dis ibu ion. In pa icula ,
we apply he nume ical p ocedu e desc ibed in Appendix I.1. This p o ides, hence, a
andomiza ion o he links’ weigh s associa ed o he phenological o e lap, and will pe mi
assessing he obus ness o he esul s wi h espec o possible a ia ions in he in e ac ion
s eng h due o shi s in he s a ing da es. In Fig. 6.2 we plo , oge he wi h he esul s o
he agg ega ed and he weigh ed ne wo ks, he a e age biodi e si y ob ained by applying
he model o Eq. 6.2 o e a se o null con igu a ions gene a ed by syn he ically shi ing he
empi ical s a ing da es.
The esul s summa ized in Fig. 6.2 poin ou se e al aspec s ha a e wo h discussing.
Fi s , in gene al e ms we eco e a compa able pa e n o he one ound by G acia-Láza o
e al. (2018), in he sense ha inc easing he mu ualism posi i ely a ec s biodi e si y in
he low-compe i ion egime, bu u ns ou o be de imen al o species’ pe sis ence when
compe i ion o mu ualis ic esou ces is se e e. Mo eo e , simila ly o wha we obse ed
when alking abou he ne wo k’s s uc u e, he e ec o phenology on he ne wo k is highly
sys em-dependen . Indeed, he wo da ase s exhibi almos opposi e beha io s: while o he
Bu kle da ase he egion o maximum pe sis ence inc eases d ama ically in he model wi h
phenology in compa ison o he agg ega ed case, in bo h yea s o he Kan sa da ase his
egion dec eases. This can be unde s ood i we conside he subs an ial di e ences be ween
he wo da ase s, ho oughly de ailed in sec ion 5.2.
In o de o a emp o be e unde s and hese esul s, we in es iga e he ela ion be ween
he pe sis ence o a species and i s indi idual cha ac e is ics. In pa icula , we conside wo
undamen al p ope ies o , espec i ely, he s uc u al and he empo al dimension o he
sys em, namely he species deg ee and i s pe iod o ac i i y. As shown in Fig. 6.3, in he h ee
empi ical examples unde s udy he deg ee is a s ong de e minan o he pe sis ence o a
species. In de ail, ha ing a small deg ee is a necessa y condi ion –al hough no su icien – o
a educed egion o pe sis ence, which means ha specialis s species a e mo e ulne able o
100
6.3. Resul s in wo empi ical da ase s
Figu e 6.2: Biodi e si y pe sis ence as a unc ion o he mu ualism and in e -species
compe i ion pa ame e s. Panels show he le els o biodi e si y in he s a iona y s a e
o he popula ion model gi en by Eqs. 6.4-6.3, as a unc ion o he s eng h o mu ualis ic
and compe i i e in e ac ions – ha is, pa ame e s
γ0
and
β0
. The colo scale ep esen s
he numbe o species in he s eady s a e. The le panels show he esul s ob ained when
phenology is neglec ed; he middle panels depic he esul s ob ained when conside ing he
obse ed phenological coe icien s; and he igh pannels show he esul s ob ained when
conside ing he phenological coe icien s co esponding o 100 independen null con igu a ions,
gene a ed by shi ing he ini ial imes o ac i i y and lowe ing as explained in Appendix I.1.
ex inc ion. This esul is obse ed o bo h he agg ega ed ne wo k and he ne wo k weigh ed
wi h he empi ical phenology, as well as o he null model whe e he s a ing da es a e shi ed.
F om a ecological iewpoin , his should come as no su p ise since specialis s species ha e
been adi ionally ega ded as mo e agile (McKinney, 1997), gi en he highe speci ici y
o hei mu ualis ic esou ces, specially pollina o s (Memmo e al., 2007; Ramos-Jilibe o
e al., 2018).
On he o he hand, conside ing he ela ion wi h he species’ phenological pe iod yields
di e gen esul s depending on he da ase , as can be seen in Fig. 6.3. In e es ingly, he
analysis o he Bu kle da ase e eals ha species wi h sho pe iod a e bene i ed om he
in oduc ion o he empo al dimension, in compa ison o bo h he agg ega ed case and he
null model. This could explain he sha p inc ease in he egion o maximum pe sis ence,
o he empi ical case, depic ed in Fig. 6.2. None heless, his inding is no gene al. In
he Kan sa da ase we ind ins ead ha in oducing he phenology is de imen al o he
101
6. Dynamics one mo e ime
pe sis ence o species wi h sho pe iods. In pa icula , in his da ase he e is a bulk o
pollina o species wi h jus one o wo days o obse ed ac i i y (see Fig. 5.3) which ep esen
he majo i y o ex inc ions in Fig. 6.3.
Figu e 6.3: Rela i e size o he egion o pe sis ence pe species as a unc ion o hei deg ee
(le ) and pe iod ( igh ). We plo he esul s o he agg ega ed case (in ed), he empi ical
case (in ligh blue) and he null moodel (in da k blue), in which case we depic he a e age
and he s anda d de ia ion o he ela i e egion o pe sis ence o each species o e he null
ensemble.
The dynamical unce ain y associa ed o species wi h sho pe iod migh be be e
unde s ood by compa ing he empi ical esul s wi h he es ima ions p o ided by he null
model. In pa icula , le us in oduce a p obabili y o su i al
ps,i
o each species
i
. In he
null model,
ps,i
is de ined as he a io be ween he numbe o imes ha species
i
pe sis s ali e
on he whole ange o explo a ion o
β
and
γ
, among he o al numbe o null con igu a ions.
In he s udy o he empi ical ne wo k his quan i y is bina y, such ha , i we call i
xs,i
, we
ha e
xs,i
= 1 i species
i
ne e ge s ex inc and
xs,i
= 0 o he wise. Using hese quan i ies,
we can de ine he z-sco e associa ed o he empi ical obse a ion when compa ed wi h he
null ensemble as:
z-sco ei=xs,i −pi,s
σi
,(6.5)
whe e he s anda d de ia ion
σi
is compu ed as he one co esponding o a Be noulli
p ocess, since he su i al/ex inc ion e en is bina y. Thus, he z-sco e will be posi i e i
he species su i ed in he empi ical case bu go ex inc a ce ain numbe o imes in he
null ensemble, and nega i e o he wise. We plo ed his quan i y as a unc ion o he deg ee
102
6.4. Conclusions and pe spec i es
o he species and as a unc ion o i s pe iod in Fig. 6.4. He e again we obse e di e gen
esul s among da ase s, in he sense ha he pa icula co ela ions a e di e en o each
ne wo k. Howe e , we also ind ha in gene al he species wi h sho pe iod end o ha e a
la ge z-sco e –in absolu e e ms– which means ha hei pe sis ence is mo e a ec ed by
he in oduc ion o phenology –ei he in a posi i e o in a nega i e way– and hence hei
long- e m obus ness is mo e di icul o p edic .
Figu e 6.4: Rela ion be ween he z-sco e as de ined in Eq. 6.5 and he deg ee (on he le ) o
he pe iod (on he igh ), o he h ee da ase s.
On he whole, we ha e obse ed ha , oge he wi h being a specialis , ha ing a sho
pe iod is ypically a sign o highe ulne abili y o ex inc ions. On he o he hand, hese
esul s pose p obably mo e ques ions han answe s, as we ind once again no uni e sal pa e n
in how phenology speci ically a ec s ecosys em’s dynamics, pa icula ly species pe sis ence.
T u h o be old, his should come as no su p ise i we ake in o accoun he subs an ial
inhe en dissimila i ies be ween he wo da ase s. How o cope wi h his p oblems and wha
could be done as p ospec i e wo k o y o answe some o he emaining ques ions is wha
we discuss in he nex sec ion, he las o his chap e .
6.4 Conclusions and pe spec i es
Along his chap e , we ha e b ie ly add essed he ques ion o he in e play be ween phenology,
mu ualism and species pe sis ence, by aking as a pa adigma ic example he case o plan -
pollina o sys ems. In de ail, we ha e de eloped a p e ious dynamical model ha accoun s
o bo h compe i i e and mu ualis ic in e ac ions, in o de o include he e ec o phenology.
We ha e applied a i s -o de app oxima ion, consis ing in weigh ing he s eng h o ecological
in e ac ions by he empo al o e lap among he pa ne s. A e examining his model, ou
103

APPENDIX A
Da ase s
A.1 Ecological ne wo ks
The Web o Li e eposi o y
1
con ains a la ge da ase o ecological ne wo ks, including e y
di e se species, in e ac ion ypes, and ecosys ems loca ion and clima e. We de ail below he
main cha ac e is ics o he da a exploi ed in Chap e s 3 and 4 o his hesis. The numbe
wi hou b acke s ep esen s he o al numbe o ne wo ks ex ac ed om he eposi o y,
while he numbe be ween b acke s ep esen s he quan i y o ne wo ks ha ha e a minimum
size o 20 nodes, which co esponds o he minimal cu o used in he s udy p esen ed in
Chap e 4.
•
133 (118) Plan -pollina o ne wo ks. He e, he links among guilds ep esen mu ualis ic
ela ionships, cha ac e ized by bene i ing bo h in e ac ing agen s. In his case,
animals, including mainly insec s, pollina e lowe ing plan s. This ac i i y p o ides
he pollina o s wi h nu ien s while pollina ed plan s enhance i s ep oduc i e success.
The wo se o nodes o he bipa i e ne wo k ep esen he species o he plan and
pollina o guilds.
•
30 (23) Seed-dispe se ne wo ks. He e, he links also ep esen mu ualis ic in e ac ions,
consis ing now o bi ds eeding on he ui s o ce ain plan s and hen con ibu ing
o hei ep oduc ion and dispe sal by dissemina ing hei seeds. Hence, one guild is
o med by he plan species and he o he by he bi d species.
•
51 (43) Hos -pa asi e ne wo ks. He e, he links depic a pa asi ic ela ionship, whe e
one o he species ob ains bene i s in de imen o he o he . Explici ly, hese ne wo ks
a e o med by di e en lea species which eed on di e se mammal species. Al hough
his is no a mu ualis ic in e ac ion, he sys em may s ill be ep esen ed by a bipa i e
ne wo k whe e he wo guilds co espond o lea and mammals species.
•
4 (4) Plan -he bi o e ne wo ks. He e, he links ep esen a consume - esou ce
in e ac ion be ween insec species (one guild) and plan species ( he o he guild).
In de ail, he ne wo ks depic di e en communi ies whe e mac olepidop e an species
eed on se e al P unus species.
•
4 (3) Plan -an ne wo ks. These ne wo ks include wo examples o di e se ypes o
communi ies: a ne wo k depic ing an s which eed on plan nec a , his being a consume -
esou ce in e ac ion, and wo ne wo ks ep esen ing communi ies whe e an species li e
in a mu ualis ic associa ion wi h ce ain plan species known as My mecophy es.
A.2 Economic ne wo ks
In Chap e 4 we s udied as well a se o economic ne wo ks, which a e publicly a ailable
in He nandez e al. (2018). In pa icula , hese da a consis o :
1Publicly a ailable a : www.web-o -li e.es
111
A. Da ase s
•
8 economic ne wo ks ep esen ing buye s-selle s in e ac ions in he Boulogne-su -Me
Fish Ma ke in F ance (He nández e al., 2018). These a e mu ualis ic ne wo ks aken
om a e y di e en con ex . Each ne wo k desc ibes he ansac ions obse ed in
di e en days in he bila e al o in he auc ion Fish Ma ke . These daily ne wo ks a e
ypically much dense han ecological ones.
A.3 Phenology and plan -pollina o ne wo ks
We used wo public da ase s which con ain obse a ions o mu ualis ic in e ac ions in plan -
pollina o sys ems oge he wi h hei phenology. We de ail he peculia i ies o each da ase
below.
The Illinois da ase
The da ase ga he ed by Bu kle e al. (2013) co esponds o a se o woodland si es in
Ca lin ille, Illinois (USA), obse ed du ing he sp ings o 2009 and 2010 om Ma ch o
May
2
. The obse ed ne wo k o mu ualis ic in e ac ions con ained o iginally 26 sp ing-
blooming he baceous plan s and 54 bee species. Howe e , 2 plan species did no exhibi
any in e ac ion in he empi ical ne wo k, so we emo ed hem o he analysis and ob ained
e en ually a 24 ×54 ne wo k.
The da ase also included he obse ed phenology, speci ically he s a ing and ending
da es o he ac i e s ages o bo h plan s and pollina o s. Using hose, we ex ac ed as well
hei pe iods o ac i i y and he middle da es.
The Les os Island da ase
The da ase eco ded by Kan sa e al. (2018) is based on a wo yea s s udy conduc ed in
Aglios S e anos, Les os Island (G eece) du ing he sp ings o 2011 and 2012 om Ap il
o July. The published da ase
3
includes he agg ega ed ne wo k o in e ac ions o e he
wo yea s, composed by 41 plan species and 168 pollina o s. Mo eo e , i p o ides as well
mixed in o ma ion o phenology, including one-yea obse a ions and wo-yea s a e ages.
None heless, Kan sa e al. sha ed wi h us he obse ed phenology o each yea sepa a ely,
ha is, he s a ing and ending da e o ac i i y o each species in bo h seasons. Using hese,
we ob ained as well hei pe iods and middle da es.
Rema kably, a la ge po ion o he pollina o species a e p esen in only one o he
seasons. In pa icula , ou o he o al 168 pollina o species obse ed along he wo yea s,
only 67 species (39
.
9%) we e pe sis en ly ound in bo h seasons, while 53 species (31
.
5%)
we e obse ed jus du ing he i s yea and 48 species (28
.
6%) jus du ing he second yea o
obse a ion. This implies ha no only he e is an impo an u no e o in e ac ions among
he wo consecu i e yea s, bu also he e is a conside able species u no e , as sugges ed as
well by p e ious s udies (Olesen e al., 2008; Chaco e al., 2018).
In o de o ob ain he ne wo k o in e ac ions speci ic o each yea , we emo ed hose
in e ac ions p esen in he agg ega ed ne wo k ha , a e including he co esponding
in o ma ion on phenology, occu among species whose pe iods o ac i i y do no o e lap. In
such cases, i is clea ha he in e ac ion was no possible due o he lack o phenological
o e lap and hence we can emo e i om he yea ly ne wo k. A e doing so o each season,
we emo ed as well hose species ha do no longe hold mu ualis ic in e ac ions –i.e. hey
ha e ze o deg ee. As a esul , bo h yea -speci ic ne wo ks sizes’ a e educed o 34 plan s
species and 113 pollina o s in he i s yea , and 38 plan species and 104 pollina o s in he
second yea o obse a ions.
This me hod pa ly app oxima es he ne wo k o in e ac ions co esponding o each
yea , in he sense ha we migh be sligh ly o e es ima ing he numbe o in e ac ions
and unde es ima ing he u no e by keeping all he plausible in e ac ions p esen in he
2The o iginal da ase is publicly a ailable a : h ps://doi.o g/10.5061/d yad. p321
3
The o iginal da ase is publicly a ailable pa ly as Supplemen a y Ma e ial o he a icle, pa ly as a
eposi o y a : h ps:// igsha e.com/a icles/da ase /Da a_ om_Disen angling_ he_ ole_o _ lo al_senso y_s imuli_
o_pollina ion_ne wo ks_/5663455
112
A.3. Phenology and plan -pollina o ne wo ks
agg ega ed ne wo k. Howe e , in he absence o a mo e de ailed da ase , i is a guably a
alid es ima ion since i en i ely espec s he ecological niches.
113
APPENDIX B
Compu a ional implemen a ion o he
null model
He e I gi e he nume ical de ails on how I ob ained he Lag ange mul iplie s
~x∗
and
~y∗
ha
de ine he co esponding s a is ical ensembles o he empi ical ne wo ks. The de e mina ion o
hese mul iplie s migh be achie ed ollowing ei he o wo p ocedu es: by di ec ly maximizing
he log-likelihood in Eq. 3.11 o by sol ing he non-linea , coupled se o equa ions in 3.12-
3.13. While in his sec ion I p esen ou pa icula implemen a ion o bipa i e g aphs,
he e is a Ma lab package de eloped by Squa ini e al. (2015) which nume ically sol es his
op imiza ion p oblem o a a ie y o ypes o unipa i e g aphs and cons ain s
1
. The main
di e ence be ween hei implemen a ion and ou app oach lies in he nume ical unc ions
used o ind he op imal Lag ange mul iplie s. Whe eas hei package uses a local op imizing
unc ion
2
, I made a special e o o ensu e ha he maxima ound a e global, in pa icula ,
by combining he use o a global sea ch algo i hm wi h a local op imiza ion unc ion epea ed
o e a la ge se o pseudo- andom ini ial condi ions.
B.1 Cons ained maximiza ion o he en opy
I nume ically op imized he log-likelihood by simula ed annealing (Co ana e al., 1987; Go e
e al., 1994, 1996). Gi en he pseudo-alea o y cha ac e o his app oach, which allows o
o e come he ba ie s sepa a ing local minima, i is ex endedly used in si ua ions in which
he co-exis ence o se e al local op ima is expec ed.
Mo e p ecisely, in ou case we need o ake in o accoun ha in Eq. 3.11 he deg ees
may be degene a e. This means ha nodes o he same guild ha ing iden ical deg ees sa is y
equi alen equa ions, hence necessa ily bea ing he same solu ion. To accoun o his, I
in oduced a mul iplici y ac o
mp
o plan s and
ma
o animals. I we call
edP
and
edA
he edundancy o plan s and o animals (namely, he co esponding numbe s o epea ed
deg ees), hen he sys em can be edimensionalized o
N0
P
=
NP− edP
and
N0
A
=
NA− edA
.
This p ocedu e is an ex ension o he bipa i e case o he edimensionaliza ion p oposed
in Ga laschelli and Lo edo (2008) o a unipa i e ne wo k. Consequen ly, he log-likelihood
migh be ew i en in o:
L(~x) =
N0
P
X
p=1
mp pln(xp) +
N0
A
X
a=1
mahaln(ya)−
N0
A
X
a=1
N0
P
X
p=1
mpmaln(1 + xpya)(B.1)
1
The package implemen s he en i e ‘Max&Sam’ me hodology o a gene al unipa i e se ing. I p o ides
he unc ions o ind he nume ical solu ion o he maximum en opy ensemble unde a numbe o di e en
cons ain s o bo h undi ec ed o di ec ed ne wo ks, bina y o weigh ed. Mo eo e i pe mi s o sample
he esul ing ensemble. The package is eely a ailable online a h ps://i .ma hwo ks.com/ma labcen al/
ileexchange/46912-max-sam-package-zip. The bipa i e case can be ep oduced, using his package, by
ede ining he di ec ed unipa i e case in such a way ha one guild solely has ou -going links, while he o he
only has incoming links. The o mal analogue o p ese ing he a e age deg ee sequences o bo h guilds
would be now o cons ain, in a e age, he in-deg ee and ou -deg ee sequences.
2
One can ind he documen a ion o he Ma lab unc ion hey used o sol e he op imiza ion p oblem
in h ps://es.ma hwo ks.com/help/op im/ug/ mincon.h ml.
115

B. Compu a ional implemen a ion o he null model
Al hough, in analy ical e ms, he o iginal exp ession in Eq. 3.11 and his la e one a e
ob iously equi alen , om a compu a ional poin o iew educing he numbe o a iables
enhances he algo i hm’s e iciency. Besides, imposing om he beginning such iden i y
be ween a iables imp o es he accu acy o he p og am.
I ha e p og ammed a s anda d e sion o he simula ed annealing algo i hm. The
andom numbe gene a o used is he one by To al and Chak aba i (1993), wi h a s a ing
empe a u e o
T
= 10
3
, a educ ion ac o o he empe a u e o
RT
= 0
.
85, and a o al
numbe o upda es pe ixed empe a u e o 2
·
10
4
. The algo i hm s ops when i e consecu i e
i e a ions di e in less han a pa ame e
ol
= 10
−6
. Fu he mo e, I an he algo i hm 10
imes pe ne wo k wi h di e en andom seeds, in o de o p oduce independen sequences
o explo a ions. I is conside ed ha he global op imum has been eached when all he uns
con e ge o he same solu ion.
B.2 Local solu ion o he sys em o equa ions
I ha e sol ed he se o equa ions by means o a local, de e minis ic algo i hm known as he
modi ied Powell hyb id me hod. In pa icula , I used he MINPACK lib a y (Mo é e al., 1980)
o FORTRAN, a ailable online (Bu on S. Ga bow, Bu on S. Ga bow). This me hod inds
he ze o o a non-linea sys em by exploi ing i s Jacobian, which I analy ically calcula ed
and implemen ed in o he p og am.
As be o e, we may e-dimensionalize he p oblem o
N0
P
equa ions o plan s and
N0
A
equa ions o animals, which now ead:
p=
N0
A
X
a=1
maxpya
1 + xpya
o p= 1, ..., N0
P,(B.2)
ha=
N0
P
X
p=1
mpxpya
1 + xpya
o a= 1, ..., N0
A.(B.3)
I implemen ed hese equa ions and hei Jacobian and an he algo i hm wi h a ole ance
ol
= 10
−11
(as de ined in he sou ce code). The possibili y o exploi ing he g adien p o ides,
in gene al, a g ea e local accu acy han he simula ed annealing echnique. Howe e , i s
sho coming lies on he isk o ge ing apped in local op ima, om which, due o i s
de e minis ic na u e, i is unable o escape. To compensa e his d awback I pe o med
a signi ican sampling o he space o ini ial condi ions, by unning 10
4
i e a ions o he
algo i hm, each wi h a di e en andom selec ion o s a ing poin s, co e ing as well dis inc
anges. Howe e , due o he encoun e o ough, a he acciden al con igu a ion su aces, he
modi ied Powell hyb id me hod was no always able o con e ge o a solu ion. The a e o
success was app oxima ely 50%.
To inally ensu e ha he maximum ound is global, I compa ed he ou comes o he
a ious independen uns. Mo eo e , o he cases when he Powell algo i hm con e ged, I
could also compa e he solu ions ob ained o bo h me hods which amoun s o a o al o 10
uns o he simula ed annealing and 10
4
o he Powell hyb id me hod. In all hese cases
same maximum was ound.
I ha e also checked ha he cons ain s a e co ec ly me wi h a ela i e p ecision be ween
0
.
01% and 10%, by compu ing he expec ed deg ees om Eq. 3.12-3.13 and compa ing o
he co esponding alues o he obse ed ne wo ks. The wo s case o 10% was ypically
caused by disc epancies in low deg ees, gene ally he mos sensi i e o imp ecisions in he
elemen s o he andomized ma ix gi en ha he ma ix elemen s o low deg ee nodes a e
usually e y small (see Fig. 3.4 in main ex as an example).
116
APPENDIX C
S a is ical measu es o s able-NODF
C.1 Analy ical exp essions o he i s wo momen s o s able-NODF
The analy ical exp ession o he a e age o s-NODF o e he ensemble is:
hs-NODF(B)i∗=1
K
NP
X
i<j







·
NA
P
a=1 hbiaihbjai
NA
P
a=1 hbjai







+1
K
NA
X
k<l









·
NP
P
p=1 hbpkihbpli
NP
P
p=1 hbpli









.(C.1)
The s anda d de ia ion o s-NODF is gi en by he analogous o Eqs. 3.26-3.27, whe e he
pa ial de i a i es in Eq. 3.27 co espond o:
K∂s-NODF(B)plan s
∂b c
=
NP
X
j= +1
bjc
j
+
−1
X
i=1
bic
−
−1
X
i=1
NA
X
a=1
bia b a
2(C.2)
K∂s-NODF(B)animals
∂b c
=
NA
X
l=c+1
b l
hl
+
c−1
X
k=1
b k
hc−
c−1
X
k=1
NP
X
p=1
bpk bpc
hc2,(C.3)
C.2 S a is ical measu es o s able-NODF
We ha e compu ed he eal nes edness and i s s a is ical signi icance using s able-NODF, o
he 167 ne wo ks in ou da ase . In pa icula , o each eal ne wo k we ha e calcula ed he
es ima ed a e age and he s anda d de ia ion using he analy ical exp essions in Eq. C.1 and
Eqs. C.2-C.3. Fig. C.1 and Table C.1 show ha eal nes edness is no s a is ical signi ican
when measu ed by s able-NODF ei he .
F ac ion o n ws wi h |z-sco e| ≤ 1 F ac ion o n ws wi h |z-sco e| ≤ 2
118 ou o 167 70.7% 162 ou o 167 97.0%
Table C.1: F ac ion o ne wo ks whose disc epancy be ween he eal and andomized
nes edness is less o equal han one o wo sigma, o nes edness measu es pe o med wi h
s able-NODF.
117
C. S a is ical measu es o s able-NODF
0
20
40
60
80
100
120
140
0 10 20 30 40 50 60 70 80 90 100 110
analy ic a e age S-NODF
S-NODF eal ne wo k
wo sigma in e al
one sigma in e al
iden i y cu e
Figu e C.1: Compa ison be ween he eal measu e o s able-NODF and i s a e age in he
ensemble es ima ed using an analy ical exp ession, o he 167 ne wo ks o ou s udy.
118
APPENDIX D
Addi ional me hods o assess he
signi icance o nes ed pa e ns
D.1 Signi icance es s
I quan i ied he signi icance o he nes edness measu es using he z-sco e index, which o
a gene al p ope y
x
eads:
x∗−hxi
σx
. Fo us,
hxi
is he a e age nes edness compu ed in he
ensemble, ei he analy ically o by explici sampling, and we compa e i wi h he empi ical
obse a ions
x∗
. The s anda d de ia ion is
σx
. Gi en ha he NODF alues a e Gaussian
dis ibu ed in he andom ensemble, he z-sco es can be di ec ly ela ed o p- alues.
Mo eo e , I pe o med a mul iple es co ec ion which allows accoun ing o he ac
ha as he numbe o s a is ical es s inc eases, so does he p obabili y o inding a e
e en s (Benjamini and Hochbe g, 1995). Thus, conside ing he mul iple compa isons p e en s
o e s a ing he numbe o signi ican disco e ies. I is pe inen o apply his echnique
he e since he 167 cases s udied a e e alua ed unde he same null hypo hesis and all o
hem ollow a no mal dis ibu ion. I employed he alse disco e y a e me hod, in pa icula
he Benjamini-Hochbe g p ocedu e which applies o independen es s (Benjamini and
Hochbe g, 1995). The co ec ion was nume ically ca ied ou using he
S a sModel
package
in Py hon (Seabold and Pe k old, 2010).
D.2 Sel o ganizing ne wo k model
In o de o eo ganize he o iginal ne wo k in o an e en mo e nes ed s uc u e, we nume ically
implemen ed he sel -o ganizing ne wo k model p oposed by Bu gos e al. (2007). This
me hodology keeps cons an many aspec s suscep ible o a ec he measu e o nes edness, like
he size and ill, bu modi ies he deg ee sequences h ough he edis ibu ion o connec ions.
We ewi ed he links among species ollowing wo simple ules: i) when changing an
in e ac ion, he new pa ne mus ha e highe deg ee han he old neighbo ii) i he
p oposed edis ibu ion lea es one o he wo nodes wi h no in e ac ions a all, we ejec
he change. This ope a ion was epea ed un il he sys em achie ed a ozen s a e in which
no mo e econnec ions we e accep ed (we conside ed his happened when 10
3N
consecu i e
ejec ions occu ed, being
N
he numbe o nodes o he ne wo k). The inal ozen s a e is
no mally no pe ec ly nes ed, since condi ion (ii) ypically leads o con igu a ions which a e
no u e ly op imal. To compensa e his, we ca ied ou 10
3
independen ewi ing ope a ions
o each ne wo k. We hen a e aged he a ge p ope ies, namely, nes edness (measu ed
using NODF) and he a iance o he join deg ee sequence o he wo guilds.
D.3 S a is ical measu es o deg ee asso a i i y
Asso a i i y is a ne wo k ea u e ha quan i ies o wha ex en nodes end o ma ch
o he nodes ha a e simila (o dissimila ) o hem. He e, we used he no ion o deg ee
asso a i i y. We ollowed he de ini ion p oposed by Newman (2002), which consis s o a
no malized co ela ion coe icien be ween deg ees. This e en ually co esponds o he Pea son
119
F. Me hods o assessing nes edness’ me ics pe o mance
NIR
We implemen ed a p og am in FORTRAN90 ha calcula es he NIR alue o he eal ne wo k
and o he co esponding se o null ne wo ks. In each case, he esul ing alue o nes edness
is mul iplied by 100 in o de o p ese e he same scale o all he me ics.
In o de o accoun o he possible e ec s o he degene acy in he o de ing, ha is,
he ac ha mul iple con igu a ions a e possible when we o de ows and columns by hei
deg ee, we compu ed he esul ing NIR as he a e age o e a la ge numbe o equi alen ly
o de ed con igu a ions. These con igu a ions we e p oduced by andomly swapping he
ma ix posi ion o nodes wi h he same deg ee. In mo e de ail, o gene a e a new o de ing
we un o e all he nodes wi h degene a e deg ee and, o each node, we accep a posi ion
swap wi h p obabili y 1
2.
Fo each eal ne wo k, we calcula ed he degene acy, ideg, he numbe o epea ed deg ees.
Then, we p oduced a o al o 10
·ideg
con igu a ions wi h he same deg ee o de bu di e se
ow and column posi ions. This p ocedu e was ca ied ou bo h o he eal ne wo k and o
each null ne wo k in he sampling wi h he excep ion o he Robe son’s ne wo k (Robe son,
1929) o which, due o i s e y la ge size (1500 species), only 10 degene a e con igu a ions
ha e been compu ed.
Spec al adius
We compu ed he la ges eigen alue using he Rso wa e (R Co e Team, 2013), in pa icula
he eigs_sym unc ion om he ARPACK package.
F.2 Co ela ions among me ics and ne wo k ea u es
The s a is ical co ela ions we e nume ically calcula ed using Py hon. The Spea man ank
co ela ion coe icien
s
and i s
p
- alue we e calcula ed using he Scipy package (
?
), in
pa icula he scipy.s a s.spea man unc ion.
We pe o med he linea i s using he S a smodels package (Seabold and Pe k old, 2010),
which ca ies ou a mul i-linea leas -squa e eg ession and p o ides mul iple in o ma ion,
including he adjus ed
R2
, he pa ial eg ession coe icien s, hei s anda d de ia ion and
hei associa ed
p
- alue. The
- a io
i,j
co esponding o each pa ial eg ession coe icien ,
βi,j, is calcula ed as ollows:
− a ioi,j =βi,j
σi,j
(F.1)
whe e
σi,j
is he s anda d de ia ion associa ed o ha coe icien . This index p o ides,
hence, in o ma ion on how signi ican ly di e en om ze o is a ce ain eg ession coe icien .
126

APPENDIX G
The nullnes eposi o y
In o de o complemen he heo e ical wo k p esen ed in Chap e s 3 and 4, I eleased an open
gi hub eposi o y named nullnes
1
ha aims a being a p ac ical ool, o bo h ecologis s
and ne wo k scien is , o assess he nes edness o eal and null ne wo ks. The eposi o y is
ho oughly documen ed, wi h examples and eady- o-use p og ams, and allows pe o ming
he analysis discussed in Pay a ó-Bo às e al. (2019) and Pay a ó-Bo às e al. (2020).
The nullnes eposi o y is di ided in o wo main blocks. The i s pa is ela ed o he
cons uc ion o he null model. In pa icula , i p o ides a p og am o compu e he maximum-
en opy and maximum-likelihood ensemble discussed in sec ion 3.2 o any bipa i e ne wo k
in oduced by he use . I also con ains he eady- o-use p obabili ies o in e ac ion in he
null ensemble, o he whole da ase o empi ical ne wo ks desc ibed in he Appendix A.
On he o he hand, he second pa o he package is conce ned wi h he measu emen o
nes edness. In de ail, we p o ide he codes o quan i y he eal deg ee o nes edness o
a gi en ma ix, oge he wi h he i s wo momen s o i s null dis ibu ion, using any o
he six me ics discussed in he i s pa o he hesis. This can be done ei he by using
he analy ical exp essions de i ed in Pay a ó-Bo às e al. (2019) (only o NODF and he
spec al adius) o by nume ically sampling he null ensemble, o any o he six me ics
s udied in Pay a ó-Bo às e al. (2020).
1
The codes and he accompanying documen a ion can be downloaded a : h ps://gi hub.com/cclaualc/
nullnes .
127
APPENDIX H
S a is ical es s on phenology
H.1 Quali y o a i using he Kolmogo o -Smi no es
In o de o assess he quali y o he i and ob ain i s co esponding p- alue, we s a by
pe o ming a Kolmogo o -Smi no one-sample es be ween he i ed unc ion and he
empi ical dis ibu ion, using he s a s.ks es so wa e om he SciPy package (Vi anen
e al., 2020) in Py hon. Nex , in o de o de e mine i s p- alue we explici ly compa e he
ob ained KS- alue wi h he expec ed s a is ics, sampled using Mon e Ca lo simula ions.
In mo e de ail, we p oceed as ollows:
•
We calcula e he K-S es s a is ics –using he SciPy so wa e–, ha we may call
Dobs
, be ween he empi ical dis ibu ion and he i ed unc ion, whose pa ame e s a e
es ima ed using he maximum likelihood i ing so wa e om he s a s unc ion in
SciPy.
•
Using Mon e Ca lo me hods, we gene a e a sample o size
Nsampl
composed by syn he ic
dis ibu ions, each sampled om he i ed unc ion and ha ing he same size as he
obse ed, empi ical dis ibu ion.
•
Fo each syn he ic sample indexed by
i
, we i a unc ion using he same unc ional
o m as in he empi ical case and using he maximum likelihood es ima o om he
s a s unc ion in SciPy.
•
We hen calcula e he K-S es s a is ics be ween his no el i and he co esponding
syn he ic sample, which p oduces a new dis ance ha we will call Dsyn,i.
•
Once his p ocedu e has been epea ed o all he syn he ic dis ibu ions gene a ed,
we can compa e he obse ed K-S s a is ics
Dobs
wi h he dis ibu ion o
Dsyn,i
in he
sample.
A e ca ying ou his compu a ion, he p- alue is hen calcula ed as:
p- alue =numbe o samples wi h Dsyn,i > Dobs + 1
Nsampl
,(H.1)
which, as no mally, quan i ies whe he he obse ed KS dis ance
Dobs
di e s signi ican ly
om wha would be expec ed i he obse ed dis ibu ion o pe iods was eally gene a ed by
he i ed unc ion. Hence, when he p- alue is su icien ly la ge he i is compa ible wi h
his assump ion.
H.2 Kolmogo o -Smi no wo sample es
We p og ammed a Kolmogo o -Smi no es on wo samples by using he ks_2samp unc ion
om he SciPy package. In de ail, he K-S es o wo samples pe mi s challenging he null
hypo hesis ha wo pa icula samples come om he same s a is ical dis ibu ion. The
smalle he KS dis ance, he smalle he disc epancy be ween he wo samples. In he case
whe e he p- alue is ela i ely small (e.g. p- alue
<
0
.
05) he di e ence among dis ibu ion
129
H. S a is ical es s on phenology
is signi ican and hence he null hypo hesis is ejec ed, which in ou case implies ha he
dis ibu ion p oduced by he gi en syn he ic model is incompa ible wi h he empi ical
obse a ions.
H.3 Kullback-Leible di e gence
The Kullback-Leible di e gence quan i ies how one p obabili y dis ibu ions
Q
(
x
)is di e en
om a second, e e ence dis ibu ion called
P
(
x
). In pa icula , he KL di e gence om Q
o P is de ined as:
DKL =X
x
P(x)log P(x)
Q(x),(H.2)
and i measu es he amoun o in o ma ion los when app oxima ing
P
(
x
)by
Q
(
x
). In
ou case, we calcula e his quan i y aking as he e e ence unc ion
P
(
x
) he empi ical
dis ibu ion o phenological o e lap, while
Q
(
x
)co esponds o dis ibu ion o phenological
o e laps p oduced by he null model unde conside a ion. We implemen his calcula ion in
a p og am in Py hon ha uses he s a s.en opy so wa e om he SciPy package.
H.4 Pea son co ela ion wi h ime lag
In o de o calcula e he Pea son co ela ion o di e en alues o ime lag, we ake he
e olu ion o a gi en s uc u al p ope y in he syn he ic model, and shi i al e na i ely
backwa ds o ahead in ime. Fo each di e en alue o ime lag, we calcula e he
co esponding Pea son coe icien be ween he syn he ic, shi ed a ay, and he unchanged
empi ical ec o . In pa icula , we use he s a s.pea son so wa e om he SciPy package
o compu e he Pea son co ela ion. We epea his p ocedu e o a wide ange o di e en
empo al lags, and e en ually ex ac he maximum alue. This also p o ides he ime lag a
which he co ela ion among he empi ical and he syn he ic sequences is maximized.
130
APPENDIX I
Nume ical implemen a ion o syn he ic
models
In his appendix we explain he di e en nume ical me hods used o encoun e he s a ing
da es o species’ ac i i y, on he basis o di e en hypo hesis and unde he cons ain ha
mu ualis ic pa ne s sha e a minimum phenological o e lap.
Fi s , we desc ibe wo nume ical app oaches o sol e he p oblem o main aining he
mu ualis ic o e lap: (i) by shi ing some gi en s a ing da es, (ii) by op imizing a ce ain
quan i y wi hin he cons ain s imposed by he non-ze o o e lap. Secondly, we explain how
we implemen ed hese me hods o each pa icula syn he ic model.
I.1 Shi o s a ing da es
He e we in oduce a me hod o shi a gi en se o ini ial imes o ac i i y, wi h he condi ion
ha a minimal mu ualis ic o e lap is p ese ed. The pe iods o ac i i y, ha we will deno e
as
pP
i
o a plan
i
and
pA
k
o an animal
k
, a e kep unchanged. We solely modi y he iming
o he beginning o he ac i i y o each species, whe e we no e he ini ial ime by
P
0,i
o each
plan iand A
0,k o each animal k.
The p og am akes as inpu he a o emen ioned in o ma ion oge he wi h he s a ing
da es o ac i i y, ha , impo an ly, need o al eady ul ill he se o inequali ies w i en in
Eqs. 5.7-5.8. This can be achie ed by aking, depending on he case, a i ial solu ion –e.g.
he same s a ing da es o all species– o he empi ical da ase –i a ailable. Then, he
p oposal o he no el s a ing da es is done unde he condi ion ha a non-ze o amoun o
empo al o e lap be ween in e ac ing mu ualis ic pa ne s mus be p ese ed. In de ail, we
impose, a leas , 1 day o o e lap, which is he minimal uni o phenology in he s udied
da ase s. To wa an ha his occu s, we ix he ollowing bounda ies o he new ini ial
imes. Le us call 0P
0,i he no el ime p oposed o plan species i:
•
The
uppe
bound is gi en by he neighbo species whose ac i i y ceases ea lie . Tha
is, by he minimum o he se o imes
{ A
0,k
+
pA
k}
, whe e
k
uns o e he indexes
o he animal species in e ac ing wi h plan
i
(so, i
B
is he bipa i e ma ix, hose
ul illing ha
Bi,k
= 1). Le us illus a e his wi h he example on Fig. I.1. The uppe
bounda y o he ini ial lowe ing ime o plan numbe 1 is gi en by he pollina o
ha inishes pollina ing ea lie , which is animal numbe 1. The e o e, as can be seen
in he igu e, he uppe limi o { P
0,1}coincides wi h { A
,1}.
•
The
lowe
bound is ound by sub ac ing he pe iod o plan i o he ime a which
s a s he las species. Thus, by he maximum o he se o imes
{ A
0,k}
, minus
pP
i
.
In he example o Fig. I.1, he pollina o ha begins i s ac i i y he la es is animal
numbe 3, a
{ A
0,3}
. To
{ A
0,3}
, we need o sub ac he pe iod o plan numbe 1 (in
yellow) in o de o we eco e he lowe limi .
131

I. Nume ical implemen a ion o syn he ic models
Acco ding o hese c i e ia, he new s a ing da es o ac i i y a e ex ac ed om he
ollowing uni o m dis ibu ion:
0P
0,i ⊂Umax{ A
0,k}−pP
i+ 1,min{ A
0,k +pA
k}−1,(I.1)
whe e he 1 is espec i ely sub ac ed and added o he bounda ies in o de o ensu e a
leas one day o o e lap be ween mu u alis ic pa ne s.
Figu e I.1: Example o a plan wich in e ac s mu ualis ically wi h h ee pollina o species.
We calcula e he uppe and lowe bounda ies o he new ini ial ime o he plan
0P
0,1
using,
espec i ely, he minimum inal ime o ac i i y o i s pollina o s (he e
A
,1
=
A
0,1
+
pA
1
) and
he maximal ini ial ime (he e
A
0,3
) minus he pe iod o he plan (
pP
1
). As can be seen in
he igu e, o his bounda ies we co espondingly added and subs ac ed one day in o de o
man ain a leas one day o mu ualis ic o e lap.
No e again ha his p ocedu e equi es an ini ial con igu a ion o s a ing da es, ei he
empi ical o hypo he ical, ha is hen andomized. Subsequen ly, he esul ing se o s a ing
da es show some dependency on he ini ial con igu a ion. As explained below, we exploi
his ea u e o gene a e syn he ic phenologies ha a e app oxima ely cen e ed.
I.2 Cons ained op imiza ions
He e we desc ibe a second p ocedu e o gene a e a se o s a ing da es ha ul ills he
inequali ies desc ibed in Eqs. 5.7-5.8. In con as wi h he algo i hm desc ibed in sec ion I.1,
his me hod does no equi e any o he inpu han he ne wo k o mu ualis ic in e ac ions
and he pe iods, and hence he e is no ini ial condi ion o he s a ing da es. Ins ead, we
seek a solu ion ha , on he hand, e i ies he sys em o coupled inequali ies, and on he
o he hand, op imizes a gi en quan i y, desc ibed by an objec i e unc ion.
In pa icula , gi en he cons ain s in Eqs. 5.7-5.8, we may use linea p og amming
echniques o ind a easible solu ion. Since
L >> N
, he p oblem is o e de e mined, ye we
know al eady ha i is consis en because he e a e some i ial solu ions which sa is y all he
cons ain s -e.g., all species s a ing o inishing he same da e-. Howe e , we a e pa icula ly
in e es ed in non- i ial solu ions which can yield o ealis ic o s a is ically in e es ing
con igu a ions o phenology. This leads o he in oduc ion o he objec i e unc ion, which
we will seek o op imize o e he possible se o solu ions. In sec ions I.4-I.6 we speci y he
nume ical implemen a ion o di e en o ms o his objec i e unc ion, bu he gene al o m
o he algo i hm is main ained.
In de ail, we sol e he cons ained op imiza ion p oblem de ined by Eqs. 5.7-5.8 h ough
a wo-s ep p ocess. Fi s , we pe o m mul iple local op imiza ions wi h di e en ini ial
seeds, using he unc ion minimize om he SciPy package in Py hon (Vi anen e al., 2020)
–speci ically he me hod ’ us -cons ’. Second, we se hese local solu ions as he s a ing
popula ion o a di e en ial e olu ion algo i hm, which pe o ms a global s ochas ic sea ch.
In pa icula we use he di e en ial_e olu ion unc ion om SciPy. This combina ion o
an i e a i e local sea ch plus a global op imiza ion aims a ensu ing a obus inding o he
global op imum.
132
I.3. Synch onized phenology
I.3 Synch onized phenology
In o de o p oduce a con igu a ion whe e species’ pe iods a e ela i ely synch onized, we
p oceed in se e al s eps. Fi s , we se he middle da es by sampling a no mal p obabili y
dis ibu ion. Once he medium da es a e de e mined, he loca ion o he pe iod o each
species’ pe iods along he seasons is andomized using an algo i hmic p ocedu e which
conse es he mu ualis ic in e ac ions p o ided by he empi ical ne wo k.
In mo e de ail, we p og ammed a so wa e in Py hon ha ope a es as ollows:
•
We selec one o he guilds ( o ins ance, he pollina o s) and de e mine hei s a ing
da es by ex ac ing hei middle da es om a no mal dis ibu ion wi h s anda d
de ia ion σ= 0.5days and mean µ= 120 days.
•
We hen se he s a ing da es o he o he guild, by using he andomiza ion p ocess
explained in sec ion I.1 o his appendix. In de ail, we de e mine he ange -maximum
and minimum da es- wi hin which we can ex ac a andom s a ing da e while ensu ing
ha he mu ualis ic o e lap will no be los . This no el s a ing da e is ex ac ed om
a uni o m dis ibu ion.
•
To inish, we e- andomize again he i s guild, using he same p ocedu e as in he
p e ious s ep, in o de o elax he assump ion o he no mali y o middle imes o
ac i i y.
Indeed, we epea s eps (ii) and (iii) i e a i ely se e al imes in o de o imp o e
he andomiza ion. This e en ually yields a non-pe ec ly synch onized con igu a ion o
phenology.
I.4 Minimiza ion o he compe i ion
In o de o ind he con igu a ion ha , unde he men ioned cons ain s, minimizes he
compe i ion, we apply he nume ical algo i hm desc ibed in sec ion I.2 whe e he objec i e
unc ion is he o al compe i i e o e lap. In de ail, a each i e a ion o he op imiza ion
algo i hm, we calcula e he o al sum o unno malized phenological o e lap among all
compe i o s, including bo h compe i ion among plan s and among pollina o s.
I.5 Maximiza ion o he a iance
We pe o m a cons ained sea ch ha maximizes he a iance o he middle da es by using
he algo i hm desc ibed in sec ion I.2. In his case, he objec i e unc ion co esponds o he
a iance, and i is calcula ed a each i e a ion o he op imiza ion p ocess.
I.6 Maximiza ion o he en opy
In his model, we ind he phenological con igu a ion ha maximizes he en opy by using
he algo i hm explained in sec ion I.2. The objec i e unc ion co esponds he e o he
Shannon-Gibbs en opy o he middle da es, as de ined in Eq. 5.11, which we explici ly
compu ed using he Py hon so wa e.
133
APPENDIX J
Nume ical in eg a ion o he popula ion
model
We simula ed he communi y dynamics as p o ided by Eq. 6.1 and i s coun e pa o
animals o he agg ega ed ne wo k, and by Eqs. 6.3-6.4 o he ne wo k weigh ed wi h he
phenological o e laps. Each simula ion s a s om andom ini ial condi ions o he ela i e
abundances ha a e aken a andom om a uni o m dis ibu ion in he in e al [0.05, 0.95].
Following e e ences Bas olla e al. (2009); Roh e al. (2014); G acia-Láza o e al.
(2018) we ake he alues o
αP,A
i
om a uni o m dis ibu ion in [0.9, 1.1], he in a-species
compe i ion pa ame e is ixed o
βP,A
j
= 5 and he Holling e m o
hP,A
= 0
.
1. Fo simplici y,
we assume ha compe i ion and mu ualism pa ame e s ake he same alues o plan s
and animals. A e a ansien , equilib ium is assumed when all he species’ equencies
emain cons an : a ha poin , we keep he in o ma ion on inal abundances and numbe o
su i ing species. A species is conside ed ex inc when i s ela i e abundance is lowe han
10
−9
. This p ocess is epea ed o di e en alues o he in e ac ion s eng h, ha is, we
modula e
β
o compe i ion and
γ
o mu ualism homogeneously o all plan and all animal
species. E en ually, his p o ides a hea map o biodi e si y o each pai o
β
and
γ
alues,
as plo ed in Fig. 6.2.
135