Chaos, Soli ons & F ac als: X 3 (2019) 10 0 021
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Spec al and localiza ion p ope ies o andom bipa i e g aphs
C.T. Ma ínez-Ma ínez
a
,
b
, J.A. Méndez-Be múdez
a
,
c
,
∗, Yami Mo eno
b
,
d
,
e
,
Jai J. Pineda-Pineda
, José M. Siga e a
g
a
Ins i u o de Física, Benemé i a Uni e sidad Au ónoma de Puebla, Apa ado Pos al J-48, Puebla 72570, Mexico
b
Ins i u e o Biocompu a ion and Physics o Complex Sys ems (BIFI), Uni e si y o Za agoza, 50018 Za agoza, Spain
c
Depa amen o de Ma emá ica Aplicada e Es a ís ica, Ins i u o de Ciências Ma emá icas e de Compu ação, Uni e sidade de São Paulo - Campus de São
Ca los, Caixa Pos al 668, São Ca los 13560-970, SP, B azil
d
Depa men o Theo e ical Physics, Uni e si y o Za agoza, 50 0 09 Za agoza, Spain
e
ISI Founda ion, Tu in, I aly
Ecology and Su i al o Mic oo ganisms Resea ch G oup (ESMRG), Labo a o io de Ecología Molecula Mic obiana (LEMM), Cen o de In es igaciones en
Ciencias Mic obiológicas (CICM), Ins i u o de Ciencias (IC), Benemé i a Uni e sidad Au ónoma de Puebla (BUAP), Puebla, México.
g
Facul ad de Ma emá icas, Uni e sidad Au ónoma de Gue e o, Ca los E. Adame No.54 Col. Ga i a, Acalpulco G o. 39650, Mexico
a i c l e i n o
A icle his o y:
Recei ed 30 Decembe 2019
Accep ed 19 Janua y 2020
A ailable online 1 Feb ua y 2020
PACS:
64.60.aq
89.75.Da
05.45.M
73.20.Jc
Keywo ds:
Bipa i e g aphs
Delocaliza ion ansi ion
Spec al p ope ies
a b s a c
Bipa i e g aphs a e o en ound o ep esen he connec i i y be ween he componen s o many sys ems
such as ecosys ems. A bipa i e g aph is a se o n nodes ha is decomposed in o wo disjoin subse s,
ha ing m and n −m e ices each, such ha he e a e no adjacen e ices wi hin he same se . The con-
nec i i y be ween bo h se s, which is he ele an quan i y in e ms o connec ions, can be quan ified by
a pa ame e α∈ [0, 1] ha equals he a io o exis en adjacen pai s o e he o al numbe o possible
adjacen pai s. He e, we s udy he spec al and localiza ion p ope ies o such andom bipa i e g aphs.
Specifically, wi hin a Random Ma ix Theo y (RMT) app oach, we iden i y a scaling pa ame e ξ≡ξ( n, m,
α) ha fixes he localiza ion p ope ies o he eigen ec o s o he adjacency ma ices o andom bipa -
i e g aphs. We also show ha , when ξ< 1/10 ( ξ> 10) he eigen ec o s a e localized (ex ended), whe eas
he localiza ion– o–delocaliza ion ansi ion occu s in he in e al 1/10 < ξ< 10. Finally, gi en he po en-
ial applica ions o ou findings, we ound o he s udy by demons a ing ha o fixed ξ, he spec al
p ope ies o ou g aph model a e also uni e sal.
©2020 The Au ho s. Published by Else ie L d.
This is an open access a icle unde he CC BY license. ( h p://c ea i ecommons.o g/licenses/by/4.0/ )
1. In oduc ion
The la es de elopmen s in ne wo k science ha e la gely con-
ibu ed o a be e unde s anding o he s uc u e and dynamics
o many eal-wold complex sys ems [1–3] . As a ma e o ac , e-
sea ch done du ing he las 20 yea s ha e allowed o ake key s eps
in ou comp ehension o seemingly di e se phenomena such as he
la ge-scale sp eading o diseases [4,5] , in o ma ion dissemina ion
[2] , cascading ailu es [6] , di usion dynamics [7–9] and mo e e-
cen ly, on how mul ilaye sys ems wo k [10–12] . These ad ances
a e no only a a heo e ical le el. The inc easing a ailabili y o
new and ich da a as well as ou compu a ional capabili ies ha e
made i possible o mo e om s udying syn he ic models, o cha -
ac e ize and model ealis ic sys ems.
∗Co esponding au ho a : Ins i u o de Física, Benemé i a Uni e sidad Au ónoma
de Puebla, Apa ado Pos al J-48, Puebla 72570, Mexico.
E-mail add ess: [email p o ec ed] (J.A. Méndez-Be múdez).
Du ing hese yea s, ne wo ks ha e been s udied om many di -
e en angles, anging om mo e heo e ically-g ounded s udies (in
he bes adi ion o g aph heo y) o ully da a-d i en models.
Some imes, he a chi ec u e o he subs a e ne wo k is known and
hus, i could be modeled explici ly. Howe e , i is o en he case in
which he ne wo ks a e syn he ic ei he because we do no know
he eal connec ion pa e ns o because we need o simpli y he
s uc u e o he sys em o enable analy ical app oxima ions. In he
la e scena io, one easonable assump ion is o gene a e andom
g aphs, so ha one ge s id o possible co ela ions and isola es he
impac o he connec i i y among he sys em’s cons i uen s on i s
dynamics. Besides, andom e sions a e o en e y use ul as null
models, ha allow o indi idua e which p ope ies o he sys em
a e uly unexpec ed and which a e no [13,14] .
Among he many esul s ha can be highligh ed, pe haps he
mos use ul ones a e hose ha ela e he s uc u e o ne wo ks
wi h hei dynamics h ough he analysis o he spec al p ope -
ies o he adjacency o Laplacian ma ices o such ne wo ks. Fo
ins ance, i has been shown ha i is possible o cha ac e ize he
h ps://doi.o g/10.1016/j.cs x.2020.10 0 021
2590-0544/© 2020 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY license. ( h p://c ea i ecommons.o g/licenses/by/4.0/ )
2 C.T. Ma ínez-Ma ínez, J.A. Méndez-Be múdez and Y. Mo eno e al. / Chaos, Soli ons & F ac als: X 3 (2019) 10 0 021
c i ical p ope ies o a disease sp eading p ocess in e ms o he
la ges eigen alue o he adjacency ma ix o he ne wo k on op o
which he dynamics akes place [4,5] . Admi edly, he ac ha he
epidemic h eshold, i.e., he poin beyond which he sys em expe i-
ences a mac oscopic ou b eak, can be exp essed in e ms o opo-
logical p ope ies makes i possible o s udy wha a e he e ec s
o he opology on he dynamics o complex ne wo ked sys ems.
Ano he impo an example o he p e ious ela ionship be ween
s uc u e and dynamics is gi en by synch oniza ion phenomena,
whe e one finds ha he s abili y o a ully synch onized sys em
can be s udied in e ms o he spec al p ope ies o he subs a e
ne wo k [1–3] .
In his pape , we ollow he line o esea ch men ioned abo e
and s udy a class o ne wo ks ha is o en ound in na u al and a -
ificial sys ems, namely, bipa i e g aphs. Wi hin he classes o ne -
wo ks ha ha e been analyzed in he las wo decades, bipa i e
g aphs ha e gone unno iced in many ega ds, o ins ance, in ela-
ion o hei spec al p ope ies. We in end o fill his gap by s udy-
ing he localiza ion and spec al p ope ies o andom bipa i e
g aphs wi hin RMT app oaches. This iewpoin has been success-
ully used o s udy some opological [15] , spec al [16–18] , eigen-
ec o [16,17] , and anspo [19] p ope ies o ER– ype andom
ne wo ks wi h a special ocus on uni e sali y. Mo eo e , we ha e
also pe o med scaling s udies on o he andom ne wo k models,
such as mul ilaye and mul iplex ne wo ks [20,21] and andom–
geome ic and andom– ec angula g aphs [22] .
The es o he pape is o ganized as ollows. In Sec ion 2 we
define he andom bipa i e g aph model we shall use in ou s udy.
Then, in Sec ion 3 we pe o m a scaling analysis o he eigen ec-
o p ope ies (cha ac e ized by he Shannon o in o ma ion en-
opy) o ou bipa i e g aph model. The scaling analysis allows
o define a uni e sal pa ame e o he model ha we alida e in
Sec ion 4 wi h he scaling o he spec al p ope ies (cha ac e ized
by he dis ibu ion o a ios o consecu i e ene gy-le el spacings).
We summa ize ou esul s in Sec ion 5 also discussing possible ap-
plica ions wi hin he domain o ecosys ems and hei s abili y.
2. Bipa i e g aph model
We conside bipa i e g aphs composed by wo disjoin se s
wi h m and n −m e ices each such ha he e a e no adjacen
e ices wi hin he same se , being n he o al numbe o e -
ices in he bipa i e g aph. The connec i i y be ween bo h se s
is quan ified by he pa ame e αwhich is he a io o cu en
adjacen pai s o e he o al numbe o possible adjacen pai s;
ha is, e ices a e isola ed when α= 0 , whe eas he bipa i e
g aph is comple e o α= 1 . Ve ices a e connec ed andomly. We
add o ou bipa i e g aph model sel -edges and u he conside
all edges o ha e andom s eng hs, which allows ha ou bipa -
i e g aph model becomes a RMT model. The e o e, we define he
co esponding adjacency ma ices as membe s o he ensemble
o n ×n spa se eal symme ic ma ices whose non- anishing ele-
men s a e s a is ically independen andom a iables d awn om a
no mal dis ibu ion wi h ze o mean
A
ij
= 0 and a iance
| A
ij
|
2
=
(1 + δij
) / 2 . Acco ding o his defini ion, a diagonal adjacency an-
dom ma ix is ob ained o α= 0 , which is known as he Pois-
son ensemble in RMT e ms. In Fig. 1 , we show examples o ad-
jacency ma ices o andom bipa i e g aphs wi h n = 100 e ices
and some combina ions o m and α. No e ha when labeling he
e ices acco ding o he se hey belong o, he adjacency ma ices
o bipa i e g aphs ha e a block s uc u e.
He e we define m ( esp. n −m ) as he numbe o e ices o he
smalle (bigge ) se . In his espec , he case m = n/ 2 is a limi ing
case whe e bo h se s ha e he same numbe o e ices, m = n −m .
Mo eo e , he case m = 1 is ano he limi ing case in which he
smalle se consis s o a single e ex. Thus, in wha ollows we
will conside andom bipa i e g aphs cha ac e ized by he pa am-
e e se ( n, m, α) wi h 1 ≤m ≤n /2 and 0 ≤α≤1. No ice ha he
case m > n /2 is edundan because i is equi alen o he in e -
change o he se s.
3. Eigen ec o p ope ies. Scaling and uni e sali y
In his s udy, we cha ac e ize he eigen ec o s o andom bipa -
i e g aphs by using in o ma ion o Shannon en opy, which o he
eigen ec o k is gi en as
S
k
= −
n

j=1

k
j


2
ln

k
j


2
. (1)
S
k measu es he numbe o p incipal componen s o he eigen-
ec o k in a gi en basis. The e o e, he la e quan i y is a
good measu e o eigen ec o localiza ion/delocaliza ion. In ac ,
his quan i y has al eady been used o cha ac e ize quan i a i ely
he complexi y and localiza ion p ope ies o he eigen ec o s o
he adjacency ma ices o se e al andom ne wo k models (see ex-
amples in [16,17,20–22] and e e ences he ein). Below we use ex-
ac nume ical diagonaliza ion o compu e he eigen ec o s k and
eigen alues λk
( k = 1 . . . n ) o he adjacency ma ices o la ge en-
sembles o andom bipa i e g aphs cha ac e ized by he pa ame e
se ( n, m, α).
In Fig. 2 , we p esen he Shannon en opies S
k o he eigen-
ec o s o en ealiza ions o he adjacency ma ices shown in
Fig. 1 . No e ha o m = n/ 2 all ows o he adjacency ma ix ha e
he same a e age numbe o nonze o o -diagonal elemen s, see
Fig. 1 (a), he e o e he co esponding eigen ec o s a e expec ed o
be equi alen and hey should ha e simila en opies; his can be
e ified in Fig. 2 (a). In con as , o any m < n /2, m ows o he ad-
jacency ma ix ha e a la ge numbe o nonze o o -diagonal ele-
men s han he emaining n −m ows, see Fig. 1 (b-d). Hence, as
i can be seen in Fig. 2 (b-d), he en opies o he co esponding
eigen ec o s can be g ouped in o wo se s cha ac e ized by di e -
en a e age alues  S  (see he dashed lines in hese panels, which
sepa a e he wo se s ha ing di e en a e ages). Despi e hese di -
e ences, aking in o accoun ha we wan o use he a e age en-
opy o find scaling p ope ies in andom bipa i e g aphs, and
ha o his pu pose we need a single quan i y ega dless o he
specific g aph, we compu e a e ages o e all a ailable eigen ec-
o s, hus aking in o accoun he con ibu ion o bo h eigen ec o
se s.
F om defini ion (1) , i ollows ha 
S

= 0 when α= 0 , since
he eigen ec o s o he (diagonal) adjacency ma ices o ou an-
dom bipa i e g aph model ha e only one non- anishing compo-
nen wi h magni ude equal o one. On he o he hand, o α= 1
he bipa i e g aph is comple e and  S  ge s i s maximal alue,
S
MAX
, o a gi en combina ion o n and m . Thus, when 0 < α< 1
we should obse e 0 <  S  < S
MAX
.
In Fig. 3 we p esen he a e age Shannon en opy  S  as a
unc ion o he connec i i y pa ame e α o he eigen ec o s o
andom bipa i e g aphs and o se e al pa ame e combina ions.
We obse e ha he cu es o  S  , o any combina ion o n and
m , ha e a e y simila unc ional o m as a unc ion o α: The
cu es  S  show a smoo h ansi ion om app oxima ely ze o o
S
MAX
when αinc eases om α∼0 (mos ly isola ed e ices) o
one (comple e bipa i e g aphs). Recall ha when  S  ≈0 he co -
esponding eigen ec o s a e localized (i.e.,  S  ≈0 defines he lo-
calized egime). In con as , when  S  ≈S
MAX
, he co esponding
eigen ec o s a e delocalized. Thus, he cu es o  S  e sus αin
Fig. 3 display he delocaliza ion ansi ion o he eigen ec o s o
ou andom bipa i e model. As a complemen a y in o ma ion, in
Fig. 4 we epo S
MAX
, i.e., he alue o  S  a α= 1 , o andom
bipa i e g aphs o se e al combina ions o n and m .
C.T. Ma ínez-Ma ínez, J.A. Méndez-Be múdez and Y. Mo eno e al. / Chaos, Soli ons & F ac als: X 3 (2019) 10 0 021 3
Fig. 1. Nonze o adjacency ma ix elemen s o andom bipa i e g aphs o some combina ions o m and α: (a) m = n/ 2 and α= 0 . 2 , (b) m = n/ 4 and α= 0 . 75 , (c) m = n/ 5
and
α= 0 . 5 , (d) m = n/ 10 and α= 0 . 25 . In all cases n = 100 .
Fig. 2. Shannon en opies S
k o he eigen ec o s o en ealiza ions o he adjacency ma ices shown in Fig. 1 . Dashed lines in panels (b-d) sepa a e g oups o en opies
cha ac e ized by di e en a e age alues.
Fig. 3. A e age Shannon en opy
 S  as a unc ion o he connec i i y α o an-
dom bipa i e g aphs (o sizes anging om n = 10 0 o 80 0) o se e al alues o
m (as indica ed in he panels). Each symbol was compu ed by a e aging o e 10
6
eigen ec o s.
I is impo an o s ess ha in ou g aph model wi h fixed n
he maximal numbe o nonze o adjacency ma ix elemen s is ob-
ained when α= 1 and m = n/ 2 , bu s ill in his case hal o he
o -diagonal adjacency ma ix elemen s a e equal o ze o. The e-
o e he adjacency ma ices o ou andom bipa i e g aphs ne e
ep oduce he Gaussian O hogonal Ensemble (GOE) o RMT - he
GOE is a andom ma ix ensemble o med by eal symme ic an-
dom ma ices A whose en ies a e s a is ically independen an-
dom a iables d awn om a no mal dis ibu ion wi h ze o mean
and a iance | A
ij
|
2
= (1 + δij
) / 2 , see e.g. [23] . Acco dingly, one
should expec S
MAX
<  S 
GOE
, whe e  S 
GOE
≈ln ( n /2.07) is he a -
Fig. 4. Maximum alues o he Shannon en opy S
MAX
as a unc ion o he bipa i e
g aph size n o se e al alues o m . The hick black line co esponds o ln ( n /2.07),
he app oxima e alue o
 S 
GOE
. The a ow indica es dec easing m .
e age en opy o he ( andom and delocalized) eigen ec o s o he
GOE. Howe e , su p isingly, we obse e ha S
MAX
≈ S 
GOE
o m =
n/ 2 , while S
MAX
<  S 
GOE
indeed occu s o any m < n /2, see Fig. 4 .
Also, om Fig. 4 , we can clea ly see ha
S
MAX
∝ ln (n ) . (2)
The e o e, we can conclude ha he maximal en opy se up in
ou andom bipa i e g aph model co esponds o m = n/ 2 and
α= 1 o which GOE s a is ics is obse ed o  S  and expec ed o
o he quan i ies.
Now, o ease ou analysis, in Fig. 5 we plo again  S  bu no -
malized o S
MAX
. The ac ha hese cu es, plo ed in semi-log
scale, a e jus shi ed o he le on he α-axis when inc eas-
ing n makes i possible o hypo hesize he exis ence o a scal-
ing pa ame e ha depends on n . In o de o check his hypo h-
esis and find such a scaling pa ame e , we fi s define a quan i y
ha allows cha ac e izing he posi ion o he cu es  S  / S
MAX
on
he α-axis: We choose he alue o α, ha we label as α∗, o
4 C.T. Ma ínez-Ma ínez, J.A. Méndez-Be múdez and Y. Mo eno e al. / Chaos, Soli ons & F ac als: X 3 (2019) 10 0 021
Fig. 5. A e age in o ma ion en opy  S  no malized o S
MAX as a unc ion o he
connec i i y
α. Same da a o Fig. 3 .
Fig. 6. Localiza ion– o–delocaliza ion ansi ion poin α∗(defined as he alue o α
o which  S  / S
MAX
≈0.5) as a unc ion o he bipa i e g aph size n o se e al al-
ues o m . Dashed lines a e he fi ings o he da a wi h Eq. (3) . The a ow indica es
dec easing m .
which  S  / S
MAX
≈0.5. No ice ha α∗cha ac e izes he localiza ion–
o–delocaliza ion ansi ion o he eigen ec o s o ou g aph model.
Fig. 6 shows he localiza ion– o–delocaliza ion ansi ion poin
α∗as a unc ion o n o se e al alues o m . The linea end o
he da a (in log-log scale) in Fig. 6 implies a powe -law ela ion o
he o m
α∗= Cn
δ. (3)
In ac , Eq. (3) p o ides e y good fi ings o he da a. The al-
ues o δ om he fi ings a e e y close o -0.978 o all he al-
ues o m conside ed he e (see hick ull lines in Fig. 6 ). F om his
obse a ion we can p opose he ollowing scaling o he cu es
Fig. 7. A e age in o ma ion en opy
 S  no malized o S
MAX as a unc ion o he
scaling pa ame e
ξ, see Eq. (4) . Same da a o Fig. 3 . Dashed e ical lines indica e
he wid h o he ansi ion egion
defined as he ull wid h a hal maximum o
he unc ions d
 S  / d ξ s. ξ.
 S  / S
MAX
s α: By plo ing again he cu es o  S  / S
MAX
now as a
unc ion o ξ, ha we define as he a io be ween he connec i i y
pa ame e and he localiza ion– o–delocaliza ion ansi ion poin
ξ=
α
α∗∝
α
n
δ≈αn
0 . 978
, (4)
we obse e ha cu es o di e en bipa i e g aph sizes n col-
lapse on op o a single cu e, see Fig. 7 . Tha is, we conclude
ha , o a gi en a io m / n, ξfixes he localiza ion p ope ies o
he eigen ec o s o he adjacency ma ices o he andom bipa -
i e g aphs, such ha , when ξ< 1/10 [10 < ξ] he eigen ec o s a e
localized [ex ended], while he localiza ion– o–delocaliza ion an-
si ion occu s in he in e al 1/10 < ξ< 10.
E en hough we we e able o scale he Shannon en opy cu es
o andom bipa i e g aphs, as shown in Fig. 7 , he e is s ill a
dependence o hose uni e sal cu es on he a io m / n . To clea ly
show his, in Fig. 8 we epo scaled cu es o he Shannon en opy
o se e al alues o m / n in he localiza ion– o–delocaliza ion an-
si ion egion. He e we can obse e ha he la ge he a io m / n ,
he sha pe he localiza ion– o–delocaliza ion ansi ion. Thus, we
cha ac e ize he wid h o he ansi ion egion, ha we call , as
he ull wid h a hal maximum o he unc ions d  S  / d ξ s. ξ. In
he inse o Fig. 8 we epo as a unc ion o m / n . F om his
figu e, we obse e a clea inc ease o when dec easing he a-
io m / n , an inc ease ha seems o sa u a e o a ios as small as
m / n ∼1/100.
I is wo h s essing ha once we ha e ound ha ξexis s and
ha his pa ame e scales he eigen ec o p ope ies (cha ac e ized
by hei Shannon en opy) o he model o andom bipa i e g aphs
he e s udied, i is na u al o expec ha o he p ope ies (i.e., spec-
al p ope ies, dynamical p ope ies, anspo p ope ies, e c.) o
he g aph model would also scale wi h he same pa ame e . This is
wha we explo e nex , when we alida e he p e ious su mise by
closely inspec ing he co esponding eigen alues.
C.T. Ma ínez-Ma ínez, J.A. Méndez-Be múdez and Y. Mo eno e al. / Chaos, Soli ons & F ac als: X 3 (2019) 10 0 021 5
Fig. 8. Scaled cu es o he Shannon en opy o andom bipa i e g aphs wi h
se e al alues o m / n . A ows indica e dec easing m / n . All cu es co espond o in-
e pola ed da a wi h n = 800 . Inse : Wid h o he ansi ion egion
as a unc ion
o m / n .
4. Spec al p ope ies
In Fig. 9 , we p esen he spec a o he adjacency ma ices o
andom bipa i e g aphs o se e al combina ions o he pa ame-
e s m, n , and α. Each panel is cha ac e ized by a fixed a io m / n
and a fixed scaling pa ame e ξ. So, om he esul s in he p e i-
ous Sec ion, one should expec he ou spec a, epo ed in each
o he panels o Fig. 9 and co esponding o di e en g aph sizes
n , o all one on op o he o he . This is in ac he case, excep o
a small-size e ec clea ly obse ed in Fig. 9 (d,g) when n = 100 . I
is also in e es ing o no e ha he block s uc u e o he adjacency
ma ix clea ly e eals i sel in he spec a, o la ge ξand small
a io m / n , see Fig. 9 (h-i).
To cha ac e ize he spec al p ope ies o he andom bipa i e
g aph model, we use he a ios o consecu i e ene gy-le el spac-
ings , which a e defined as ollows. Le { λ} be a se o o de ed
eigen alues, he co esponding spacings s
k
a e
s
k
=
λk +1
−λk

λ
(5)
whe e  λ is he local mean eigen alue densi y, while he a ios
k
a e defined as [24]
k
=
min (s
k
, s
k −1
)
max (s
k
, s
k −1
)
(6)
such ha
k
∈ [0, 1] ∀ k . Mo eo e , he p obabili y dis ibu ion unc-
ion o in he Poisson limi (which is ep oduced by ou andom
bipa i e g aph model when α= 0 ) is [25]
P
P
( ) =
2
(1 + )
2
. (7)
Ano he impo an limi , ha we will use as a e e ence, is he GOE
case o which P ( ) ge s he o m [25]
P
GOE
( ) =
27
4
+
2
(1 + +
2
)
5 / 2
. (8)
I is impo an o s ess ha he nea es -neighbo ene gy-le el
spacing dis ibu ion P ( s ) [23] is al eady a well accep ed quan i y
o measu e he deg ee o chaos o diso de in complex sys ems
and has been ex ensi ely used o cha ac e ize spec al p ope ies
o complex ne wo ks (see examples in [16,21,22] and e e ences
he ein). Howe e , he use o P ( ) is mo e con enien he e since
i does no equi e he p ocess known in RMT as spec al un old-
ing [23] , whose implemen a ion o spec a wi h kinks as hose in
Fig. 9 (h-i) could be cumbe some.
Fig. 10 p esen s his og ams o P ( ) o andom bipa i e g aphs
wi h se e al combina ions o pa ame e s ( m, n, α). As well as in
Fig. 9. Eigen alues λk
o he adjacency ma ices o andom bipa i e g aphs o se e al pa ame e combina ions ( m, n,
α). Columns [ ows] a e cha ac e ized by a fixed m / n
[
ξ]. A single g aph ealiza ion is conside ed o each cu e. Dashed lines in panels (h) and (i) coincide wi h hose in Figs. 2 (b) and 2 (d), espec i ely.

6 C.T. Ma ínez-Ma ínez, J.A. Méndez-Be múdez and Y. Mo eno e al. / Chaos, Soli ons & F ac als: X 3 (2019) 10 0 021
Fig. 10. Dis ibu ion o a ios o consecu i e ene gy-le el spacings P ( ) o he eigen alues o he adjacency ma ices o andom bipa i e g aphs wi h se e al pa ame e s
combina ions ( m, n,
α). Columns [ ows] a e cha ac e ized by a fixed m / n [ ξ]. Each his og am is cons uc ed wi h 10
6
a ios. Dashed lines in panels (a-c) [(g-i)] co espond
o he RMT p edic ion o P ( ) in he Poisson [GOE] limi , see Eq. (7) [ Eq. (8) ]. In panels (d- ) bo h equa ions, Eqs. (7) and (8) , a e shown in dashed lines. Inse s a e
enla gemen s o he main panels o close o ze o.
Fig. 9 , each panel is cha ac e ized by a fixed a io m / n and a fixed
scaling pa ame e ξ. Wi h his figu e we e i y he in a iance o
P ( ) o fixed ξ, excep o a small size e ec ha is enhanced a
→ 0; see he inse s in panels (a-c,g-i) whe e he con e gence o a
s eady P ( ) is ob ained o la ge enough n . Besides, om Fig. 10 , we
obse e he Poisson o GOE ansi ion in he shape o P ( ) when
inc easing ξ. Also, a he ansi ion bo de s, i.e. a ξ= 0 . 1 and
ξ= 10 , he shape o P ( ) is well desc ibed by he co esponding
RMT p edic ions in he Poisson and GOE limi s, espec i ely. This
confi ms ou defini ion o he localiza ion– o–delocaliza ion ansi-
ion egion: 0.1 < ξ< 10. While, as expec ed, o in e media e al-
ues o ξ, see e.g., Fig. 10 (d- ), P ( ) has a shape which is in e medi-
a e be ween P
P
( ) and P
GOE
( ).
Finally, we would like o add ha i is qui e su p ising ha e en
o m/n = 1 / 10 he P ( ) is e y close o P
GOE
( ) when ξis la ge,
see Fig. 10 (i). Recall ha o any m / n < 2 he co esponding adja-
cency ma ices ha e mo e null han no null o -diagonal ma ix
elemen s (see Fig. 1 ), he e o e, being e y di e en om membe s
o he GOE. Mo eo e , we would also like o ecall ha we ound
ha S
MAX
≈ S 
GOE
only o m/n = 1 / 2 , while S
MAX
<  S 
GOE
o any
m / n < 1/2. The e o e, o ou andom bipa i e g aph model, we can
claim ha P ( ) is less sensi i e o de ia ions om GOE s a is ics
han  S  .
5. Conclusions
In his pape we ha e nume ically s udied he p ope ies e-
la ed o he eigen ec o s and eigen alues o he adjacency ma i-
ces o andom bipa i e g aphs. Specifically, we ha e conside ed
andom bipa i e g aphs wi h sel -loops, whe e all non- anishing
adjacency ma ix elemen s a e Gaussian andom a iables. Ou
andom bipa i e g aph model depends on h ee pa ame e s: The
g aph size n , he g aph connec i i y α, and he size o he smalle
se m composing he bipa i e g aph.
Fi s , h ough a p ope scaling analysis o he Shannon en opy
o he eigen ec o s o he adjacency ma ices o such a andom bi-
pa i e g aph model, we defined a scaling pa ame e ξ≡ξ( n, m, α)
ha fixes he localiza ion p ope ies o he eigen ec o s o a gi en
a io m / n . Mo eo e , ou analysis p o ides a way o p edic he lo-
caliza ion p ope ies o he andom bipa i e g aphs: Fo ξ< 0.1
he eigen ec o s a e localized, he localiza ion– o–delocaliza ion
ansi ion occu s o 0.1 < ξ< 10, whe eas when 10 < ξ he eigen-
ec o s a e ex ended. Nex , o b oaden he applicabili y o ou find-
ings, we demons a ed ha o a fixed ξ, he spec al p ope ies
(cha ac e ized by he dis ibu ion o a ios o consecu i e ene gy-
le el spacings) o he g aph model a e also uni e sal, namely, hey
do no depend on he specific alues o he bipa i e g aph pa am-
e e s.
The esul s he e de i ed a e impo an in a leas one applied
field o esea ch. Admi edly, he s udy o he s abili y o ecologi-
cal sys ems makes use o he wo main ing edien s o ou s udy.
On he one hand, many ecosys ems, including p ey-p eda o and
mu ualis ic sys ems, a e ai h ully ep esen ed by bipa i e g aphs,
which a e assumed o be andom ma ices when no in o ma ion
abou he eal s uc u e is known. On he o he hand, he anal-
ysis o he s abili y o such sys ems is o en educed o unde -
s and he eigen alues and eigen ec o s s uc u e o he in e ac-
ion ma ices (o hei Jacobian). Ou esul s a e impo an in so
a hey show ha he e a e uni e sal p ope ies in such an-
dom bipa i e ne wo ks, which migh help o unde s and, in i s
u n, obus dynamical pa e ns o such sys ems ega dless o hei
specific de ails such as size and in e ac ion s eng hs. We plan
o explo e in mo e de ail his po en ial applica ion in he nea
u u e.
C.T. Ma ínez-Ma ínez, J.A. Méndez-Be múdez and Y. Mo eno e al. / Chaos, Soli ons & F ac als: X 3 (2019) 10 0 021 7
Decla a ion o Compe ing In e es
The au ho s decla e ha hey ha e no known compe ing finan-
cial in e es s o pe sonal ela ionships ha could ha e appea ed o
influence he wo k epo ed in his pape .
Acknowledgemen s
JAM-B acknowledges financial suppo om FAPESP (G an
No. 2019/06931-2 ), B azil, and VIEP- BUAP (G an No. 100405811-
VIEP2019 ) and PRODEP- SEP (G an No. 511-6/2019.-11821 ), Mex-
ico. YM acknowledges pa ial suppo om he Go e nmen o
A agon, Spain h ough g an E36-17R (FENOL), by MINECO and
FEDER unds (FIS2017-87519-P) and by In esa Sanpaolo Inno a ion
Cen e . The unde s had no ole in s udy design, da a collec ion,
and analysis, decision o publish, o p epa a ion o he manusc ip .
JMS was suppo ed in pa by wo g an s om he Minis e io de
Economía y Compe i i idad, Agencia Es a al de In es igación (AEI)
and Fondo Eu opeo de Desa ollo Regional (FEDER) (MTM2016-
78227-C2-1-P and MTM2015-69323-REDT), Spain.
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