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on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
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Mandala on Wes enholz
m onwes enho@uni-osnab ueck.
de
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Simplicial complexes in ne wo k in usion
p o iling: pa e n cons uc ion h ough
simplicial cen ali ies
Mandala on Wes enholz1,4*, Ma inA zmuelle 2,3,4 and TimRöme 1,4
In oduc ion
Ne wo k in usion de ec ion (see, e.g.,(Mukhe jee e al. 1994; Somme and Paxson
2010)) is a p ominen and impo an esea ch di ec ion due o g owing challenges in
cybe secu i y, e.g., ela ing o he isk o all ha pe sonal da a will be assaul ed (see,
e.g.,(Rosenbe g 2017; Wal e s and No ak 2021)), as well as inc eased cybe secu i y
measu es in gene al (Ti umala e al. 2019). Hence, i is o pa icula in e es o enhance
ou unde s anding o making sense o such si ua ions and da a, e.g.,by iden i ying and
conside ing g oups o special IP add esses, like he ones ha a e a acked as well as he
a acke s hemsel es du ing a se ies o in usion assaul s.
Fo ha , we apply g aph-based me hods. In gene al, g aph-based app oaches ha e
ound a ious applica ions in compu e science, ma hema ics, and neighbou ing disci-
plines (see, e.g.,(Ma isca e al. 2024; Kosk 2024; Tang 2023)). In A zmuelle e al. (2023),
such an idea was used o model ne wo k in usion de ec ion da a, in o de o pe o m
g oup-based analysis o c i ical si ua ions, i.e.,speci ic ne wo k in usion e en s. Mo e
Applied Ne wo k Science
Abs ac
Fo s udying in usion de ec ion da a we conside da a poin s e e ing o indi idual
IP add esses and hei connec ions. We build ne wo ks ep esen ed by g aphs
associa ed wi h hose da a poin s, such ha e ices in a g aph a e cons uc ed
o deno e he espec i e IP add esses, wi h he key p ope y ha a acked da a
poin s a e pa o he s uc u e o he ne wo k. Mo e p ecisely, his pape p oposes
a no el app oach using simplicial complexes o model he desi ed ne wo k and
he espec i e in usions in e ms o simplicial a ibu es, hus gene alizing p e ious
g aph-based app oaches. Applying adap ed ne wo k cen ali y measu es ela ed o
simplicial complexes yields pa e ns associa ed o e ices, which hemsel es con ain
a se o ea u es. These a e used o desc ibe he a acked o he a acke e ices,
espec i ely. Compa ing his new s a egy wi h classical concep s demons a es he
ad an ages o he p esen ed app oach using simplicial ea u es o de ec ing and
cha ac e izing in usions.
Keywo ds Simplicial complex, Ne wo k in usion p o iling, Simplicial pa e ns
Page 2 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
p ecisely, IP add esses co espond o e ices o an associa ed di ec ed g aph modeled
using he ne wo k in usion da a. In his g aph, he e is an edge be ween wo e ices
whene e he e exis s any kind o da a exchange be ween he espec i e IP add esses.
Gi en such a g aph, he au ho s in A zmuelle e al. (2023) aimed o ind pa e ns asso-
cia ed o e ices, which a e hemsel es se s o well-chosen ea u es o he IP add esses
and he espec i e nodes. The goal is o ind common p ope ies (i.e.,se s o ea u es)
o g oups o e ices ha bes cha ac e ize g oups o IP add esses o in e es as good as
possible, such as he ones ela ed o a acke s o a acked da a poin s. Fo his pu pose,
g aph cen ali y measu es a e used o cons uc speci ic ea u es as p ope ies o in e -
es ing subse s o he e ices o he g aph.
I u ns ou ha g aph-based pa e ns desc ibed in A zmuelle e al. (2023) desc ibe
c i ical si ua ions o IP add esses much be e han non-g aph-based pa e ns ela ed
o classical cons uc ions. Howe e , he e is oom o imp o emen , since g aphs only
enable he de ec ion o possible pai wise in e ac ions be ween IP add esses. In con-
as , highe o de s uc u es, like he in e ac ion o h ee o mo e add esses simul ane-
ously a e hen no di ec ly accessible. The e is he need o gene alizing he g aph-based
app oach om abo e in a sui able way, which s ill is simple enough o p ac ical applica-
ion. This need is mo i a ed by he la e obse a ion.
Simplicial complexes a e ul illing his aim, and ha is why we p opose hese o mod-
eling, analysis and in e p e a ion. Recall ha gi en a se o e ices V, such a simplicial
complex is a se o subse s o V, called aces, which is closed unde aking subse s. Geo-
me ically, each o hese subse co esponds o he e ices o a simplex in a gi en eal
ec o space.
In his wo k, we usually use Vie o is–Rips complexes (see, e.g.,in gene al (Edelsb un-
ne and Ha e 2022; Gilbe 1961; Hug and Rei zne 2016; Pen ose 2003) and o speci ic
con ex s, e.g.,(Akinwande and Rei zne 2020; Rei zne e al. 2024; G ygie ek e al. 2020)
whe e
V⊂Rn
and a se o e ices induces a ace i and only i pai wise each wo e -
ices a e close enough wi h espec o a gi en me ic. Gi en such a simplicial complex,
we can associa e pa e ns o ea u es o i s e ices using he simplicial s uc u e o he
complex. Such a simplicial app oach u ns ou o be e ec i e o desc ibing and s udying
he s uc u e o a ne wo k based on aces in a ious dimensions. Fo example, we migh
be in e es ed in selec ing a ace o he highes dimension con aining a gi en e ex as
a ea u e o a pa e n. Then, is connec ed o he o he e ices o his ace ; in his way,
we can hen desc ibe he connec i i y o conce ning he whole ne wo k.
Obse e ha g aphs appea exac ly as 1-dimensional skele ons o simplicial com-
plexes, only con aining he aces o dimensions 0 and 1 – hus only p o iding a 1-dimen-
sional pe spec i e. In con as , ou new app oach gene alizes he me hods in A zmuelle
e al. (2023) using simplices in gi en complexes o highe dimensions. We ocus on cen-
ali y measu es ela ed o simplicial complexes o yield new pa e ns associa ed o e i-
ces, which a e no isible in he wo ld o g aphs.
The es o he pape is s uc u ed as ollows. Sec ion“P elimina ies” p esen s ele an
basics on simplicial complexes. Nex , Sec ion“Adjacencies and deg ees o simplicial
complexes” p o ides sui able no ions o adjacencies and deg ees o aces o simplicial
complexes. No e ha he e a e se e al ways adjacencies can be de ined (see (Se ano
and Gómez 2020)). In his pape , we use he a ian gene alizing g aph adjacencies.
Wi h hese de ini ions, cen ali ies o e ices a e cons uc ed in Sec ion“Simplicial
Page 3 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
cen ali y measu es”, which a e gene aliza ions o g aph-based cen ali y measu es. In
Sec ion“Building pa e ns”, i is ou lined how we can cons uc simplicial pa e ns wi h
ea u es, in pa icula , hose ones elying on simplicial cen ali y measu es. Nex , pu ely
g aph-based pa e ns and simplicial ones a e analyzed. Fo his, quali y unc ions a e
in oduced, which compa e measu es o e ices o in e es (e.g.,a acke s in he ne -
wo k) in ela ion o all e ices, and measu es o e ices o in e es inside he suppo
o pa e ns in ela ion o i s comple e suppo . I u ns ou ha simplicial pa e ns based
in high dimensional s uc u es a e signi ican ly be e o se e al complex ne wo ks
han pu ely g aph-based pa e ns, as s udied in A zmuelle e al. (2023). We conclude
he pape wi h an ou look in Sec ion“Ou look”, p esen ing a discussion wi h espec o
u he wo k and in e es ing esea ch di ec ions how o in es iga e simplicial pa e ns as
ela ed o a ne wo k u he .
P elimina ies
In his sec ion, de ini ions and no a ion ela ed o simplicial complexes a e gi en, which
a e used all o e he manusc ip . Fo mo e de ails, o example, see Munk es (2018),
Bob owski and Kahle (2018), Kahle and Meckes (2013), S anley (2007).
De ini ion 2.1 Le
V={ 1,...,
m}
be a ini e se . A amily
∆
o subse s o V is an
abs ac simplicial complex, i
o any σ∈∆and τ⊆σholds τ∈∆.
Fo
0≤k≤m−1
, an elemen
σ∈∆
wi h
|σ|=k+1
is called an abs ac k-simplex
and k is called he dimension o
σ
.
The highes possible dimension k o an abs ac k-simplex in an abs ac simplicial com-
plex is called he dimension o he complex. Obse e ha e e y abs ac simplicial com-
plex can be ealized as a geome ic simplicial complex (see, e.g.,(Munk es 2018)) in
Rn
wi h a su icien ly la ge n. In he ollowing we a e always conside ing abs ac simplicial
complexes. Fo b e i y, we o en skip he wo d “abs ac ” o he co esponding objec s.
Example 2.2 The (abs ac ) simplicial complex
{∅
,
{
x1
}
,
{
x2
}
,
{
x3
}
,
{
x4
}
,
{
x5
}
,
{
x1,x
2
}
,
{
x1,x
3
}
,
{
x2,x
3
},
{x4,x
5},{x2,x
5},{x2,x
4},{x1,x
2,x
3},{x4,x
5,x
2}}
can be ealized in
R2
. A possible ealiza ion is illus a ed in Fig. 1. Though in e es ing
ques ions in esea ch a e ela ed o such embeddings, hey a e no discussed in his
manusc ip , excep implici ly when illus a ing complexes in pic u es.
Rema k 2.3 Fo his wo k, i is impo an o no e ha simplicial complexes a e e y
good models o desc ibing ne wo ks. Mo e p ecisely, objec s in a ne wo k co espond
o e ices. Such e ices a e connec ed by an edge i he ela ed objec s ul ill a gi en
ela ionship (which is based on he con ex o he ne wo k). A k-simplex wi h
k>1
in he
simplicial complex speci ies whe he a se o a leas
k+1
e ices ul ills some kind o
pai wise ela ionship o in e es .
Page 4 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
The e a e se e al ways o build a simplicial complex on a se o e ices. A key cons uc-
ion in a ious a eas (see, e.g.,( Edelsb unne and Ha e 2022; Gilbe 1961; Hug and
Rei zne 2016; Pen ose 2003)) is gi en in he ollowing de ini ion.
De ini ion 2.4 Le
V={
1
,...,
m}∈Rn
,
∈R>0
, and a gi en me ic d on
Rn
. The
Vie o is–Rips complex
R(V, )
is he simplicial complex, whose k-simplices
σ(
k
)
a e he
se s
{
i
1,...,
i
k+1 }⊆V
wi h
d(
il
,
ij)≤
o all
j, l
∈{1
,...,k
+1}.
The Vie o is–Rips complex is he clique complex (see, e.g.,(Kozlo 2008)) o he unde -
lying g aph G consis ing o i s 0- and 1-dimensional aces. This means ha whene e
he e exis s a
(k+ 1)
-clique in G, he co esponding k-simplex belongs o he simplicial
complex. No e ha he e a e many o he ways o c ea e simplicial complexes, like Cěch
complexes.
The Vie o is–Rips cons uc ion is he only one ha is used in his pape . To cons uc
a Vie o is–Rips complex, i is equi ed o de ine he dis ance be ween wo da a poin s
x1,x
2∈Rn
ia a me ic d. The e a e se e al easonable op ions o ha . Below, he e a e
some sugges ions, whe e he objec s acco ding o Rema k2.3 a e IP add esses and one is
in e es ed in hei in e ac ion in he in e ne .
•Euclidean dis ance
d1(x1,x
2)=∥x2−x1∥2
•O he spa ial dis ances like
d2(x1,x
2)=∥x2−x1∥∞
o
d3(x1,x
2)=∥x2−x1∥1
•
d
4(x1,x
2)=
{1
ϵi he e is a possible da a exchange o size ϵbe ween x1,x
2
2o he wise
•
d
5(x1,x
2)=
{0i he e is a possible da a exchange be ween x
1
and x
2
1
o he wise
Example 2.5 The simplicial complex in Example2.2 is a Vie o is–Rips complex wi h
V={x1,x
2,x
3,x
4,x
5}
,
d=d1
, and
=3
.
I one is in e es ed in ela ionships and in ac i i ies ha a e aking place a a special loca-
ion, e.g.,in a company, i migh be help ul o conside spa ial dis ances, like he Euclid-
ean one. The e a e si ua ions whe e i migh be mo e easonable o use dis ances which
a e mo e closely ela ed o possible da a exchange, like
d4
and
d5
.
Recall ha he cons uc ions o g aphs in A zmuelle e al. (2023) a e based on placing
an (undi ec ed) edge be ween wo e ices and w in a ini e e ex se V co esponding
Fig. 1 Geome ic ealiza ion o a simplicial complex
Page 5 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
o IP-add esses, when hey in e ac (in one o wo di ec ions). These a e exac ly Vie o-
is–Rips g aphs (which a e also called Gilbe g aphs o andom geome ic g aphs Gil-
be (1961)) using he me ic
d5
.
Example 2.6 Le us econside he simplicial complex o Example2.2. Some exempla y
dis ances a e gi en by
d1,...,d
5
om abo e, which a e
d
1(x1,x
2)=
√10
4
,d
2(x1,x
2)=
3
2
,d
3(x1,x
2)=2,d
4(x1,x
2)depends on e, d5(x1,x
2
)=0.
No e ha one can also dis inguish be ween incoming and ou coming da a by add-
ing di ec ions o ou cons uc ions so a . In his wo k, we always conside undi ec ed
si ua ions.
Adjacencies and deg ees o simplicial complexes
No ions like adjacencies and deg ees in g aphs a e well known (see, e.g.,(Dies el 2025)).
Gene alizing his o simplicial complexes and based on he wo k in Se ano and Gómez
(2020), he main goal o his sec ion is o in oduce a ious ypes o adjacencies and
deg ees in such complexes. La e , hese no ions a e hen used, in pa icula , o cons uc
ea u es o e ices, which use simplicial cen ali y measu es.
F om now on a simplicial complex is always deno ed by
∆
and
σ(
q
)
is a simplex in
∆
o
dimension q. Fi s we gi e he ollowing de ini ions om Se ano and Gómez (2020,Sec-
ion 2.1) which will be impo an in he upcoming pa s o his pape .
De ini ion 3.1 Le
σ(q)=∅
and
σ′(
q
′)=∅
be di e en simplices in
∆
and
p∈N
wi h
1≤p≤dim∆
. Then, we de ine he ollowing:
(i)
σ(
q
)
and
σ′(
q
′)
a e called p-uppe adjacen , deno ed by
σ(q)∼Up
σ
′(q′)
i he e exis s
ap−simplex τ(
p
)∈∆,ha ing bo h σ(
q
)and σ′(
q
′)as aces.
(ii)
σ(
q
)
and
σ′(
q
′)
a e called s ic ly p-uppe adjacen , deno ed by
σ(
q
)∼
U∗
p
σ
′(
q
′),
i
σ(
q
)∼Up
σ′
(
q
′)and
σ
(
q
)∼Up+1
σ′
(
q
′).
(iii) The p-uppe deg ee o
σ(q)
is
degp
U(
σ
(
q
))=|{
σ′′
(
q
′′ )∈∆|
σ
(
q
)∼
U
p
σ′′
(
q
′′ )}|.
(i ) The (h,p)-uppe deg ee o
σ(q)
1
is
deg(
h,p
)
U(
σ
(
q
))=|{
σ′′
(
q
+
h
)∈∆|
σ′′
(
q
+
h
)∼
U
p
σ
(
q
)}|.
( ) The s ic (h,p)-uppe deg ee o
σ(q)
is
deg(
h,p
)∗
U(
σ
(
q
))=|{
σ′′
(
q
+
h
)∈∆|
σ′′
(
q
+
h
)∼
U
p∗
σ
(
q
)}|.

Page 6 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
( i) The maximal simplicial uppe deg ee o
σ(q)
is
deg
∗
U(σ(q))=
dim ∆
−q
∑
h=1
deg(h,(q+h))∗
U(σ(q))
.
No ice ha he maximal simplicial uppe deg ee
deg∗
U(
σ
(q))
coun s he numbe o max-
imal simplices, which ha e dimension s ic ly highe han q, ha con ain he simplex
σ(
q
)
.
Example 3.2 (i) Two (abs ac ) simplices, which a e aces o a solid e ahed on a e
3-uppe adjacen .
(ii) Two e ices which sha e a common edge a e 1-uppe adjacen ; hey a e e en s ic
1-uppe adjacen , i hey a e no lying in a common iangle.
The e a e also o he a ia ions o he concep o adjacency and deg ee, which a e also
in oduced in Se ano and Gómez (2020,Sec ion 2.1), ha a e lis ed below.
De ini ion 3.3 Le
σ(q)=∅
and
σ′(
q
′)=∅
be di e en simplices in
∆
and
p∈N
wi h
0≤p≤dim ∆−1
. Then, we de ine he ollowing:
(i)
σ(q)
and
σ′(
q
′)
a e called p-lowe adjacen , i
|
σ
(
q
)∩
σ
′(
q
′)|≥
p
+1.
This is deno ed by
σ(q)∼Lp
σ
′(q′).
(ii)
σ(q)
and
σ′(q′)
a e called s ic ly p-lowe adjacen , i
|σ(q)∩σ′(q′)|=p+1.
This is
deno ed by
σ(
q
)∼
L
p∗
σ′
(
q
′).
(iii) The p-lowe deg ee o
σ(q)
is
degp
L(
σ
(
q
))=|{
σ′′
(
q
′′ )∈∆|
σ
(
q
)∼
L
p
σ′′
(
q
′′ )}|.
(i )
σ(q)
and
σ′(q′)
a e called p-adjacen , i hey a e s ic ly p-lowe adjacen and no
p′
-uppe adjacen o
p′=q+q′−p
. This is deno ed by
σ(
q
)∼Ap
σ′
(
q
′).
( ) The p-adjaceny deg ee o
σ(q)
is
degp
A(
σ
(
q
))=|{
σ
′′(
q
′′ )∈∆|
σ
(
q
)∼Ap
σ
′′(
q
′′ )}|.
( i)
σ(q)
and
σ′(
q
′)
a e called maximal p-adjacen , deno ed by
σ(
q
)∼
A
∗
p
σ
′(
q
′)
i and only i
σ(
q
)∼
A
p
σ
′(
q
′)and
σ
′(
q
′)⊈
σ
′′(
q
′′ )whene e
σ
(
q
)∼
A
p
σ
′′(
q
′′ ).
( ii) The maximal p-adjaceny deg ee o
σ(q)
is
Page 7 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
degp∗
A(
σ
(
q
))=|{
σ′′
(
q
′′ )∈∆|
σ
(
q
)∼
A∗
p
σ′′
(
q
′′ )}|.
( iii) The maximal simplicial deg ee o
σ(q)
is
deg∗(
σ
(q)) = deg∗
A(
σ
(q)) + deg∗
U(
σ
(q)),
whe e
deg
∗
A(σ(q))=
q
−1
∑
p=0
degp∗
A(σ(q))
.
The ollowing examples a e some illus a ions o he de ini ions in oduced so a .
Example 3.4 Conside he simplicial complex om Example2.2. The simplices
x1,x
2,x
3
and
x2,x
4,x
5
exhibi he p ope ies ou lined below:
(i)
x2
is a e ex o bo h simplices. Hence, hey a e 0-lowe adjacen . They a e s ic ly
0-lowe adjacen as well, as hey sha e no simplex o highe dimension.
(ii) Bo h simplices a e no included in a common simplex o highe dimension, so hey a e
no p-uppe adjacen o any p.
(iii) They a e 0-adjacen , since hey a e s ic ly 0-lowe adjacen and hey a e no 4-uppe
adjacen .
(i ) The 0-adjacency deg ee o
{x1,x
2,x
3}
is 3, since besides
{x2,x
4,x
5}
, also
{x2,x
4}
and
{x2,x
5}
a e 0-adjacen o
{x1,x
2,x
3}
.
In he upcoming sec ions, we will ocus on hose si ua ions, whe e one simplex is
always a e ex, al hough De ini ion3.3 is o cou se also usable o simplices o highe
dimensions.
In his special case we obse e o g aphs, ha wo e ices a e called adjacen , when
hey bo h sha e a common edge. This g aph-adjacency is co e ed by he de ini ions o
adjacencies in simplicial complexes abo e as well. Mo e p ecisely, we ha e he ollowing
esul .
Lemma 3.5 Le
G=(
V,E
)
be a g aph and
∈V
be a e ex. Then, he ollowing p ope -
ies hold:
(i) Le
w∈V
be ano he e ex in G. Then, w and a e g aph-adjacen i and only i hey
a e 1-uppe adjacen .
(ii) I
=w∈V
, hen ,w a e no p-lowe adjacen and hey a e no p-adjacen o any
0≤p≤dim ∆ −1
.
P oo (i) I and w a e g aph-adjacen , hen i ollows immedia ely om he
de ini ions, ha hey a e 1-uppe adjacen . On he o he hand, i and w a e 1-uppe
adjacen , hen hey a e bo h a ace o a 1-simplex (i.e.,an edge) o G. (ii) The e ices
and w ha e no non- i ial ace in common and a e no p-lowe adjacen o any
possible p. This implies ha hey a e also no p-adjacen o such a p.
□
Lemma 3.6 Le
∈∆
be a e ex. Then
Page 8 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
(i)
degp
L( ) = degp
A( )=0
o all
p>0
,
(ii)
degp
L(
)=0=deg
p
A(
)
o
p=0
,
(iii)
deg∗(
) = deg∗
U(
).
P oo (i) Since a e ex has no o he non- i ial simplices as subse s, he conclusion
ollows.
(ii) By de ini ion is no 0-lowe adjacen o i sel . Hence,
degp
L( )=0=deg
p
A( )=0
.
(iii) Since
degp
A(
)=0
, we ha e
degp∗
A(
σ
(
q
))=0
.
This concludes he p oo .
□
Below, we a e mainly in e es ed in simplices in which a gi en e ex is included. In
his con ex , only p-uppe adjacencies and ela ed deg ees a e o in e es . The uppe -
adjacency is also easonable o use, as i is a gene aliza ion o g aph-adjacencies due o
Lemma3.5. Hence, in he ollowing we only use p-uppe adjacencies.
Simplicial cen ali y measu es
In his sec ion, we in oduce cen ali y measu es o simplicial complexes which a e
mo i a ed by he same concep ela ed o g aphs. In he ollowing we ocus on he
deg ee and closeness cen ali ies. These gene aliza ions o well-known g aph deg ee and
closeness cen ali ies (see, e.g.,(Bo ga i and E e e 2006; Newman 2018)) ha e all been
p oposed in Se ano and Gómez (2020). Wi h hese cen ali y measu es we c ea e ea-
u es o e ices o a complex as announced abo e in he in oduc ion in Sec ion“In o-
duc ion”. To be mo e o mal, a cen ali y measu e is a unc ion
c:I→R,
whe e I is he se , which is usually called he se o indi iduals. I I is a simplicial com-
plex, hen a mo e impo an simplex
σ∈I
, i.e.,one wi h a signi ican ly high cen ali y
c(
σ
)
, con ains way mo e aluable in o ma ion ega ding he conside ed ne wo k han
simplices wi h smalle cen ali ies. The e a e many possibili ies o de ine such a cen al-
i y measu e. He eby, he no a ion
cσ
is an abb e ia ion o he unc ion alue
c(σ)
.
Mo i a ed by de ini ions and sugges ions in Se ano and Gómez (2020), we in oduce
he ollowing cen ali y measu es o simplicial complexes. Obse e ha in ha wo k,
he au ho s p opose cen ali y measu es wi h images in [0,1]. Bu , o be consis en wi h
he conside ed cen ali y measu es in A zmuelle e al. (2023), such no maliza ions a e
neglec ed in his manusc ip . Then, cen ali y measu es in A zmuelle e al. (2023) a e
special cases o he simplicial e sions gi en below. Fo simplici y, he ollowing concep s
a e only de ined o e ices o simplicial complexes, since, in his pape in conside ed
applica ions indi iduals a e always e ices. As al eady men ioned, in he end o Sec-
ion“Adjacencies and deg ees o simplicial complexes”, only uppe deg ees a e consid-
e ed. The i s cen ali y measu e o impo ance is he one om he ollowing de ini ion.
De ini ion 4.1 Le
σ∈∆
be a e ex and
p∈N>0
. Then
cDp
σ= degp
U(σ)
is he p-deg ee cen ali y o
σ
and
Page 9 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
cD∗
σ= deg∗(σ)
is he maximal simplicial deg ee cen ali y o
σ
.
See (Se ano and Gómez 2020, De . 12, 13) o a ela ed de ini ion. An impo an p op-
e y o his cen ali y measu e is discussed in he ollowing heo em.
Theo em 4.2 The p-deg ee cen ali y o a e ex
σ∈∆
ul ills
c
Dp
σ≤
p
+1
∑
j=1
(
deg(0,1)
U(σ)+1
j
)
−1
.
(1)
P oo Obse e ha
(
j
−1)
-dimensional aces in
∆
wi h
2≤j≤p+1
a e p-uppe adja-
cen o
σ
, i hey lie in a common p-dimensional simplex wi h
σ
. Gi en such a ace, all
i s e ices no equal o
σ
ha e he p ope y ha hey induce an edge in
∆
oge he wi h
σ
, i.e., hey a e (0,1)-adjacen o
σ
. No e ha
σ
migh be a e ex o such a ace o no .
Hence, i su ices o coun all possible
(j−1)
- aces induced by e ices which a e ei he
σ
o (0,1)-adjacen o
σ
, accep ing an o e -coun o such si ua ions. Finally, he e a e a
mos
deg(0,1)
U(σ)
many 0-dimensional aces in
∆
which a e p-uppe adjacen o
σ
. This
concludes he p oo .
□
The e is some signi ican compu a ional e o equi ed o p o ide an uppe bound o
he p-deg ee cen ali y o
σ
h ough Theo em4.2. Nex , we de ine a u he deg ee cen-
ali y which allows al e na i e uppe bounds.
De ini ion 4.3 Le
σ∈∆
be a e ex and
p∈N>0
. Then
c
D
(p,p)
σ= deg(
p,p
)
U(
σ
)
is he (p,p)-deg ee cen ali y o
σ
and
c
D
(
p,p
)∗
σ= deg(
p,p
)∗
U(
σ
)
is he (p,p)-s ic deg ee cen ali y o
σ
.
See (Se ano and Gómez 2020, De . 10, 11) o a ela ed de ini ion. No e ha he (p,p)-
deg ee cen ali y o
σ
is he numbe o p-dimensional aces o
∆
ha con ain
σ
, and he
(p,p)-s ic deg ee cen ali y is he co esponding s ic e sion. Since
σ
is always a sim-
plex o dimension 0, we ha e
c
D∗
σ=
dim ∆
∑
p
=1
deg(p,p)∗
U(σ)
.
(2)
Rema k 4.4 A new uppe bound o
c
D
p
σ
is
c
Dp
σ≤deg(
p,p
)
U(σ)·(2
p
+1 −2) = c
D
(
p,p
)
σ·(2
p
+1 −2).
(3)
This bound ollows om he ac ha e e y non- i ial ace, excep o
σ
, o a p-simplex,
con aining
σ
, is p-uppe adjacen o
σ
.
The uppe bound in (3) is o en be e han he bound in Theo em4.2. Fo example,
conside a simplicial complex
∆
which has exac ly one e ahed on ha con ains
σ
, and
Page 16 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
is called a k-pa e n wi h espec o F. We deno e he se o k-pa e ns by
F(k)
. Then, we
say ha
p∈F(k)
is ue wi h espec o
i∈I
, i
p(
i
)={
1
b1(
i
)
,...,
k
bk(
i
)}={1}.
The suppo o p is
sp={i∈I|p(i)={1}} ⊆ I,
which is also called he p- ul illing indi iduals. Se
ip:= |sp|
.
Rema k 5.3 (i) Le
∆
be an a ibu ed simplicial complex and p be a compa ible pa e n.
This pa e n can also be conside ed as ue o alse wi h espec o a e ex
σ∈∆
by
aking he alue p(i), whe e i is he i s componen o
D(σ)
.
(ii) In he li e a u e, e.g.,in A zmuelle e al. (2023), he e exis a ious ways o de ining
pa e ns. Fo example, one could equip he pa e ns wi h mo e s uc u e by using
uples ins ead o se s. This p o ides he oppo uni y o conside mul iplici ies and
o o de he ea u es. In his pape , pa e ns a e always se s, which is sui able o he
conside ed applica ions below.
Fo bina y unc ions as al eady conside ed abo e, i is use ul o in oduce he ollowing
no a ion.
De ini ion 5.4 Le p be a pa e n. A a ge is a bina y unc ion
:I→{0,1}.
The a ge
sha e o wi h espec o p is
p:=
|{i∈s
p
| (i)=1}|
i
p
=
|{i∈I|p(i)=1and (i)=1}|
i
p
.
Fo a gi en a ge , we sea ch o pa e ns wi h a high a ge sha e. To ind a use ul mea-
su emen o he pa e n quali y and, hus, o i s in e es ingness, we conside a quali y
unc ion (see (A zmuelle 2015; A zmuelle e al. 2023; G ossk eu z e al. 2008) o u -
he de ails).
De ini ion 5.5 Gi en a a ge , a k-quali y unc ion is a eal- alued unc ion
q :F(k)→R.
The quali y o a k-pa e n p wi h espec o is gi en by
q (p)
.
Le
0
=|{
i
∈
I
|
(
i
)=1}|
|I|
be he sha e o he indi iduals ha ul ill a a ge wi h espec o
all indi iduals and choose
a∈R
such ha
(ip)a
is well de ined. Fo example, he qual-
i y o a k-pa e n p wi h espec o can now be de e mined by he ollowing k-quali y
unc ion.
qa
(p)=(ip)a·( p− 0).
(6)
He e, we a ge si ua ions, o example, whe e
0
is a he small in compa ison o
p
and
p
i sel should be much la ge . The size o he pa e n in e ms o con ained ins ances is
hen weigh ed by pa ame e a. Based on his app oach, se e al well-es ablished quali y
unc ions can be ound in he li e a u e A zmuelle (2015), as shown below:
•The gain quali y unc ion
q0
,

Page 17 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
•The binomial es quali y unc ion
q0.5
,
•The Pia e sky–Shapi o quali y unc ion
q1
.
Then, he goal is o ind pa e ns wi h a high quali y alue, i.e.,a high
qa
(p)
o a pa e n
p, wi h a gi en a and he a ge as he concep o in e es .
In he emaining pa o his sec ion, s a egies a e discussed o build and hen o e al-
ua e simplicial-based pa e ns. Fo a sample da a se , we in es iga e he p oblem whe he
simplicial ea u es help o inc ease he quali y o co esponding pa e ns in compa ison
o g aph-based ea u es. Fo his pu pose, we ocus on pa e ns in he con ex o spe-
ci ically cons uc ed syn he ic da a, which speci ically ea u es analysis op ions o ou
e alua ion s a egies.
Fo gene a ing syn he ic da a in ou applica ion con ex , we ely on s anda d
app oaches om he ield o complex ne wo ks, applying he suscep ible in ec ious
(SI) model, e.g.,(C epey e al. 2006; Li 2018), o gene a ing he non-a acke /a acke
s uc u es.
Thus, a i s , we c ea e syn he ic da a by building a simplicial complex, whose e i-
ces a e assigned as a acke s o non-a acke s (which a e pa ially a acked). Using he
ob ained syn he ic da ase , we c ea e ea u es using he a ailable me ics on simplicial
complexes and ne wo ks, espec i ely. Using hese ea u es, we can hen cons uc pa -
e ns wi h espec o he a ge s a acke /non-a acke , o s udying he impac o me -
ics on simplicial complexes.
Rema k 5.6 The s a egy o cons uc ing he syn he ic da a is gi en h ough he ollow-
ing s eps.
(i) Choose an exis ing and eal wo ld social ne wo k which has no oo many indi iduals.
Then, hese co espond o e ices, which a e connec ed by an edge i he e is
in e ac ion on he le el o indi iduals.
(ii) Choose a numbe
k∈N
and selec k andom e ices o be a acke s.
(iii) C ea e s ime pe iods o a acks on he gi en social ne wo k. Indi iduals who ha e
been a acked may mu a e in o new a acke s.
(i ) De e mine a esul ing ne wo k om (iii) o u he in es iga ions.
( ) Build he Vie o is–Rips complex o he unde lying ne wo k om (i ) using he me ic
d5
om Sec ion2.
In he ollowing we apply his algo i hm on one speci ic social ne wo k o illus a e ou
modeling and analysis app oach using simplicial complexes. In pa icula , o s ep (i),
he e we use he cong ess ne wo k om Fink (2023), which is educed o he i s 20 indi-
iduals. Thus, all o he e ices (wi h numbe s 21–475), as well as all edges in ol ing a
leas one o such e ices a e dele ed. The esul ing ne wo k is illus a ed in Fig.4.
Fo s ep (ii), ou da a poin s a e andomly chosen wi h espec o he disc e e uni-
o m dis ibu ion. Then, we pe o m s eps (iii) and (i ) o
s=1
and use he SI model
app oach desc ibed below. Finally, he Vie o is–Rips dis ance is chosen as
=1/2
o
s ep ( ); see De ini ion2.4.
Fo (iii) i emains o discuss sho ly he SI model o modelling disease in ec ions a a
ime , which is well-known in he li e a u e (see, e.g.,(Li 2018)). Le
Page 18 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
Sdeno e he numbe o suscep ible indi iduals (non-a acke s), and
V
he numbe o i us in ec ious indi iduals (a acke s),
whe e
S= ( )and V=g( )
o unc ions and g wi h alues in
N
a a ime
≥0
.
The e a e he ollowing necessa y assump ions o he SI model:
(i) No bi hs and dea hs happen, i.e., he numbe o o al indi iduals N is cons an a any
ime . This means
( )+g( )=N o all ≥0.
(7)
(ii) As sugges ed in Li (2018, Equa ion 1.21), we choose he in ec ion ep oducing a e as
=P·λ/N,
whe e P is he p obabili y o a con ac o p oduce an in ec ion and
λ
is he a e age
con ac numbe which is also gi en by he a e age (1,1)-deg ee o he e ices in he
co esponding ne wo k.
The change o he suscep ibles is modeled by a mul iple o he p oduc o he numbe o
he wo g oups (a acke s and non-a acke s), since his model assumes ha he a e o
change is p opo ional o he numbe o suscep ibles and he numbe o in ec ious indi-
iduals. Mo eo e ,
−dS
d =dV
d
due o Eq. (7).
Hence, he change o in ec ious and suscep ible indi iduals is desc ibed by
dS
d
=
−
cV
·
Sand
dV
d
=cV
·S
Fig. 4 Modi ied cong ess ne wo k
Page 19 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
o a cons an
c∈R
. Thus, one has
dV
d
=cNV
·
(1
−
V /N )
.
Assuming ha
V≪N
i ollows
V
N∼0
. Hence,
dV
d ≈
cN
·
V= V
wi h he change a e
=cN
. Since he esul ing di e en ial equa ion
dV
d
= V (1
−V
N)
(8)
is sepa able, one ob ains he in eg al equa ion
∫1
V(1
−
V
N
)dV =
∫ d ,
which can be sol ed by using pa ial ac ion decomposi ion.
∫
d =
∫
A
V+B
1−
V
N
dV =
∫
A
(1
−
V
N
)
V
(1 −
V
N)
+BV
V
(1 −
V
N)
dV
leads o he solu ion
A=1
and
B=1/N
. So, we ob ain
∫
d =
∫1
N(1
−
V
N
)dV +
∫1
V
dV.
Obse e ha by calcula ing he in eg als we ha e
+c1= ln(V)
−
ln(N
−
V) = ln
V
N−V.
Finally, one ecei es
e
ec1=
V
N−V.
This equa ion can be ans o med in o
V=
e
c
2
N
−
e
c
2
V
by de ining
c2:= ec1
. Then, V can be ew i en as
V
=e
c2N
1+e c
2
=e
c2N
e
c2(1+e−
c
−1
2)
=N
1+e
−
c (9)
by de ining ano he cons an
c:=
c
−1
2
. This cons an c can be compu ed by inse ing
=0
in o he p e ious equa ion. The equa ion
g
(0) =
N
1+e−
·
0
c
leads o
Page 20 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
c
=
N
g(0) −
1
.
Equa ion (9) implies ha
g
( )=
N
1+( N
g(0) −
1)e−
.
(10)
An analogous conside a ion o S shows ha
( )=
N
1+( N
g
(0) −
1)e
.
(11)
In he limi one has
lim
→∞
g( ) = lim
→∞
N
1+( N
g(0) −
1)
·
e− =Nand lim
→∞
( ) = lim
→∞
N
1+( N
g(0) −
1)
·
e
=0,
since
e−
→∞
→0
. This means ha in he limi e e yone is in ec ed.
Applying he SI model and, in pa icula , Eqs. (10) and (11) we employ he ollowing
algo i hm o s ep (iii) in Rema k5.6 using he modi ied cong ess da a se .
Algo i hm 5.7 (i) Choose andomly a s a popula ion o
k=
g
(0) = 4
a acke s.
(ii) Compu e he numbe o i us in ec ious indi iduals g(1) whe e he p obabili y o a
con ac o p oduce an in ec ion is chosen as
P=0.2
.
(iii) F om he se o suscep ible indi iduals, which a e connec ed o a leas one
in ec ious indi idual, choose andomly
g(1) −g(0)
many indi iduals wi h espec o
he disc e e uni o m dis ibu ion.
(i ) The se o i us in ec ious indi iduals a ime 1 is hen gi en by he in ec ious
indi idual a ime 0 oge he wi h he new ones.
A e applying Algo i hm5.7 on he modi ied cong ess ne wo k da a se , one ob ains
he a acke da a se which is shown in Fig.5 below ( he blue e ices co espond o he
a acke s).
Wi h he gi en syn he ic da a, one is able o build simplicial pa e ns using measu es
om Sec ion4. To keep he discussion simple, we ocus on pa e ns o leng h 1, which
means ha we only concen a e on one ea u e in each pa e n.
Mo e p ecisely, he ollowing ea u es a e discussed, whe e

is he indica o - unc ion
and
k,l ∈R
.
(i)

(
cD
(p,p)
σ
<k
)and

(
cD
(p,p)
σ
>l
),
(ii) 

(
c
Ep
σ
<k
)and

(
c
Ep
σ
>l
),
(iii)

(cCp
σ<k)and

(cCp
σ>l).
We compa e he chosen ea u e pa e ns o leng h 1 o
p=1
, which co esponds o
he g aph-based case ha is al eady discussed in A zmuelle e al. (2023), and o la ge
p wi h each o he by compu ing hei quali y
q0
i
wi h espec o he a ge o inding
Page 21 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
a acke s and non-a acke s. This means ha a ge s
1
and
2
a e conside ed, whe e
1
maps an indi idual o 1 i i is a non-a acke and o he wise o 0, and
2
maps an indi-
idual o 1 i i is an a acke and o he wise o 0.
Fo (i) he quali y alues o he ea u es
p1=(c
D
(1
,
1)
σ<k
1),p
2=(c
D
(2
,
2)
σ<k
2)and p3=(c
D
(3
,
3)
σ<k
3)
a e analyzed wi h espec o
1
. Mo eo e , he quali y alues o he ea u es
p3=(c
D
(1
,
1)
σ>l
1),p
4=(c
D
(2
,
2)
σ>l
2)and p5=(c
D
(3
,
3)
σ>l
3)
a e analyzed wi h espec o
2
.
One could also cons uc leng h-one pa e ns using ea u es
c
D
(p,p)
σ
o
p≥4
, bu in
his example i su ices o conside
p<4
. As illus a ed in Fig.5 no simplices exis in he
ne wo k wi h dimensions highe han 3 (see Table1).
He e we choose
k1=4,k
2=2and k3=1
as well as
l1=5,l
2= 14 and l3=7.
These choices u n ou o deli e he bes quali y alues
q0
1
( esp.
q0
2
) o
c
D
(i,i)
σ>l
i
( esp.
c
D
(i,i)
σ<k
i
) compa ed o o he possible numbe s.
The ea u es as well as he p ope y o being an a acke o a non-a acke a e shown in
Table1.
Mo e p ecisely, he conside ed pa e ns ha e he quali y alues which a e de e mined
ia
Fig. 5 Modi ied cong ess ne wo k wi h a acke s a e one ime s ep o SI

Page 22 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
Table 1 (p, p)-deg ee cen ali ies
Ve ex
c
D
(1,1)
c
D
(2,2)
c
D
(3,3)
c
D
(4,4)
A acke ? Non-a acke ?
0 6 5 1 0 0 1
1 3 2 0 0 0 1
2 3 1 0 0 0 1
3 5 6 2 0 0 1
4 6 5 0 0 1 0
5 2 1 0 0 0 1
6 1 0 0 0 0 1
7 5 7 3 0 0 1
8 8 12 6 0 0 1
9 6 3 0 0 1 0
10 2 0 0 0 0 1
11 8 9 3 0 0 1
12 6 8 2 0 1 0
13 10 14 6 0 0 1
14 4 2 0 0 1 0
15 6 6 2 0 1 0
16 4 4 1 0 0 1
17 12 21 8 0 1 0
18 5 3 0 0 1 0
19 4 5 2 0 0 1
Ve ex
p1
p2
p3
p4
p5
p6
0 0 0 0 1 0 0
1 1 0 1 0 0 0
2 1 1 1 0 0 0
3 0 0 0 0 0 0
4 0 0 1 1 0 0
5 1 1 1 0 0 0
6 1 1 1 0 0 0
7 0 0 0 0 0 0
8 0 0 0 1 0 0
9 0 0 1 1 0 0
10 1 1 1 0 0 0
11 0 0 0 1 0 0
12 0 0 0 1 0 0
13 0 0 0 1 0 0
14 0 0 1 0 0 0
15 0 0 0 1 0 0
16 0 0 0 0 0 0
17 0 0 0 1 1 1
18 0 0 1 0 0 0
19 0 0 0 0 0 0
Page 23 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
q0
1
(
p1
)=
i
0
p·
(
p− 0
)=5
0
·
(5
/
5
−
13
/
20) = 7
/
20,
q
0
1(p2)=4
0·(4/4−13/20) = 7/20,
q
0
1(p3)=9
0·(5/9−13/20) = 17/180,
q
0
2(p4)=9
0·(5/9−7/20) = 37/180,
q
0
2(p5)=1
0·(1 −7/20) = 13/20,
q
0
2
(p6)=1
0
·
(1
−
7/20) = 13/20.
Thus, he (1,1)-deg ee cen ali y is as good as he (2,2)-deg ee cen ali y, and hey ha e
bo h a highe quali y han he (3,3)-deg ee cen ali y, wi h espec o he pa e n
1
and
he chosen quali y unc ion. The (2,2)-deg ee cen ali y is as good as he (3,3)-deg ee
cen ali y and hey ha e bo h a highe quali y han he (1,1)-deg ee cen ali y wi h
espec o he pa e n
2
and
q0
2
. Hence,
p=2
is he bes choice o conside ing a simpli-
cial deg ee cen ali y ea u e pa e n o only leng h 1, gi en he desc ibed se up.
The ea u es in (ii) a e conside ed in Table 2. He e we analyze he ollowing pa e ns
wi h espec o
1
.
p1=(
cE
1
σ
<k
1)
,p
2=(
cE
2
σ
<k
2)and
p
3=(
cE
3
σ
<k
3)
wi h
k1=0.5,k
2=0.2and k3=2.
Mo eo e , we in es iga e he ollowing pa e ns wi h espec o
2
.
p4=(c
E
1
σ>l
1),p
5=(c
E
2
σ>l
2)and p6=(c
E
3
σ>l
3)
wi h
l1=2.5,l
2=2.5and l3=2.5.
Table 2 p-eigen ec o cen ali ies
Ve ex
c(
E
1)
c(
E
2)
c(
E
3)
c(
E
4)
A acke ? Non-a acke ?
p1
p2
p3
p4
p5
p6
0 1.421 1.279 0.913 0.0 0 1 0 0 1 0 1 0
1 0.962 0.973 0.0 0.0 0 1 0 0 1 0 0 0
2 0.663 0.122 0.0 0.0 0 1 0 1 1 0 0 0
3 1.384 1.430 1.338 0.0 0 1 0 0 1 0 0 0
4 1.442 1.455 0.0 0.0 1 0 0 0 1 0 0 0
5 0.648 0.661 0.0 0.0 0 1 0 0 1 0 0 0
6 0.298 0.0 0.0 0.0 0 1 1 1 1 0 0 0
7 1.564 1.617 1.760 0.0 0 1 0 0 1 0 0 0
8 1.935 1.962 2.134 0.0 0 1 0 0 0 0 0 0
9 0.908 0.384 0.0 0.0 1 0 0 0 1 0 0 0
10 0.421 0.0 0.0 0.0 0 1 1 1 1 0 0 0
11 1.828 1.803 1.564 0.0 0 1 0 0 1 0 0 0
12 1.643 1.640 1.299 0.0 1 0 0 0 1 0 0 0
13 2.380 2.336 2.293 0.0 0 1 0 0 0 0 0 0
14 0.772 0.617 0.0 0.0 1 0 0 0 1 0 0 0
15 1.601 1.514 1.475 0.0 1 0 0 0 1 0 0 0
16 1.321 1.362 1.238 0.0 0 1 0 0 1 0 0 0
17 2.816 2.772 2.565 0.0 1 0 0 0 0 1 1 1
18 1.014 0.384 0.0 0.0 1 0 0 0 1 0 0 0
19 1.187 1.244 1.381 0.0 0 1 0 0 1 0 0 0
Page 24 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
He e he chosen
k1,k
2,k
3
and
l1,l
2,l
3
deli e he bes quali y alues wi h espec o
q0
1
and
q0
2
. These numbe s a e no unique wi h his p ope y. e.g., o
l3
one can also choose
2.4.
I is no help ul o conside eigen ec o cen ali ies o
p≥4
o he modi ied con-
g ess ne wo k, since hen he eigen ec o cen ali ies a e 0 o all e ices (see Table2 o
examples). Thus, i is no possible o use such
p≥4
o c ea e pa e ns o dis inguishing
a acke s and non-a acke s.
A compu a ion yields
q0
1
(
p1
)=2
0
·
(2
/
2
−
13
/
20) = 7
/
20
,
q
0
1(p2)=3
0·(3/3−13/20) = 7/20,
q
0
1(p3) = 170·(12/17 −13/20) = 19/340
,
q
0
2(p4)=1
0·(1/1−7/20) = 13/20,
q
0
2(p5)=1
0·(1/1−7/20) = 13/20,
q
0
2
(p
6
)=1
0
·
(1/1
−
7/20) = 13/20.
Thus, he 1-eigen ec o cen ali y is as good as he 2-eigen ec o cen ali y, and hey
ha e bo h a highe quali y han he 3-eigen ec o cen ali y wi h espec o he pa -
e n
1
and he chosen quali y unc ion. The 1-eigen ec o cen ali y is as use ul as he
2-eigen ec o cen ali y and he 3-eigen ec o cen ali y wi h espec o he pa e n
2
.
So, in o al, he eigen ec o -cen ali ies o
p=1
and
p=2
a e he bes choices ega d-
ing he simplicial-eigen ec o - ea u e-pa e ns o leng h 1.
Fo (iii) only ea u es o
c
C
1
σ
a e conside ed, since he closeness cen ali y measu e is
always 0 o o he p, see Table 3. The bes quali y o cha ac e izing non-a acke s ( a ge
1
) is achie ed wi h he pa e n
p1=(c
C
1
σ<0.024).
Table 3 p-closeness cen ali ies
Ve ex A acke ? Non-a acke ?
c
C
1
p1
p2
0 0 1 0.029 1 0
1 1 0 0.026 0 0
2 1 0 0.025 0 0
3 0 1 0.028 1 0
4 0 1 0.028 1 0
5 1 0 0.023 0 1
6 1 0 0.02 0 1
7 1 0 0.029 1 0
8 1 0 0.031 1 0
9 1 0 0.027 1 0
10 1 0 0.023 0 1
11 1 0 0.032 1 0
12 0 1 0.030 1 0
13 1 0 0.036 1 0
14 1 0 0.024 0 0
15 0 1 0.031 1 0
16 1 0 0.028 1 0
17 1 0 0.039 1 0
18 0 1 0.029 1 0
19 1 0 0.025 1 0
Page 25 o 27 on Wes enholz e al. Applied Ne wo k Science (2025) 10:35
Mo eo e , he ea u e
p2=(c
C
1
σ>0.027)
has he highes quali y wi h espec o a ge
2
, i.e., o cha ac e izing a acke s. As in
(ii) he chosen numbe s a e no unique wi h espec o his p ope y.
Mo e p ecisely,
p1
and
p2
ha e he ollowing quali y alues.
q0
1
(
p1
)=
ip·
(
p− 0
)=3
0
·
(3
/
3
−
13
/
20) = 7
/
20
,
q
0
2
(p
2
)=i
p·
(
p−
0
) = 130
·
(6/13
−
7/20) = 29/260
.
The pa e n
p1
has a a he high quali y and is hus use ul o u he in es iga ions. The
cons uc ed ea u e pa e n wi h espec o he a acke a ge
2
is no ha bene icial,
bu he e may be addi ional o he possibili ies o using his ea u e (which a e no con-
side ed he e in his manusc ip ), like combining i wi h o he ea u es in longe pa e ns.
Obse e ha , in con as o (i) and (ii), in (iii) o he modi ied cong ess ne wo k, i was
no possible o cons uc help ul simplicial ea u es o complexes o highe dimensions
han 1, bu only g aph-based ea u es using he closeness cen ali y. In consequence, i
would be easonable o conside he e in he u u e o he ne wo ks, whe e he e migh be
a dense s uc u e o highe dimensional simplices o ind e ices ha ha e non-ze o
p-closeness cen ali y o
p>1
.
Rema k 5.8 To a oid he syn he ic da a used in his sec ion one has o eco d non- i -
ial eal in usion da a, which con ains no only he a acks, bu also he in e ac ion in-
be ween g oups o a acke s and o a acked IP-add esses. This is le as an u u e esea ch
p oblem.
Ou look
In his wo k we applied simplicial complexes o de ec ing highe -o de pa e ns as sim-
plicial-based pa e ns, in he con ex o ne wo k in usion de ec ion se ings. In Sec ion
5 we conside ed a syn he ic da a se and assessed how good he conside ed simplicial
ea u e pa e ns wo ked o i .
In he ollowing we ou line pe spec i es on u he eligible s a egies o e alua e he
usage o simplicial-based pa e ns o desc ibing a ge s, e.g., o di e en ia e a acke
and non-a acke s in ne wo ks based on eal da a. The e a e se e al op ions o con inue
he in es iga ions om ou wo k, including he ollowing.
(i) Using a ious ( u he ) da ase s o analyze he u ili y o simplicial ea u e pa e ns. Fo
his i is easonable o a y syn he ic o eal la ge o small da a as well as da a om
di e en applica ion domains.
(ii) Cons uc ing pa e ns ha use mo e simplicial complex ea u es a once. I is easonable
o conside pa e ns ha ha e leng h
>1
o s udy u he cen ali y measu es which
we e no ye men ioned in his manusc ip .
(iii) Equipping simplicial pa e ns wi h u he ea u es which a e no o igina ing om
simplicial cen ali y measu es. I is also possible o cons uc u he ea u es om
one gi en cen ali y measu e by using new condi ions. In he ollowing we p o ide
some examples o he la e sugges ion.