Simplicial complexes in network intrusion profiling: pattern construction through simplicial centralities

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Applied Ne wo k Science (2025) 10:35 h ps://doi.o g/10.1007/s41109-025-00720-z *Co espondence: Mandala on Wes enholz m onwes enho@uni-osnab ueck. de Full lis o au ho in o ma ion is a ailable a he end o he a icle 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.