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1 Computer Communcatons 3 (011) Contents lsts avalable at ScenceDrect Computer Communcatons ournal homepage: Detectng communtes of trangles n complex networs usng spectral optmzaton Belacem Serrour a, Alex Arenas b, *, Sergo Gómez b a Lab. LIESP, Bâtment Nautbus (ex 710), Unversté Claude Bernard Lyon 1, 696 Vlleurbanne Cedex, France b Departament d Engnyera Informàtca Matemàtques, Unverstat Rovra Vrgl, 3007 Tarragona, Span artcle nfo abstract Artcle hstory: Avalable onlne May 010 Keywords: Complex networs Communtes Trangle modularty Spectral optmzaton The study of the sub-structure of complex networs s of maor mportance to relate topology and functonalty. Many efforts have been devoted to the analyss of the modular structure of networs usng the qualty functon nown as modularty. However, generally speang, the relaton between topologcal modules and functonal groups s stll unnown, and depends on the semantc of the lns. Sometmes, we now n advance that many connectons are transtve, and as a consequence, trangles have a specfc meanng. Here we propose the study of the modular structure of networs consderng trangles as the buldng blocs of modules. The method generalzes the standard modularty and uses spectral optmzaton to fnd ts maxmum. We compare the parttons obtaned wth those resultng from the optmzaton of the standard modularty n several real networs. The results show that the nformaton reported by the analyss of modules of trangles complements the nformaton of the classcal modularty analyss. Ó 010 Elsever B.V. All rghts reserved. 1. Introducton The study of the modular (or communty) structure of complex networs has become a challengng subect [1] wth potental applcatons n many dscplnes, rangng from socology to computer scence, see revews [ ]. Understandng the modular unts of graphs of nteractons (lns) between nodes, representng people and ther acquantances, documents and ther ctaton relatons, computers and ther physcal or logcal connectons, etc., s of utmost mportance to graspng nowledge about the functonalty and performance of such systems. One of the most successful approaches to dentfy the underlyng modular structure of complex networs, has been the ntroducton of the qualty functon called modularty [5,6]. Modularty encompasses two goals: () t mplctly defnes modules as those subgraphs that optmze ths quantty, and () t provdes a quanttatve measure to fnd them va optmzaton algorthms. It s based on the ntutve dea that random networs are not expected to exhbt modular structure (communtes) beyond fluctuatons [7]. A lot of effort has been put nto proposng relable technques to maxmze modularty [8 16], see revew [17]. To a large extent, the success of modularty as a qualty functon to analyze the modular structure of complex networs reles on ts ntrnsc smplcty. The researcher nterested n ths analyss s endowed wth a * Correspondng author. E-mal addresses: bserrour@bat710.unv-lyon1.fr (B. Serrour), alexandre.arenas@ urv.cat (A. Arenas), sergo.gomez@urv.cat (S. Gómez). non-parametrc functon to be optmzed: modularty. The result of the analyss wll provde a partton of the networ nto communtes such that the number of edges wthn each communty s larger than the number of edges one would expect to fnd by random chance. As a consequence, each communty s a subset of nodes more connected between them than wth the rest of the nodes n the networ. The user has to be aware of some aspects about resoluton lmtatons that avod graspng the modular structure of networs at low scales usng modularty [18]. The problem can be solved usng multresoluton methods [19,0]. The mathematcal formulaton of modularty was proposed for unweghted and undrected networs [5] and generalzed later to weghted [6] and drected networs [1]. The generalzed defnton s as follows QðCÞ ¼ 1 X N X N w ¼1 ¼1 w n w wout w! dðc ; C Þ; where w s the strength of the ln between the nodes and of the networ, ¼ P w s the strength of lns gong from, w n ¼ P w s the strength of lns comng to, and the total strength of the networ s w ¼ P w. Fnally, C s the ndex of the communty to whch node belongs to, and d(x,y) s the Kronecer functon assgnng 1 only f x = y, and 0 otherwse. A close loo at Eq. (1) reveals that the buldng bloc of the communty structure we are loong for, wthn ths formulaton, s the ln between two nodes. Every term n Eq. (1) accounts for the dfference, wthn a module, between the actual exstence of a ln ð1þ /$ - see front matter Ó 010 Elsever B.V. All rghts reserved. do: /.comcom

2 630 B. Serrour et al. / Computer Communcatons 3 (011) wth weght w and the probablty of exstence of such a ln ust by chance, preservng the strength dstrbuton. However, n many cases the mnmal and functonal structural entty of a graph s not a smple ln but a small structure (motf) of several nodes []. Motfs are small subgraphs that can be found n a networ and that correspond to a specfc functonal pattern of that networ. Statstcal over-representaton of motfs (compared wth the random occurrence of these sub-structures) has been a useful technque to determne mnmum buldng blocs of functonalty n complex networs, and several wors explot ther dentfcaton [ ]. Among the possble motfs, the smplest one s the trangle whch represents the basc unt of transtvty and redundancy n a graph, see Fg. 1. Ths motf s over-represented n many real networs, for example motfs 1 and 13 n Fg. 1, the feedbac wth two mutual dyads and the fully connected trad respectvely, are characterstc motfs of the WWW. Motf 7 (feed-forward loop) s over-represented n electronc crcuts, neurons connectvty and gene regulatory transcrpton networs. The reason for ths over-representaton reles on the functonalty of such small subgraphs on the evoluton and performance of the specfc networ. In the WWW as well as n socal networs, the fully connected trad s probably the result of the transtvty of contents or human relatons, respectvely. The feed-forward loop s related to the relablty or fal tolerance of the connectons between mportant elements nvolved n communcaton chans. The dea we propose here s that fndng modules contanng such motfs as buldng blocs could mprove our nformaton about the modular structure of complex networs. The mportance of transtvty s traced bac to the semnal paper [5] where t s proposed the clusterng coeffcent, a scalar measure quantfyng the total number of trangles n a networ through the average lelhood that two neghbors of a vertex are neghbors themselves. The man goal of our wor s to determne communtes usng as buldng blocs trangular motfs. We propose an approach for trangle communty detecton based on modularty optmzaton usng the spectral algorthm decomposton and optmzaton. The resultng algorthm s able to dentfy effcently the best partton n communtes of trangles of any gven networ, optmzng ther correspondent modularty functon.. Spectral decomposton for trangle communty detecton Let G =(V, A) be a weghted undrected graph representng a complex networ, where V represents the vertces set and A the edges set. The obectve s to dentfy communtes of trangles,.e. a partton wth the requrement that the densty of trangles formed by any three nodes, and nsde the same module s larger than the trangles formed outsde the module. We wll defne ths obectve usng a proper adaptaton of modularty..1. Trangle modularty tensor In [6] some of us ntroduced a mathematcal formalsm to cope wth modularty of motfs of any sze. Captalzng on ths wor, here we study the specfcty of trangle modularty Q M (C) of a certan partton C of an undrected graph (the extenson to drected graphs s straghtforward, although a lttle bt more ntrcate, we present ths extenson n the Appendx). The mathematcal defnton s Q M ðcþ ¼ X X X B dðc ; C ÞdðC ; C ÞdðC ; C Þ; where C s the ndex of the communty whch node belongs to, and B B ¼ 1 w w w 1 ðw w Þðw w Þðw w Þ; ð3þ T G T N s a three ndces mathematcal obect (trangle modularty tensor, from now on) that evaluates for each trad,,, the dfference between the actual densty of strength of the trangle n the graph and the expected densty of ths trangle n a random confguraton wth the same strength dstrbuton (null case). The normalzaton constant T G s the total number of trads of nodes formng trangles n the networ, T G ¼ X X X w w w ; ðþ and ts counterpart T N for the null case term s T N ¼ X X X ðw w Þðw w Þðw w Þ: It s straghtforward to chec that the trangle modularty tensor satsfes: B ¼ B ¼ B ; X X X B ¼ 0: ðþ ð5þ ð6þ ð7þ Fg. 1. Lst of all possble three-nodes motfs.

3 B. Serrour et al. / Computer Communcatons 3 (011) Spectral optmzaton of trangle modularty The computaton of the trangle modularty s demandng due to the combnatoral number of trads that can be formed. The proposal of any optmzaton algorthm for ths functon must be aware of ths cost. Among the possbltes already stated n the lterature we devse that the spectral optmzaton scheme, frst proposed n [16], sa canddate to perform ths tas effcently. The dea behnd ths algorthm s to use the egenspectrum of the modularty matrx, whch plays a role n communty detecton smlar to that played by the graph Laplacan, and use a recurson splttng remnscent of graph parttonng calculatons. The problem we have s that a drect mappng to the usual spectral modularty optmzaton s not straghtforward gven the structure of Eq. (). Bascally we need to transform Eq. () n a functon wth the followng structure: QðCÞ / X X s M s ; ð8þ where the leadng egenvector of M, the modularty matrx, wll nduce the frst recurson step, splttng the networ n two parts. We propose the followng transformaton: let us assume a partton of the networ n two communtes, ntroducng the varables s, whch are +1 or 1 dependng on the communty to whch node belongs to, and tang nto account that dðc ; C Þ¼ 1 ð1 þ s s Þ; then dðc ; C ÞdðC ; C ÞdðC ; C Þ¼ 1 8 ð1 þ s s Þð1 þ s s Þð1 þ s s Þ ¼ 1 ð1 þ s s þ s s þ s s Þ; ð9þ ð10þ where we have made use of s ¼þ1. Therefore, usng Eqs. (6) and (7), Q M ðsþ ¼ 1 X X X B ð1 þ s s þ s s þ s s Þ ¼ 3 X X X B s s : Defnng the trangle modularty matrx M ¼ X B ¼ 1 T G w X then Q M ðsþ ¼ 3 X X s M s : w w 1 T N ðw w Þðw w Þ X ðw w Þ; ð11þ ð1þ ð13þ Thus, we have been able to reduce the optmzaton of the trangle modularty nto the standard spectral algorthm gven n [16]. For the case of undrected networs, ths matrx s symmetrc and the computaton of ts egenspectra gves real values. However, f the networ s drected, ths property s not necessarly true, and then a symmetrzaton of the matrx s needed before computng ts spectrum (see Appendx). Once a frst dvson of the networ n two parts has been obtaned, t s possble to terate the process, whle modularty mproves, by a recursve applcaton of the spectral splttng to each subgraph. To ths end, we need the value of the trangle modularty matrx for any subgraph. Supposng we have a subgraph g to be dvded nto g 1 and g, the change n trangle modularty s gven by DQ M ðg! g 1 ; g Þ¼ X X X B þ B B ;;g 1 ;;g ;;g ¼ 3 X X B s s X! B ¼ 3 X s M ðgþs ; ð1þ ;g ;g ;g g where M ðgþ ¼ X g X B d B!; ð15þ g and s s +1 for nodes n g 1 and 1 for nodes n g. Therefore, the new trangle modularty matrx s not ust a submatrx of the orgnal one, but addtonal terms appear to tae nto account the connectvty wth the rest of the networ..3. Algorthm Once the trangle modularty has been transformed to the proper form to be optmzed by spectral decomposton, we can proceed to formulate a complete decomposton-optmzaton algorthm. After the frst analyss of the egenspectra, the egenvector assocated to the largest egenvalue s used to determne the elements that wll be assgned to one of the two communtes accordng to the sgn of ther egenvector component. Ths process s recursvely executed untl no new splts are obtaned. The decomposton gven by the spectral parttonng can be mproved by a fne-tunng of the nodes asgnments after the process ends. We use the Kernghan Ln optmzaton method to mprove the modularty as explaned n [16]. The man dea s to move vertces n a group to another ncreasng the modularty. We move all vertces exactly once. At each step, we choose to move the vertex gvng the best mprovement (largest ncrease n the modularty). When all vertces are moved, we repeat the process untl no mprovement s possble. Some computatonal ssues should be consdered here: the computaton of the largest egenvalue and ts correspondng egenvector can be effcently determned usng the teratve Lanczos method [7]; the computaton of Q M (S) s, n prncple, of order O(N 3 ), however t can be done very effcently by pre-computng and storng the values of T N and T G, and the lsts of trangles to whch each node belongs to; fnally, the KL post-processng stage whch s eventually the computatonal bottlenec of the process, must be parameterzed accordng to the number of nodes we pretend to move and the relatve mprovement of modularty observed. Algorthm 1: Trangle communty detecton Requre: Connected networ G(V, E) Ensure: Trangle communtes C, Trangle modularty of the partton Q M (C) 1: Read networ : Current subgraph g G 3: Buld modularty matrx M(g) : Compute Q M (g) 5: Compute leadng egenvalue and egenvector of M(g) 6: Decomposton of group g n two groups: g1 and g, usng the sgns of egenvector components 7: Compute the modularty Q M (g1,g) of the ntal splt of group g 8: Improve Q M (g1,g) usng KL optmzaton between g1 and g 9: Compute the modularty Q M (g1,g) of the splt of group g 10: f Q M (g1,g) > Q M (g) then 11: goto 3 wth g g1 1: goto 3 wth g g 13: end f 3. Results In ths secton we show the results of the algorthm, appled to several real networs. We have used the followng networs:

4 63 B. Serrour et al. / Computer Communcatons 3 (011) Table 1 Comparson of standard and trangle modulartes. Networ Nodes Lns Q Q M D(Q,Q M ) Football Zachary Dolphns Adnoun Elec s Neurons Cortex Football [1], a networ of Amercan football games between Dvson IA colleges durng regular season Fall 000. Zachary [8], a socal networ of frendshps between 3 members of a arate club at a US unversty n the 1970s. Dolphns [9], an undrected socal networ of frequent assocatons between 6 dolphns n a communty lvng off Doubtful Sound, New Zealand. Adnoun [30], adacency networ of common adectves and nouns n the novel Davd Copperfeld by Charles Dcens. Elec s08 [], benchmar of sequental logc electronc crcut. Neurons [31], networ of neural connectvty of the nematode C. elegans. Cortex [3], networ of connectons between cortcal areas n the cat bran. To evaluate the nformaton provded by the new trangle modularty, we perform a comparson wth the standard modularty Eq. (1). We have developed a comparson n both the values of the optmal modularty, and the parttons obtaned Modulartes comparson Table 1 shows the best standard, and trangle modulartes found usng spectral optmzaton. We defne a new parameter D(Q,Q M )=(Q M Q)/Q that measures the relatve dfference between both. Postve values of D(Q,Q M ) ndcate that the contrbuton of trangles to communtes s larger than standard modularty communtes, and the contrary for negatve values. From Table 1 we observe that n Adnoun, whch s almost a bpartte networ, the standard modularty s larger than the trangle modularty, n accordance wth the absence of these motfs. On the other sde, for the Zachary networ, a human socal networ where transtvty s mplct n many acquantances, the trangle modularty becomes more nformatve than the standard modularty. Indeed, the optmal standard modularty proposes a decomposton of ths networ n four groups, whle the optmal trangle modularty s acheved for a partton n two groups plus two solated nodes (nodes 10 and 1) that do not partcpate n any trangle. Moreover the partton n two groups s n accordance wth the observed splt of ths networ after a fght between the admnstrator and the nstructor of the club, see Fg Communtes comparson A deeper comparson conssts of analyzng the dfferent modules obtaned usng the standard and trangle modularty. To ths end, we need some measures to analyze the dfference n the assgnments of nodes to modules, tang nto account that we wll also have dfferent modular parttons. Here, we use two measures, the Normalzed Mutual Informaton (NMI) and the asymmetrc Wallace ndex (AW). In [33] the authors defne the NMI to compare two clusterngs. The dea s the followng: let be a clusterng A wth c A communtes and a clusterng B wth c B communtes, and let us defne the confuson matrx N whose rows correspond to the communtes of the frst clusterng (A) and columns correspond to the communtes of second clusterng (B). The elements of the confuson matrx, N ab, represent the number of common nodes between communty a of the clusterng A and communty b of the clusterng B, the partal sums N a: ¼ P b N ab and N :b ¼ P a N ab are the szes of these communtes, and N :: ¼ P P a b N ab s the total number of nodes. The measure NMI between two clusterngs A and B s NMIðA; BÞ ¼ P c A P cb a¼1 b¼1 N ab log N abn :: N a:n :b þ Pc B a¼1 N a: log Na: N :: b¼1 N :b log N :b N :: : ð16þ Fg.. Zachary networ parttons. Best parttons found by optmzaton of (a) trangle modularty and (b) standard modularty. The real splttng of the networ s represented by the shape of the symbols (squares and crcles). Colors ndcate the assgnment of nodes to the modules found. (For nterpretaton of color mentoned n ths fgure the reader s referred to the web verson of the artcle.)

5 B. Serrour et al. / Computer Communcatons 3 (011) Table Comparson of parttons obtaned usng standard and trangles modulartes. The dfferent measures are explaned n the text. Networs NMI AW 1 AW Football Zachary Dolphns Adnoun Elec s Neurons Cortex If the parttons are dentcal, then NMI taes ts maxmum value of 1. If the parttons are totally ndependent, NMI = 0. It measures the amount of nformaton that both parttons have n common. The asymmetrc Wallace ndex [3] s the probablty that a par of elements n one cluster of partton A (resp. B) s also n the same cluster of partton B (resp. A). Usng the same defntons as for the NMI, the two possble asymmetrc Wallace ndces are: AW 1 ða; BÞ ¼ AW ða; BÞ ¼ P cb a¼1 a¼1 b¼1 N abðn ab 1Þ ; ð17þ a¼1 N a: ðn a: 1Þ P cb b¼1 N abðn ab 1Þ P cb b¼1 N : ð18þ :bðn :b 1Þ The asymmetrc Wallace ndex shows the ncluson of a partton n the other. In Table, we observe that the largest NMI s for the communtes of football networ. That means that the standard and trangle communtes found n that networ are very smlar. Indeed, the structure of the football networ s very dense and almost all nodes partcpate n trangles. For the AW of the cortex networ s equal to 1, that means that all the trangle communtes are ncluded n the standard ones.. Conclusons We have desgned an algorthm to compute the communtes of trangular motfs usng a spectral decomposton of the trangle modularty matrx. The algorthm provdes parttons where transtve relatons are the buldng blocs of ther nternal structure. The results of these parttons are complementary to those obtaned maxmzng the classcal modularty, that accounts only for ndvdual lns, and can be used to mprove our nowledge of the mesoscopc structure of complex networs. Acnowledgments The authors acnowledge J. Borge-Holthoefer and A. Fernández for useful dscussons. Ths wor was supported by Spansh Mnstry of Scence and Technology FIS C0-0 and the Generaltat de Catalunya SGR B.S. acnowledges support from he Rhone-Alpes regon for the fnancng of tranng by Explora doc exchange scholarshp. Appendx A Here we show the computaton of the trangle modularty matrx for a drected motf, n partcular motf 7 n Fg. 1, although as wll be shown the process s equvalent for any other motf confguraton. In ths case, we have Q M ðcþ ¼ X X X B dðc ; C ÞdðC ; C ÞdðC ; C Þ; ð19þ where B s B ¼ 1 w w w 1 ð T G T N w n Þð w n Þðwout w n The normalzaton constant T G s now T G ¼ X X X w w w ; and T N ¼ X X X w n w n Þ: ð0þ ð1þ w n : ðþ Usng the transformaton proposed n Eq. (10) M ¼ X then B ¼ 1 T G w X w w 1 T N Q M ðsþ ¼ 3 X X s M s : w n w out w n X w n : ð3þ ðþ Owng to the fact that the graph s drected, the modularty matrx M may be not symmetrc, whch causes techncal problems. However, t s possble to restore the symmetry thans to the scalar nature of Q M (S) [35]. A symmetrzaton of the trangle modularty matrx M, M 0 ¼ 1 M þ MT ð5þ yelds Q M ðsþ ¼ 1 Q MðSÞþQ M ðsþ T ¼ 3 X X s M 0 s ; ð6þ recoverng the necessary symmetry to apply the standard spectral optmzaton. In the same manner, we can defne the modularty matrx for all possble motfs of Fg. 1 ust by modfyng B. For example, for motf 13 n Fg. 1 we have: B ¼ 1 w w w w w w T G 1 w n w out w n w out ; w n ð7þ T N T G ¼ X T N ¼ X References X X w w w w w w ; ð8þ X X ð Þ w n w out w n w out : w n ð9þ [1] M. Grvan, M.E.J. Newman, Communty structure n socal and bologcal networs, Proceedngs of the Natonal Academy of Scences USA 99 (00) 781. [] L. Danon, A. Díaz-Gulera, J. Duch, A. Arenas, Comparng communty structure dentfcaton, Journal of Statstcal Mechancs (005) P [3] S.E. Schaeffer, Graph clusterng, Computer Scence Revew 1 (007) 7. [] A. Lancchnett, S. Fortunato, Communty detecton algorthms: a comparatve analyss, Physcal Revew E 80 (009) [5] M.E.J. Newman, M. Grvan, Fndng and evaluatng communty structure n networs, Physcal Revew E 69 (00) [6] M.E.J. Newman, Analyss of weghted networs, Physcal Revew E 70 (00) [7] R. Gumerà, M. Sales-Pardo, L.A.N. Amaral, Modularty from fluctuatons n random graphs and complex networs, Physcal Revew E 70 (00)

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