Proceedings of the IEEE Social Computing Conference, Washington DC, September 8-14, pp

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1 Proceedngs of the IEEE Socal Computng Conference, Washngton DC, September 8-14, pp On Measurng the Qualty of a Network Communty Structure Mngmng Chen Department of Computer Scence Rensselaer Polytechnc Insttute 110 8th Street, Troy, NY Emal: chenm8@rp.edu Tommy Nguyen Department of Computer Scence Rensselaer Polytechnc Insttute 110 8th Street, Troy, NY Emal: nguyet11@rp.edu Boleslaw K. Szymansk Department of Computer Scence Rensselaer Polytechnc Insttute 110 8th Street, Troy, NY Emal: szymab@rp.edu Abstract Modularty s wdely used to effectvely measure the strength of the communty structure found by communty detecton algorthms. However, modularty maxmzaton suffers from two opposte yet coexstng problems: n some cases, t tends to favor small communtes over large ones whle n others, large communtes over small ones. The latter tendency s known n the lterature as the resoluton lmt problem. To address them, we propose to modfy modularty by subtractng from t the fracton of edges connectng nodes of dfferent communtes and by ncludng communty densty nto modularty. We refer to the modfed metrc as Modularty Densty and we demonstrate that t ndeed resolves both problems mentoned above. We descrbe the motvaton for ntroducng ths metrc by usng ntutvely clear and smple examples. We also dscuss the results of applyng ths metrc, modularty, and several other popular communty qualty metrcs to two real dynamc networks. The results mply that Modularty Densty s consstent wth all the communty qualty measurements but not modularty, whch suggests that Modularty Densty s an mproved measurement of the communty qualty compared to modularty. I. INTRODUCTION Communtes are basc structures n socology ntensvely studed snce 1950 s [1, [2. Socal meda enabled onlne communtes that lnk people regardless of ther physcal locaton. Thus, computatonal methods are needed to analyze and detect such large communtes. Formally, communtes are defned as groups of nodes wthn whch connectons are denser than between them [3 and communty detecton became one of the fundamental ssues n network scence. Communty detecton has been shown to reveal latent yet meanngful structure not only for groups n onlne and contact-based socal networks, but also n functonal modules n proten-proten nteracton networks, groups of customers wth smlar nterests n onlne retaler user networks, groups of scentsts n nterdscplnary collaboraton networks, etc. [4. In the last decade, the most popular communty detecton method, proposed by Newman [5, has been to maxmze the qualty metrc known as modularty [3, [6 over all the possble parttons of a network. Ths metrc measures the dfference (relatve to the total number of edges) between the actual and expected (n a randomzed graph wth the same number of nodes and the same degree dstrbuton) number of edges wthn a gven communty. It s wdely used to measure the strength of the communty structures detected by the communty detecton algorthms. However, modularty maxmzaton has two opposte yet concurrent problems. In some cases, t tends to splt large communtes nto smaller communtes. In other cases, t tends to form large communtes by mergng communtes that are smaller than a certan threshold whch depends on the total number of edges n the network and on the degree of nter-connectvty between the communtes. The latter problem s known as the resoluton lmt problem [7. To solve these two problems smultaneously, we propose a new communty qualty metrc, that we termed Modularty Densty, as an alternatve to modularty. Frst, we show that modularty decreased by Splt Penalty, defned as the fracton of edges that connect nodes of dfferent communtes, solves the problem of favorng small communtes. Next, we demonstrate that ncludng communty densty nto modularty addresses the problem of favorng large communtes. We refer to the resultng metrc as Modularty Densty. We dscuss our experments wth Modularty Densty, modularty, and other popular communty qualty metrcs, ncludng the number of Intra-edges, Contracton, the number of Inter-edges, Expanson, and Conductance [8, on two real dynamc networks. The results show that Modularty Densty s dfferent from orgnal modularty, but consstent wth all those communty qualty measurements, whch mples that Modularty Densty s effectve n measurng the communty qualty of networks. The rest of the paper s organzed as follows. Frst, n Secton II we dscuss some related works. Then, we brefly ntroduce modularty and llustrate our motvaton to propose the new metrc wth examples n Secton III. Secton IV presents the experments that demonstrate Modularty Densty solves the two problems of modularty smultaneously. Fnally, we conclude and dscuss the future work n Secton V. II. RELATED WORK Several metrcs for evaluatng the qualty of communty structure have been ntroduced. The most popular and wdely used s modularty [3, [6. It s defned as the dfference (relatve to the total number of edges) between the actual and expected (n a randomzed graph wth the same number of nodes and the same degree sequence) number of edges nsde a gven communty. Although ntally defned for unweghted and undrected networks, the defnton of modularty has been subsequently extended to capture communty structure n weghted networks [9 and then n drected networks [10. However, recently, Fortunato and Barthélemy [7 presented a resoluton lmt problem of modularty, essence of whch s that optmzng modularty wll not fnd communtes smaller than a threshold sze, or weght [11. Ths threshold depends on the total number, or total weght, of edges n the network and on the degree of nterconnectedness between the communtes. Moreover, Good et al. [12 shown that the range of modularty values computed over all possble parttons of a graph has a

2 (a) Two very well separated communtes. (b) Two well separated communtes. (c) Two weakly connected communtes. (d) Ambguty between one and two communtes. (e) One well connected communty. (f) One very well connected communty. Fg. 1. Sx smple network examples that have two dfferent communty structures, one wth a sngle bg communty contanng all eght nodes and the other wth the two small communtes each contanng four dfferent nodes. structure n whch the maxmum modularty partton s typcally concealed among an exponentally large (n terms of the graph sze) number of structurally dssmlar, hgh-modularty parttons. To address ths resoluton lmt problem, multresoluton versons of modularty [13, [14 were proposed to allow researchers to specfy a tunable target resoluton lmt parameter and dentfy communtes on that scale. Typcally, t s not clear how to choose the correct value for ths parameter. Furthermore, Lancchnett and Fortunato [15 stated that even those mult-resoluton versons of modularty as well as ts o- rgnal verson are not only nclned to merge the smallest wellformed communtes but also to splt the largest well-formed communtes. In contrast, the Modularty Densty metrc we propose here solves those two problems of modularty wthout the trouble of specfyng any partcular parameter. III. MODULARITY DENSITY In ths secton, we frst formally ntroduce Newman s defnton of modularty and then llustrate the motvaton for modfyng modularty wth several smple network examples. Next, we propose a new communty qualty metrc, called Modularty Densty, as an alternatve to modularty by combnng modularty wth Splt Penalty and communty densty to avod the two coexstng problems of modularty. Fnally, we defne Modularty Densty for dfferent knds of networks, ncludng unweghted and undrected networks, weghted networks, and drected networks, based on the correspondng formulas of modularty. A. Newman s Modularty Modularty [3, [6 for unweghted and undrected networks s defned as the rato of dfference between the actual and expected (n a randomzed graph wth the same number of nodes and the same degree sequence) number of edges wthn the communty. For the gven communty partton of a network G = (V, E) wth E edges, modularty (Q) [3 s gven by Q = [ E n ( c 2E n c + E out ) 2 c, (1) E 2E c C where C s the set of all the communtes, c s a specfc communty n C, Ec n s the number of edges between nodes wthn communty c, and Ec out s the number of edges from the nodes n communty c to the nodes outsde c. The defnton of modularty [9 for the weghted networks has precsely the same formula, Equaton (1), as for the unweghted and undrected networks. However, for weghted networks, E s the sum of the weghts of all the edges n the network, Ec n s the sum of the weghts of the edges between nodes wthn communty c, and Ec out s the sum of the weghts of the edges from the nodes n communty c to the nodes outsde c. The formula of modularty for drected networks [10 s as follows Q = [ E n c (En c + E out,c )(E n c + E c,out) E E 2, c C (2) where E out,c s the number of edges from the nodes outsde communty c to the nodes n communty c and E c,out s the number of edges from the nodes n communty c to the nodes outsde c. For undrected networks, t s clear that E out,c = E c,out = Ec out and thus the drected modularty s reduced to undrected modularty. B. Motvaton for Introducng Splt Penalty In ths subsecton, we demonstrate the motvaton for ntroducng Splt Penalty nto modularty by usng seven ntutvely clear and smple network examples, sx of whch are presented

3 TABLE I. METRIC VALUES OF THE EXAMPLE: TWO VERY WELL SEPARATED COMMUNITIES. Modularty (Q) Splt Penalty (SP ) Q s Q ds Two communtes One communty TABLE II. METRIC VALUES OF THE EXAMPLE: TWO WELL SEPARATED COMMUNITIES. Modularty (Q) Splt Penalty (SP ) Q s Q ds Two communtes One communty TABLE III. METRIC VALUES OF THE EXAMPLE: TWO WEAKLY CONNECTED COMMUNITIES. Modularty (Q) Splt Penalty (SP ) Q s Q ds Two communtes One communty TABLE IV. METRIC VALUES OF THE EXAMPLE: AMBIGUITY BETWEEN ONE AND TWO COMMUNITIES. Modularty (Q) Splt Penalty (SP ) Q s Q ds Two communtes One communty TABLE V. METRIC VALUES OF THE EXAMPLE: ONE WELL CONNECTED COMMUNITY. Modularty (Q) Splt Penalty (SP ) Q s Q ds Two communtes One communty TABLE VI. METRIC VALUES OF THE EXAMPLE: ONE VERY WELL CONNECTED COMMUNITY. Modularty (Q) Splt Penalty (SP ) Q s Q ds Two communtes One communty TABLE VII. METRIC VALUES OF THE EXAMPLE: ONE COMPLETE GRAPH. Modularty (Q) Splt Penalty (SP ) Q s Q ds Two communtes One communty n Fgure 1. The seventh example s a complete graph wth eght nodes and one bg communty contanng all eght nodes whle the alternatve partton conssts of the two small communtes each contanng four dfferent nodes. We could easly judge that for the frst, second, and the thrd examples, the communty structure wth two small communtes s better than the communty structure n whch they are merged together. For the fourth example, the two dfferent communty structures are nearly of the same qualty. However, for the ffth, sxth, and the seventh examples, the communty structure wth one bg communty s of better qualty than the alternatve. Tables I-VII show the metrc values of the seven network examples descrbed above. Tables I-III, and Table VII demonstrate that modularty succeeds n measurng the qualty of the two dfferent communty structures n those four examples. However, from Tables IV-VI, we could observe that modularty actually fals to measure the communty qualty of those three examples because t mples that the communty structure wth two small communtes s better. In contrast, for the ffth and the sxth examples, the communty structure wth one bg communty s of better qualty. Yet, n ths case modularty gves preference to the communty structure wth two separated small communtes, demonstratng that modularty has the problem of favorng small communtes. To address the drawback of favorng small communtes, we propose that the qualty of the communty structure should take nto account the edges between dfferent communtes. We ntroduce Modularty wth Splt Penalty (Q s ) by subtractng from modularty the Splt Penalty (SP ) whch s the fracton of edges that connect nodes of dfferent communtes. More formally, Q s = Q SP. (3) The ntuton here s clear. Modularty measures the postve effect of groupng nodes together n terms of takng nto account exstng edges between nodes whle Splt Penalty measures the negatve effect of gnorng edges jonng members of dfferent communtes. Enlargng communty elmnates some Splt Penalty but f there are only a few edges across current partton, modularty of the merged communty could be lower, negatng the beneft of mergng. Splttng a communty nto two or more communtes ntroduces some Splt Penalty but f there are only a few edges between those separated communtes, an ncrease of modularty can make such splttng benefcal. Tables I-VII demonstrate that Q s can correctly measure the qualty of the communty structures of all seven network examples. C. Modularty wth Splt Penalty In ths subsecton, we extend the formula of Q s to dfferent knds of networks, such as unweghted and undrected networks, weghted networks, and drected networks, based on the correspondng formulas of modularty presented n Subsecton III-III-A. From Subsecton III-III-B, we know that Splt Penalty (SP ) s the fracton of edges that connect nodes of dfferent communtes. Thus, for undrected networks, no matter unweghted or weghted, Splt Penalty s defned as SP = [ c C c C c j c c j c E c,c j 2E. (4) where E c,c j s the number of edges from communty c to communty c j for unweghted networks or the sum of the weghts of the edges from communty c to communty c j for weghted networks. For drected networks, Splt Penalty s gven by SP = [ E c,c j. (5) E Therefore, for undrected networks, both unweghted and weghted, from Equatons (1), (3), and (4), Q s s defned as Q s = Ec n ( 2E n c + Ec out ) 2 E c,c j E 2E 2E. c C c j C c j c For drected networks, usng Equatons (2), (3), and (5), Q s can be expressed as Q s = [ E n c (En c + E out,c )(Ec n + E c,out) E E 2 c C E c,c j. E c j c (6) (7)

4 TABLE VIII. METRIC VALUES OF THE EXAMPLE: A RING OF THIRTY CLIQUES, EACH HAVING FIVE NODES AND CONNECTED BY SINGLE EDGES. Modularty (Q) Splt Penalty (SP ) Q s Q ds Thrty communtes Ffteen communtes Fg. 2. A rng network example made out of thrty dentcal clques, each havng fve nodes and connected by sngle edges. D. Motvaton for Introducng Communty Densty Modularty has the resoluton lmt problem that Q s makes even worse. Ths problem s llustrated n Fgure 2. It dsplays a rng network comprsed of thrty dentcal clques, each of whch has fve nodes and they are connected by sngle edges. In ths case, the modularty of the communty structure wth each clque formng a dfferent communty, totally thrty communtes, should be larger than that of the communty structure n whch two consecutve clques form a dfferent communty, totally ffteen communtes. However, Table VIII shows that the relaton s reversed snce the communty structure wth ffteen communtes has larger modularty than that of the communty structure wth thrty communtes. Moreover, as ponted out n [7, when m(m 1) + 2 < n, where n s the number of clques and m s the number of nodes n each clque, modularty s hgher for the large communty wth two consecutve clques nstead of the small communty wth a sngle clque. Moreover, Table VIII demonstrates that the dfference of Q s for these two communty structures s larger than the correspondng dfference of modularty. More specfcally, Q s = ( ) = > Q = ( ) = , whch means that Q s makes the resoluton lmt problem even worse. To address the resoluton lmt problem above, we propose and t s also qute ntutve to ntroduce communty densty nto modularty, ncorporatng both the number of edges and the number of nodes n the communtes and also Splt Penalty. The correspondng new metrc s called Modularty Densty (Q ds ). Table VIII shows that the Q ds of the communty structure n whch two consecutve clques form a dfferent communty s almost half of the Q ds of the alternatve n whch each clque forms a dfferent communty. Hence, n ths case, Q ds avods the resoluton lmt problem. Furthermore, Tables I-VII and Fgure 1 demonstrate that Q ds correctly measures the qualty of the communty structures of all seven network examples. Even for the network example of Fgure 1(d) n whch there s ambguty whch communty structure s of hgher qualty, the Q ds of the one bg communty s only slghtly larger than the Q ds of the two small communtes as shown n Table IV. E. Modularty Densty In ths subsecton, we wll gve the formulas for Q ds for dfferent knds of networks, ncludng unweghted and undrected networks, weghted networks, and drected networks, based on the correspondng formulas of Q s presented n Subsecton III-III-C. For undrected networks, regardless whether unweghted or weghted, we defne Q ds usng Equaton (6) as follows Q ds = [ E n ( c 2E n E d c c + Ec out ) 2 d c 2E d c = c C c j c 2E n c c (c 1), d c,c j = E c,c j c c j. Q ds = c C E c,c j d c,c 2E j, In the above, d c s the nternal densty of communty c, d c,c j s the par-wse densty between communty c and communty c j. Note that Ec n n d c and E c,c j n d c,c j are unweghted for both unweghted and weghted networks, so that those two communty denstes are always less than or equal to 1.0. For drected networks, usng Equaton (7), Q ds s gven by [ E n c E d c c j C c j c E c,c j d c,c E j (En c + E out,c )(Ec n + E c,out) E 2 Ec n d c = c (c 1), d c,c j = E c,c j c c j. IV. EVALUATION AND ANALYSIS In ths secton, we frst ntroduce two real dynamc datasets and varous other popular communty qualty measurements. Then, we show the expermental results that valdate Modularty Densty (Q ds ) ablty to address the two problems of modularty (Q) smultaneously. Formal proofs that Q ds solves the modularty shortcomngs are presented n the extended verson of ths paper [16. A. Real Dynamc Datasets In ths subsecton, we ntroduce two real dynamc datasets on whch we conduct experments n order to valdate that Q ds avods the two problems of modularty. Senate Dataset [17, [18. The Senate dataset s a tmeevolvng weghted network comprsed of Unted States senators where the weght of an edge represents the smlarty of ther roll call votng behavor. Ths dataset was obtaned from webste votevew.com and the smlartes between a par of senators were calculated followng Waugh et al. [18 as the number of blls for whch the senators of the par voted the same way, normalzed by the number of blls for whch they both voted. The dataset totally conssts of 111 d 2 c, (8) (9)

5 TABLE IX. THE AVERAGE METRIC DIFFERENCES BETWEEN LABELRANKT WITH DIFFERENT VALUES OF CONDITIONAL UPDATE PARAMETER q AND ESTRANGEMENT ON SENATE DATASET. LabelRankT condtonal update q Q Q s Q ds # Intra-edges Contracton # Inter-edges Expanson Conductance TABLE X. THE AVERAGE METRIC DIFFERENCES BETWEEN LABELRANKT WITH DIFFERENT VALUES OF CONDITIONAL UPDATE PARAMETER q AND ESTRANGEMENT ON REALITY MINING BLUETOOTH SCAN DATA. LabelRankT condtonal update q Q Q s Q ds # Intra-edges Contracton # Inter-edges Expanson Conductance snapshots correspondng to Senate s actvtes over 220 years and ncludes 1916 unque senators. Realty Mnng Bluetooth Scan Data [19. Ths dataset was created from the records of Bluetooth Scans generated among the 94 subjects n Realty Mnng study conducted from at the MIT Meda Laboratory. In the network, nodes represent the subjects and the drected edges correspond to the Bluetooth Scan records whle the weght of each edge represents the number of drect Bluetooth scans between the two subjects. In the experments, we only used the records from August 02, 2004 (Monday) to May 29, 2005 (Sunday) and we dvded them nto weekly snapshots, so each snapshot represents scans collected durng the correspondng week. There are total of 43 snapshots. B. Communty Qualty Measurements In the dscusson of the expermental results we use varous communty qualty metrcs, ncludng the number of Intraedges, Contracton, the number of Inter-edges, Expanson, and Conductance [8, whch characterze how communty-lke s the connectvty structure of a gven set of nodes. All of them rely on the ntuton that communtes are sets of nodes wth many edges nsde them and few edges outsde of them. Now, gven a network G = (V, E) and gven a communty or a set of nodes c, let c be the number of nodes n the communty c and let Ec n denote the total number of edges n c for unweghted networks or the total weght of such edges for weghted networks. We denote the total number of edges from the nodes n communty c to the nodes outsde c for unweghted networks or the total weght of such edges for weghted networks as Ec out. Then, the defntons of the fve qualty metrcs are as follows: The number of Intra-edges: Ec n ; t s the total number of edges n c or the total weght of such edges. The larger the value of ths metrc s, the better the communty qualty s. Contracton: 2Ec n /c for undrected networks or Ec n /c for drected networks; t measures the average number of edges per node nsde the communty c or the average weght per node of such edges. The larger the value of Contracton s, the better the communty qualty s. The number of Inter-edges: Ec out ; t s the total number of edges from the nodes n communty c to the nodes outsde c or the total weght of such edges. The smaller t s, the better the communty qualty s. Expanson: Ec out /c; t measures the average number of edges (per node) that pont outsde the communty c or the average weght per node of such edges. The smaller the value of Expanson, the better the communty qualty s. E Conductance: out c 2Ec n+eout c for undrected networks or E out c Ec n+eout c for drected networks; t measures the fracton of the total number of edges that pont outsde the communty for unweghted networks or the fracton of the total weght of such edges for weghted networks. The smaller the value of Conductance s, the better the communty qualty s. C. Expermental Results In ths subsecton, we report the results of performng communty detecton on the two real dynamc datasets ntroduced n Subsecton IV-A by usng the dynamc communty detecton algorthms, LabelRankT [20 and Estrangement [17. LabelRankT [20 detects communtes n large-scale dynamc networks through stablzed label propagaton. Estrangement [17 detects temporal communtes by maxmzng modularty n a snapshot subject to a constrant on the estrangement from the partton n the prevous snapshot. We chose these two algorthms because the second algorthm reles on the modularty optmzaton whle the frst one does not. In the experments, we adopted the best parameter of Estrangement but varyng the condtonal update parameter q [0, 1 of LabelRankT from 0.05 to As seen n the results, n most cases, the best q s around 0.7 n agreement wth the best value reported n [20. For the communty structures found by the two algorthms, we calculated the values of modularty (Q), Q s, Modularty Densty (Q ds ), and the fve metrcs descrbed n Subsecton IV-B. Table IX and Table X present the average metrc dfferences between LabelRankT wth dfferent values of condtonal update parameter q and Estrangement on Senate dataset and Realty Mnng Bluetooth Scan data, respectvely. That s, we frst computed the values of the eght metrcs above for the communty detecton results, detected by Estrangement, of each snapshot. Then, we calculated the eght metrcs values for the communty detecton results, dscovered by LabelRankT for all q, of each snapshot. Next, we got the metrc dfferences of all eght metrcs by subtractng the metrc values of

6 Estrangement from those of LabelRankT for all q s over each snapshot. Then, averagng those dfferences of each metrc over all the snapshots, we obtaned the correspondng average metrc dfferences. Table IX demonstrates that Q gets ts largest value when q = 0.2; Q s reaches the largest value when q = 0.6; Q ds, Intra-edges, and Contracton get ther largest values at q = 0.7 and q = 0.8; also, Inter-edges, Expanson, and Conductance reach ther smallest values at q = 0.7 and q = 0.8. Thus, Q ds s consstent wth the fve metrcs ntroduced n Subsecton IV-B on determnng the best q for LabelRankT on Senate dataset whle Q and Q s are not consstent wth them. Further, we could observe that Q s always negatve whch ndcates that LabelRankT performs below Estrangement over all q s because the goal of Estrangement s to maxmze modularty (Q). However, the other seven metrcs mply that LabelRankT performs better than Estrangement when q > 0.1. Therefore, we could explctly observe that maxmzng Q to detect communtes has problems n measurng the communty detecton qualty correctly on Senate dataset. Table X shows that sx metrcs get ther best (largest or smallest) values at q = 0.6 whle the two exceptons, Q and the number of Intra-edges, reach ther largest values when q = 0.5 and q = 0.7, respectvely. Thus, the sx metrcs, except Q and the number of Intra-edges, are consstent on determnng the best value of q for LabelRankT on Realty Mnng Bluetooth Scan data. Ths ndcates that on Realty Mnng Bluetooth Scan data, maxmzng Q to detect communtes has problems. It s also nterestng to observe that for q = 0.05 and q = 0.1 n Table IX, Inter-edges metrc mples that Label- RankT performs better than Estrangement on Senate dataset, whch s not consstent wth Q s, Q ds, Intra-edges, Contracton, Expanson, and Conductance metrcs. Moreover, we could learn from Table X that all metrcs, except Inter-edges metrc, show that LabelRankT performs below the performance of Estrangement over all q s. Thus, Inter-edges metrc has some problems. Also, as mentoned n the paragraph above, Intraedges metrc s not consstent wth the other sx metrcs on determnng the best q for LabelRankT, whch means that Intra-edges metrc has problems. We conjecture that the reason for the shortcomng of Intra-edges and Inter-edges metrcs s the same as the case of Q whch does not consder the number of nodes n the communtes. Ths reason also mples the superorty of Q ds over Q and Q s. Based on the results presented n the above two tables, we conclude that Q ds solves the two problems of modularty. We also conjecture that the dfference between the best values of q for LabelRankT determned by Q and Q s and the dfference determned by Q s and Q ds on Senate dataset s a manfestaton of the two problems of modularty maxmzaton, namely favorng small communtes and the resoluton lmt problem. Moreover, the dfference between the best values of q for LabelRankT determned by Q and Q s on Realty Mnng Bluetooth Scan data ndcates that maxmzng Q has the problem of favorng small communtes. Thus, Q s and Q ds can be used for checkng whether fndng communtes by maxmzng Q on a specfc dataset wll suffer any of the two problems. V. CONCLUSION In ths paper, we propose a new communty qualty metrc, called Modularty Densty, whch solves the shortcomngs of modularty of favorng small communtes n some crcumstances and large communtes n others. We demonstrate wth experments on real dynamc datasets that Modularty Densty s an effectve alternatve to modularty. 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