Directed Louvain : maximizing modularity in directed networks

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1 Directe Louvain : maximizing moularity in irecte networks Nicolas Dugué, Anthony Perez To cite this version: Nicolas Dugué, Anthony Perez. Directe Louvain : maximizing moularity in irecte networks. [Research Report] Université Orléans <hal > HAL I: hal Submitte on 20 Nov 2015 HAL is a multi-isciplinary open access archive for the eposit an issemination of scientific research ocuments, whether they are publishe or not. The ocuments may come from teaching an research institutions in France or abroa, or from public or private research centers. L archive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recherche, publiés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires publics ou privés. Distribute uner a Creative Commons Attribution 4.0 International License

2 Directe Louvain : maximizing moularity in irecte networks Nicolas Dugué Anthony Perez November 20, 2015 Abstract In this paper we consier the community etection problem from two ifferent perspectives. We first want to be able to compute communities for large irecte networks, containing million vertices an billion arcs. Moreover, in a large number of applications, the graphs moelizing such networks are irecte. Nevertheless, one is often force to forget the irection between the connections, either for the sake of simplicity or because no other options are available. This is in particular the case on large networks, since there are only a few scalable algorithms at the time. We thus turn our attention to one of the most famous scalable algorithms, namely Louvain s algorithm [3], base on moularity maximization. We moify Louvain s algorithm to hanle irecte networks base on the notion of irecte moularity efine by Leicht an Newman [13], an provie an empirical an theoretical stuy to show that one shoul prefer irecte moularity. To illustrate this fact, we use the LFR benchmarks by Lancichinetti an Fortunato [8] to esign an evaluation benchmark of irecte graphs with community structure. We also give some examples an insights on the situations where one shoul really consier irection when maximizing moularity. Finally, for the sake of completeness, we compare the results obtaine with Oslom [12], one of the best algorithms to etect communities in irecte networks. While the results obtaine with such an algorithm are by far better on the LFR benchmarks, we emphasize that it is still not well-suite to eal with very large networks. 1 Introuction In various omains such as social networks or bioinformatics, being able to etect communities efficiently constitutes a very important research interest [6]. In most cases, the unerlying graphs representing ata are irecte. This happens for instance when consiering some social network graphs, where relations between two users u an v can be represente by stating that u has an influence over v rather than simply saying that they both interact. It thus seems quite obvious to consier irection when etecting communities, an several algorithms were propose in this sense, such as Oslom [12] or InfoMap [19, 20]. In this article, we are intereste in etecting communities in very large networks such as the Twitter graph [4], which contains more than 50 millions vertices an almost 2 billions arcs. As we shall see Section 5, Oslom [12] fail to efficiently prouce communities when consiering such networks [4], especially if one wants to use only a few computer ressources. Due to this fact, a common solution is to simply forget irection when etecting communities in really large network an to run Louvain s algorithm [3] (which is extremely well-suite for large networks). To the best of our knowlege, there is no version of this algorithm maximizing irecte moularity [13]. This fact can also be seen in a survey comparing algorithms for community etection, where Lancichinetti an Fortunato [10] i not even consier Louvain s 1

3 algorithm for their irecte networks analysis. Instea, they use simulate annealing for moularity optimization [6] even if they confirme on unirecte networks that Louvain s algorithm performs better an runs a lot faster. Our results. In this work, we give some insights about the importance of irection while etecting communities. To that aim, we consier Louvain s algorithm [3], which is implemente for non-irecte graphs only. By moifying the existing source coe [2], we manage to eal with irecte graphs, following the notion of irecte moularity introuce by Leicht an Newman [13] (Section 2). We then generate a benchmark of irecte graphs using the framework provie by Fortunato et al. [8], an compute communities on such graphs using both versions of the Louvain s algorithm 1. Our results show strong evience that irection is important when etecting communities (Section 5). Finally, we also compare these results to communities obtaine by a recent community etection algorithm calle Oslom [12], both from the semantic an complexity viewpoints (Section 5). We emphasize that Oslom [12] cannot eal with large graphs such as the Twitter graph [4] (billions of eges), while Louvain s algorithm prouces results in a couple of hours. 2 Detecting community in large (irecte) networks Moularity. A classic way of etecting communities is to fin a partition of the vertex set that maximizes an optimization function. One of the most famous optimization function is calle moularity [16]. This function provies a way to value the existence of an ege between two vertices of an unirecte network by comparing it with the probability of having such an ege in a ranom moel following the same egree istribution than the original network. For instance, an ege between two vertices of large egree is not surprising, an thus oes not contribute much to the moularity of a given partition, whereas an ege between two vertices of small egree is more surprising. Formally, the moularity Q of a partition C of an unirecte graph G = (V, E) is efine as follows : Q = 1 [ A i ] i δ(c i, c ) i, where m stans for the number of eges of G, A i represents the weight of the ege between i an (set to 0 if such an ege oes not exist), i is the egree of vertex i (i.e. the number of neighbors of i), c i is the community to which vertex i belongs an the δ-function δ(u, v) is efine as 1 if u = v, an 0 otherwise. Leicht an Newman [13] aapte the notion of moularity for irecte graphs, motivate by the following observation: if two vertices u an v have small in-egree/large out-egree an small out-egree/large in-egree, then having an arc from v to u shoul be consiere more surprising than having an arc from u to v. Taking this into account, the efinition for irecte moularity of a partition of a irecte network can be easily formulate: 1 Louvain s algorithm is usually non-eterministic, but in orer to obtain consistent results, we always consier the vertices in the same orer. 2

4 Q = 1 m [A i in i, i out m ] δ(c i, c ) where A i now represents the existence of an arc between i an an in i for the in-egree (resp. out-egree) of i. (resp. out ) stans Louvain s algorithm. We now briefly escribe the behavior of Louvain s algorithm to maximize moularity. The algorithm is the same for both the classic an irecte versions of moularity. It relies on a greey proceure: starting from any partition of the vertices (usually the partition into singletons), the algorithm tries to increase the value of moularity by moving vertices from their community to any other neighbor one. In other wors, the algorithm computes the gain of moularity obtaine by aing vertex i to community C as follows (for the unirecte case): Q = [ in +C i = C i tot i 2 ( tot + i ) 2 ] [ ( in tot ) 2 ( ) 2 ] i where C i enotes the egree of noe i in community C, in the number of eges containe in community C an tot the total number of eges incient to community C. Actually, the first formula is the one as escribe in [3], but one can see that it reuces to the secon one. The algorithm oes a similar calculation to compute the gain obtaine by removing vertex i from its own community C i in a first place. The algorithm carries on as long as it exists a move that improves the value of moularity. The behavior of the algorithm is exactly the same in the irecte case, the main ifference lying in the calculation for the gain of moularity obtaine by aing vertex i to community C, which can now be one using the following: [ Q = C i out m i in tot +in i m 2 out tot where in tot (resp. out tot ) enotes the number of in-going (resp. out-going) arcs incient to community C. 3 Theoretical comparison between unirecte an irecte moularity Our goal is to valiate the observation mae by Leicht an Newman [13] by showing what happens if one uses the moularity Q [16] on a irecte graph instea of using its irecte version Q [13]. To that aim, we use a straightforwar case stuy where we consier two subgraphs C 1 an C 2 which both are communities of a irecte network (see Figure 1). We are basically stuying when merging these communities lea to a value increase of both moularities. To calculate the unirecte moularity on a network which is usually irecte, we have to ignore the links irection. ] 3

5 Figure 1: Figure extracte from the article of Lancichinetti an Fortunato [11]. Thus, if we are processing a irecte network D = (V, A) where (u, v) (v, u) A, then (u, v) E in the unirecte version G = (V, E). We use Q C 1\C 2 (resp. Q C 1\C 2 ) to refer to the unirecte (resp. irecte) moularity value of the network with C 1 an C 2 istinct communities. In the same way, we use Q C 1 C 2 (resp. Q C 1 C 2 ) to refer to the moularity value of the network where C 1 an C 2 are part of the same community. We name A 1,2 arcs between communities C 1 an C 2, an E 1,2 the corresponing eges in the unirecte network. Consiering the unirecte case, E 1,2 = A 1,2 if (u, v) A 1,2 then (v, u) / A 1,2. We also have that E 1,2 = 1 2 A 1,2 if (u, v) A 1,2 then (v, u) A 1,2. Thus, E 1,2 A 1,2 2 E 1, Unirecte case When C 1 an C 2 are consiere as part of the same community, E 1,2 links contribute to increase moularity value, as shown in bol in the following formula. ( ) Q C 1 C 2 int C = 1 m + int C 2 m + E 1,2 i m 4m 2 + i 4m 2 + i 2 i, C 1 i, C 2 When C 1 an C 2 are splitte in two ifferent communities, both the terms in bol before isappear. ( ) Q C 1\C 2 int C = 1 m + int C 2 i m 4m 2 + i 4m 2 i, C 1 i, C 2 Thus, if summing these bol terms results in a positive number, C 1 an C 2 are merge. At the contrary, if the sum is negative, C 1 an C 2 are consiere as two istinct communities. Therefore, stuying when these communities are merge or not consists in stuying the sum of these terms as follows. δ Q = Q C 1 C 2 Q C 1\C 2 = E 1,2 m i 2 = 1 m ( E 1,2 4 i )

6 Hence: δ Q > 0 E 1,2 > i (1) 3.2 Directe case In the irecte case, we obtain a quite similar result. Inee, when C 1 an C 2 are consiere as being part of the same community, we obtain: Hence: δ Q = Q C 1 C 2 Q C 1\C 2 = A 1,2 δ Q > 0 <=> A 1,2 > in i out 4m 2 ( in i out + out i out i in 4m 2 in ) 3.3 Comparison To compare the choices mae by both moularities, we replace the vertex egree of Equation 1 by its in- an out-going counterparts. i = ( in i We thus obtain the following equivalence : E 1,2 > ( in i out Let us efine the following terms : S = T = + out i + out i )( in + out ) in ) ( in i out ( in i in + + out i + out i in ( in i in ) + out i Thus, in the unirecte case, C 1 an C 2 are merge when E 1,2 > S + T while in the irecte case, the fusion is one when A 1,2 > S, T being absent from the equation. The term S confirms the observation mae by Leicht an Newman [13]. Furthermore, we can see that T is not relevant at all. Multiplying the incoming egrees in one sie an the outgoing egrees in the other sie oes not allow to estimate links probability to exist between communities in a ranom network. This may explain the better results obtaine with the Louvain algorithm implementing the irecte moularity. out ) out ) 5

7 4 Analysis of the ifferences We now stuy the conitions that make ifferences arise between the two versions of Louvain s algorithms. We first give some intuition on the configurations that can lea to two ifferent moves in the algorithm, an then provie several examples where the ifference is significant. Sufficient conitions to have a ifference. To complete the previous observations, we give some conitions that will influence community etection between the two methos. Recall that the gain of moularity can be easily compute (in both cases) using the following: Q C i Q = C i tot i [ out i in tot +in i m tot i m out tot ] an out i in out tot +in i tot m We thus have to stuy the behavior of the terms for a given vertex i an a given community C. In particular, we want to express the conitions when the first one is positive an the secon one negative, or vice-versa. Recall that, in the first case, the classic Louvain s algorithm will not consier aing vertex i to community C to increase moularity, while the irecte version will o so. Cases that make a ifference. We first provie some simple examples when the classic Louvain s algorithm fails at etecting communities, whereas the algorithm maximizing irecte moularity fins a perfect match with the groun truth communities. Figure 2: On the left, the three communities obtaine by maximizing stanar moularity are represente. On the right, the ones obtaine using irecte moularity. Consier the graph represente Figure 2, which contains 100 vertices an 2 communities. When maximizing irecte moularity, Louvain s algorithm succees in retrieving the communities whereas the classic moularity maximization fails to merge two communities. The explanation for this situation follows from our previous arguments. Inee, the graph contains vertices with unbalance in an out-egrees, who thus influence the greey metho of Louvain s algorithms. Such a situation can also be observe on larger graphs 2 (see Figure 3). 2 For the sake of visiblity we o not consier larger graphs, but mention that similar situations happen also. 6

8 Figure 3: On top, the grountruth communities. On the bottom left, the communities foun by the irecte version of the algorithm an on the bottom right the ones provie by the classic one. 5 Experimental results We now present empirically the ifferences that arise between classic moularity maximization an the irecte one. To that aim, we evaluate the results of both the moulairies over the so-calle irecte LFR benchmarks [10].We consier partitions (that is non-overlapping communities). 5.1 The LFR benchmarks To valiate the efficiency of Louvain algorithm aapte to irecte graphs, we use benchmarks introuce by Lancichinetti an Fortunato [9]. These benchmarks allow to test community etection algorithms on irecte graphs, an are esigne in orer to be as realistic as possible with respect to real networks. It is inee possible to set important features such as the power-law istribution of the egrees of the noes or of the communities sizes, as well as the maximum an average egrees of noes in the graphs. Another maor feaure introuce in these benchmarks is the mixing parameter. The mixing parameter allows to create graphs with communities more or less well-efine. A low mixing parameter inicates communities well-efine, an hence easy to etect. Reciprocally, a high mixing parameter allows to create graphs with communities which will be har to etect. 5.2 Measures To compare the results obtaine by the community etection methos, we use three evaluation measures. The results of the community etection algorithms are thus compare with the communities efine by the benchmarks we use. In the following, we use clustering to enote the community sets obtaine by the algorithms use. The term cluster is thus one community of these sets. We use community to talk about the grountruth communities establishe by the benchmark. 7

9 The first measure, calle V-Measure [18] is mae of two criteria: homogeneity an completeness. This may be compare to the F-measure base on precision an recall measures. A clustering maximizes the homogeneity if for each cluster, we fin only elements of a same community. Symetrically, completeness is maximize when for each community, all elements of a same community are in a single cluster. By computing the harmonic mean of these two values, we obtain the so-calle V-measure. The secon one is the NMI [21] for Normalize Mutual Information. Built upon concepts from information theory, this measure is commonly use to compare clusterings. Roughly speaking, this measure efines how much knowing one of two clusterings reuces uncertainty about the other. Thus, the higher the NMI, the more information the two clusterings share. We use the normalization introuce by Strehl an Gosh [21] efine as follows. efinition 1 (NMI [21]) Let U an V be two clusterings. Then the Normalize Mmutual Information is efine as a function of the mutual information I an the conitional entropy H: NMI(U, V) = I(U, V) H(U) H(V) Finally, to compute the Purity [24], we assign each cluster to the community which noes are more frequent in the cluster. Then, by summing all well-classifie noes for each of these clusters an iviing it by the number of vertices, we obtain the accuracy of our clustering. 5.3 Classic LFR benchmarks We begin this section by giving some results obtaine by generating LFR benchmarks using parameters escribe by Lancichinetti an Fortunato [10]. Those parameters consier two ifferent cases, namely graphs having small an big community sizes. In the first case, the communities are set to contain between 10 an 50 vertices, while they are require to contain between 20 an 100 vertices in the latter one. In such graphs, the average egree is set to 20 an the maximum egree is set to 50. We consier graphs having respectively 1000 an 5000 noes, an make the mixing parameter go from 0.1 (i.e. well-efine communities) to 0.6. Finally, we set the power law istribution to 2 in all cases. We first give a general picture of the results we obtaine w.r.t. to the number of vertices an the size of the communities (Table 1). n minc maxc # graphs Table 1: Proportion of graphs where the NMI of the classic moularity (nmi o ) is greater than the one of the irecte moularity (nmi ) by a given percentage. Louvain s algorithm maximizing irecte moularity is better in almost 75% of the cases. Moreover, we woul like to mention that if the improvement is rather low on average, there are some interesting cases where the improvement is rastic. 8

10 We then compare the outputs of both version of Louvain s algorithms accoring to the aforementione quality measures. We conucte such an experimentation by consiering ifferent networks sizes, an also by moifying the mixing parameter. As one can observe Table 2, the irecte version (which correspons to the bottom table) is better in most cases. n mu NMI V-measure Homogeneity Completeness Purity n mu NMI V-measure Homogeneity Completeness Purity Table 2: Results obtaine on the classic LFR benchmarks with the classic an the irecte versions of Louvain algorithm. Each mesure inicates the average taken over 100 graphs with the inicate parameters. 5.4 Generate LFR benchmarks In this Section, we present similar observations on a new set of benchmarks that we generate for this purpose. Recall that the average an maximum egree are respectively fixe to 20 an 50 in the classic LFR benchmarks. This seems quite unrealistic when trying to simulate social networks: on Table 3 we observe in several well-known complex networks that the average egree is in general much lower, while the maximum egree is much higher. Hence, it seems quite restrictive to impose a maximum egree of 50, when it is really common for vertices of such networks to have a egree close to n 3. Network Noes Eges Avg egree Max egree Power-law Zachary karate club [22] Openflight [17] 2, , Arxiv Astro-ph [14] 18, , Internet AS [23] 34, , , SlashDot [5] 51, , , Flickr [15] 2, 302, , 140, , DBPeia [1] 2, 152, 642 7, 494, , Table 3: Basic properties of several physical, social or reference base network from literature. To complete our analysis, we generate about graphs with ifferent parameters closely relate to those observe in real networks. In particular, we infere the average an maximum 9

11 Figure 4: The values of the normalize mutual information are represente on the Y axis, while the mixing parameter is on the X axis. egrees w.r.t. the power law an mixing parameter. For every combination of such parameters, we generate 100 ifferent graphs an ran a statistical analysis. We first give a general picture of our observations. n # graphs Table 4: Proportion of graphs where nmi o is greater than nmi by a given percentage. It arises from Table 4 that in 70% of the cases, irecte moularity provies better results than the classic Louvain s algorithm. We woul like to mention that very similar results can be observe when consiering other similarity measures such as F-measure, V-measure, purity, completeness, homogeneity. To be consistent with the results presente in [10], we have a closer look on the measures for 1000 an 5000 noes, respectively. The results presente Figure 4 were obtaine by taking the average of the measures over 100 ifferent graphs with the same parameters. We woul like to mention that Louvain s algorithm seems to be particularly well-suite to maximize irecte moularity, since the results obtaine seem to be really better than the ones presente 10

12 by Fortunato et al. LF09a, obtaine using simulate annealing for moularity optimization [6]. 5.5 A comparative analysis with Oslom [12] To conclue this part of our work, we woul like to compare Louvain s algorithm optimizing irecte moularity with another algorithm use to etect communities in irecte networks, namely Oslom [12]. We are particularly intereste in the time complexity neee to run such algorithms. We want to emphasize that such an algorithm is really not well-suite for hanling large (irecte) networks. Results. We now present the results obtaine by running Oslom [12] on the classic LFR benchmarks. We focus on both the accurency an the time complexity neee to obtain such results. n mu NMI V-measure Homogeneity Completeness Purity Table 5: Results obtaine on the classic LFR benchmarks. Each mesure inicates the average taken over 100 graphs with the inicate parameters. As one can see on Table 5, Oslom [12] is rastically better than Louvain s algorithm on the classic LFR benchmarks. This is not a surprising result, since Oslom [12] seems to provie really goo results [10]. However, as we shall see in the remaining of the paper, Oslom [12] cannot provie results in a reasonnable amount of time for networks with relatively small size. Time complexity. We now focus on the time complexity neee to obtain results when using OSLOM or Louvain s algorithm with irecte moularity. We conuct our analysis on a set of real networks extracte from Konect Database [7] (see Table 6). 11

13 Name Vertices Arcs Louvain Oslom E-Coli secons 1.8 secons Bio-Yeast secons 35 secons Spanish-Book secons 27 minutes Wor-Association secons 30 minutes Einburg secons > 10 hours Table 6: Directe networks use for time complexity analysis. As mentione previously, we o not consier here the quality of the output (OSLOM performs better than Louvain s algorithm), but we only focus on the time neee to obtain such an output. For networks with vertices, Oslom [12] alreay takes a significant amount of time 3 to compute the results. On the largest graph we consier, the classic configuration of Oslom [12] fails at proucing any output even after hours of computation. Thus, if one shoul clearly prefer using Oslom [12] on small networks, its use is absolutely impossible for networks as large as the ones that really often arise in the literature. We hope that our stuy will enlight that the version of Louvain s algorithm that maximizes irecte moularity shoul be consiere as more reliable to eal with such networks. References [1] Sören Auer, Christian Bizer, Georgi Kobilarov, Jens Lehmann, Richar Cyganiak, an Zachary Ives. Dbpeia: a nucleus for a web of open ata. In Proceeings of the 6th international The semantic web an 2n Asian conference on Asian semantic web conference, ISWC 07/ASWC 07, pages , [2] Vincent Blonel, Jean-Loup Guillaume, Renau Lambiotte, an Etienne Lefebvre. Louvain metho: Fining communities in large networks. [3] Vincent D. Blonel, Jean-Loup Guillaume, Renau Lambiotte, an Etienne Lefebvre. Fast unfoling of communities in large networks. J. of Stat. Mech.: Theory an Experiment, 2008(10):P10008, [4] Meeyoung Cha, Hame Haai, Fabricio Benevenuto, an Krishna P. Gummai. Measuring User Influence in Twitter: The Million Follower Fallacy. In ICWSM 10: Proc. of int. AAAI Conference on Weblogs an Social, [5] Vicenç Gómez, Anreas Kaltenbrunner, an Vicente López. Statistical analysis of the social network an iscussion threas in slashot. In Proceeings of the 17th international conference on Worl Wie Web, WWW 08, pages , We woul like to mention that our results iffer significantly from the ones presente in [12]. This comes from the fact that a new version of Oslom [12] has been release. While such a version provies better results than the previous ones, it seems that the time neee to obtain the results is more important. 12

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