Spectral Clustering with Epidemic Diffusion

APS/123-QED Sp ectral Clustering with Epidemic Diffusion Laura M. Smith 1 , Kristina Lerman 2 , Cristina Garcia-Cardona 3 , Allon G. Percus 3 , and Rumi Ghosh 4 1. California State University, F ul lerton, California 2. USC Information Scienc es Institute, Marina del R ey, California 90292 3. Clar emont Graduate University 4. HP L abs, Page Mil l R o ad, Palo Alto, California (Dated: Septem b er 10, 2018) Sp ectral clustering is widely used to partition graphs in to distinct mo dules or communities. Ex- isting methods for sp ectral clustering use the eigenv alues and eigenv ectors of the graph Laplacian, an op erator that is closely associated with random walks on graphs. W e prop ose a new spectral partitioning metho d that exploits the properties of epidemic diffusion. An epidemic is a dynamic pro cess that, unlike the random walk, sim ultaneously transitions to all the neigh b ors of a giv en no de. W e sho w that the replicator, an operator describing epidemic diffusion, is equiv alent to the symmetric normalized Laplacian of a reweigh ted graph with edges rew eighted by the eigenv ector cen tralities of their incident nodes. Thus, more weigh t is giv en to edges connecting more cen tral no des. W e describe a metho d that partitions the no des based on the comp onen twise ratio of the replicator’s second eigenv ector to the first, and compare its p erformance to traditional sp ectral clus- tering tec hniques on synthetic graphs with kno wn communit y structure. W e demonstrate that the replicator giv es preference to dense, clique-lik e structures, enabling it to more effectively discov er comm unities that may b e obscured by dense intercomm unity linking. P ACS num b ers: 05.45.Xt, 89.75.Hc, 89.75.-k, 89.65.Ef, 89.75.Fb, 02.10.Ud I. INTR ODUCTION Graph partitioning is used in man y applications, in- cluding comm unity detection [1], image segmen tation [2], and data mining [3], where it is necessary to partition a graph into mo dules or clusters of similar, or similarly b eha ving, no des. Spectral partitioning uses the eigen- v ectors associated with the k smallest eigen v alues of the graph Laplacian matrix (or its normalized version) to partition the graph into k clusters [4–7]. Existing metho ds for spectral partitioning are closely asso ciated with random w alks on graphs. A random w alk is a sto c hastic dynamic pro cess where transitions take place from a no de to a random neigh b or of that no de, and it is describ ed by the (normalized) graph Laplacian. The existence of a go o d partition implies that random walks tak e a long time to reach a stationary distribution on the graph [2, 8], because they sp end a long time within a mo dule and seldom pass b et w een modules [9]. This forms a basis for ob jectiv e functions used to select which edges to cut so as to partition the graph, suc h as normalized cut and conductance, though these functions hav e trouble partitioning real-w orld graphs where man y inter-module edges obscure the underlying structure [1]. Epidemic diffusion is another t yp e of dynamic pro cess on a graph. An epidemic undergo es transitions simulta- neously to all the neighbors of a giv en node, rather than a single neigh b or, and is often used to mo del the spread of a virus or an innov ation through a so cial net work [10, 11]. Recen tly , Lerman and Ghosh in tro duced the replicator matrix [12], an analog of the graph Laplacian, to describ e epidemic diffusion on graphs. They used the replicator to simulate dynamics of synchronization in a net work of oscillators, showing that oscillators coupled via epidemic diffusion sync hronize into different structures than oscil- lators coupled via random walk-lik e diffusion. W e prop ose a metho d for sp ectral graph partition- ing based on epidemic diffusion. First, w e show that the replicator is equiv alent to the symmetric normalized Laplacian of a reweigh ted graph, where new edge weigh ts are the product of old edge weigh ts and the eigen vector cen tralities of the tw o end p oints. The eigenv ector cen- tralit y [13] of a graph is given b y the eigen vector corre- sp onding to the largest eigenv alue of the adjacency ma- trix. Therefore, edges linking central no des are given a higher w eigh t b y the reweigh ting scheme. The equiv alence b et ween the replicator and symmetric normalized Laplacian of a reweigh ted graph allows us to exploit well-kno wn relationships b etw een sp ectral clus- tering and graph partitioning. T o use the replicator for sp ectral partitioning, we give a computationally efficien t pro cedure that orders no des based on the comp onen t- wise ratio of the second to first eigenv ectors and selects a partition that minimizes a quality function computed on the rew eigh ted graph. This tends to preserve dense structures, since edges linking more central no des in suc h dense clusters are less likely to b e cut. W e study the p erformance of the prop osed sp ectral partitioning metho d using syn thetic graphs with known comm unity structure. W e demonstrate that sp ectral clustering based on epidemics leads to a b etter recov ery of ground truth communities than traditional metho ds based on the graph Laplacian, esp ecially in graphs that are more challenging b ecause of the presence of many edges b et ween clusters. Our w ork suggests that epidemic diffusion can b e a useful probe of graph structure, as it can illuminate properties of graphs that are distinct from those found by metho ds based on the random walk. 2 I I. SPECTRAL CLUSTERING An unw eighted graph G = ( V , E ), with vertices (or no des) V and edges (or links) E , can be represented by a | V | × | V | adjacency matrix A , with A ij = 1 if the edge ( i, j ) ∈ E , and A ij = 0 otherwise. By con v ention A ii = 0. W e consider undirected graphs, where A ij = A j i . The degree of no de i is defined as the n umber of edges incident on it, d i = P j A ij . Other useful constructs are D , a diagonal degree matrix where D ii = d i , and the iden tity matrix I . A. Graph Laplacian and Sp ectral Clustering The graph Laplacian matrix is defined as L = D − A . The eigen v alues and eigen vectors of L capture man y prop erties of the graph. In the simplest case, if the graph has k disjoin t comp onen ts, the k smallest eigen v alues of L are zero, and the asso ciated eigenv ectors are indicator functions assigning nodes to their resp ective cluster or comm unity [7]. Ev en if the k smallest eigen v alues are not all zero, their corresp onding eigenv ectors can b e used to partition no des into k clusters b y pro jecting these no des on to a subspace of the first k eigenv ectors and using stan- dard clustering tec hniques such as k -means [2, 5]. The simplest spectral clustering metho d, spectral bisection, partitions no des based on the v alues of the second eigen- v ector v of the adjacency matrix or the graph Laplacian. A splitting v alue c is used to divide the nodes into dif- feren t clusters based on whether v i < c or v i ≥ c [6]. A range of splitting v alues ha v e b een used, including zero, the median v alue within the v ector, the largest gap, and the v alue pro ducing the b est ratio cut, b est conduc- tance [14], or another measure. In practice, normalized versions of the graph Lapla- cian pro duce better results in sp ectral clustering appli- cations [3, 5]. Two examples are the symmetric normal- ized Laplacian L s = I − D − 1 / 2 AD − 1 / 2 and the random w alk Laplacian L rw = I − D − 1 A , so named b ecause the matrix of transition probabilities for a random walk on a graph is given by D − 1 A . B. Graph Cuts and Their Quality Measures In tuitively , a cluster is a set of nodes S ⊂ V that are more tigh tly connected to each other than to nodes out- side of the cluster. W e use ¯ S = V \ S to denote the complemen t of S , which consists of no des that are not in S . In order to bisect the graph into tw o disjoint clusters, one typically wan ts to minimize the n umber of cut edges b et ween clusters, E ( S, ¯ S ) = X i ∈ S,j ∈ ¯ S A ij , while maximizing cluster size, whic h may b e measured b y the n umber of no des it contains, | S | , or the sum of the degrees of the no des in the set, ν ( S ) = P i ∈ S d i , also called vol ume of the set. Sev eral functions hav e b een prop osed for measuring the quality of a graph cut. The b est kno wn of these are ratio cut R ( S ) and normalized cut N ( S ): R ( S ) =  1 | S | + 1 | ¯ S |  E ( S, ¯ S ) (1) N ( S ) =  1 ν ( S ) + 1 ν ( ¯ S )  E ( S, ¯ S ) . (2) There is a relationship betw een graph cuts and sp ectral clustering. Deciding which edges to cut to optimize an y of these qualit y functions is an NP-complete problem. Sp ectral clustering solv es a relaxation of the problem, where the discrete indicator v ariables that assign no des to clusters b ecome contin uous. Although in general there are no useful b ounds for the appro ximation pro duced by this relaxation [7], in practice it often provides a simple and effective clustering metho d. Solutions to the relaxed optimization problem are given by the second eigen vector of the graph Laplacian L or the normalized graph Lapla- cian L s [6]. Relaxing ratio cut leads to sp ectral clustering using L , while relaxing normalized cut leads to sp ectral clustering using L s [2, 7]. Such relaxation methods hav e also been applied productively to the p opular modularity maximization metho d for comm unity detection [15, 16]. By analogy with sp ectral bisection [6], the leading eigen- v ector approac h assigns no des to clusters based on the sign of the comp onents of the leading eigenv ector of the mo dularit y matrix. C. Sp ectral Clustering and Random W alks There exists a further relationship b etw een sp ectral clustering, the partition quality function, and prop erties of random walks. A random walk on a graph is a sto c has- tic process where transitions take place to a randomly c hosen neigh b or of a giv en node. Cluster properties of the graph can b e expressed in terms of the transition matrix D − 1 A [17] of a random walk. Sp ectral clustering finds a partition suc h that a random w alk stays within the same cluster for a long time and seldom jumps b e- t ween clusters [2, 9]. Therefore, the presence of a go o d partition (low normalized cut v alue) implies that it will tak e a random walk a long time to reach its equilibrium distribution. I II. EPIDEMIC DIFFUSION ON GRAPHS An epidemic is a dynamic pro cess that sim ultaneously undergo es transitions to every neighbor of the current no de. Epidemics are used to mo del the spread of dis- ease [18] and innov ation [11] in so cial netw orks. Epi- demics differ from random walks in important wa ys. 3 First, rather than choosing a single neighbor to tran- sition to or “infect” as the random w alk do es, an epi- demic will attempt to “infect” every neighbor of a node. In a random walk, the probability of finding the w alker in a given lo cation is a conserv ed quantit y that diffuses through the graph, and the random walk transition ma- trix is a stochastic matric. Epidemics, on the other hand, replicate themselv es with each successful transmission, without follo wing a conserv ation law [12]. Lerman and Ghosh [12] introduced the replicator op- erator R = λ max I − A to describ e dynamics of syn- c hronization in a net work of no des coupled via epidemic diffusion. Here λ max is the largest eigen v alue of A , also kno wn as the epidemic threshold [19]. In this system, a dynamic v ariable u i asso ciated with node i can c hange its v alue based on the v alues of its neighbors according to: d u dt = − Ru , (3) where R replaces the Laplacian used in the analogous heat equation that gives the (diffusive) evolution of a random walk on a graph [20]. By construction, the repli- cator has a steady state given b y θ , the eigenv ector of A asso ciated with λ max : Aθ = λ max θ . θ is also known as the eigenve ctor c entr ality [13], and was introduced by Bonacic h to explain the imp ortance of actors in a so cial net work based on the imp ortance of the actors to which they were connected. Clusters of no des with similar v alues of the dynamic v ariable u emerge as the system of coupled no des ev olves to wards the steady state [12]. This motiv ates a commu- nit y detection metho d with no des classified according to the rate of conv ergence to their steady-state v alues. F or large time t , we approximate the solution to Eq. 3 using the tw o leading eigenv ectors θ and ψ of R , u i ( t ) ≈ c 1 θ i + c 2 e − λ 2 t ψ i = c 1 θ i  1 + c 2 c 1 e − λ 2 t ψ i θ i  , where c 1 and c 2 are constants, and λ 2 is the second small- est eigen v alue of R associated with eigenv ector ψ , guar- an teed to b e nonzero if the graph is connected. Therefore, con vergence depends on ψ i /θ i , the comp onen twise ratio of the second to fir st eigen v ectors. Note that eigen v e ctors of R corresp onding to R ’s t wo smallest eigen v alues are the same as the eigen vectors of A corresp onding to A ’s t wo largest eigen v alues. A. Replicator as the Symmetric Normalized Laplacian of a Rew eighted Graph In a so cial net work, one migh t exp ect nodes of high “imp ortance” to attract other no des, resulting in com- m unities forming around no des with large eigenv ector cen trality v alues θ i . In this section w e propose a mo difi- cation of our graph, conv erting the unw eighted netw ork in to a weigh ted one where weigh ts are given b y the pro d- uct of the eigenv ector cen tralities of an edge’s end p oin ts: ˜ A ij = A ij θ i θ j . Moreo ver, we show that the replicator on the un weigh ted graph given b y A is in fact exactly equiv alent to the symmetric normalized Laplacian of the rew eighted graph giv en b y e A . In the rew eighted graph, the degree of no de i is given b y ˜ d i = X j A ij θ i θ j = θ i X j A ij θ j = λ max θ 2 i . F or con venience, define Θ as the diagonal matrix whose elemen ts are the comp onents of eigenv ector θ , i.e., Θ ii . Then, from ˜ A ij and ˜ d i ab o ve, e A = Θ A Θ and e D = λ max Θ 2 . (4) W e can no w write the symmetric normalized Laplacian of the reweigh ted graph: e L s = I − e D − 1 / 2 e A e D − 1 / 2 = I −  1 √ λ max Θ − 1  Θ A Θ  1 √ λ max Θ − 1  = I − 1 λ max A = 1 λ max R . Hence, R = λ max e L s . The equiv alence betw een epidemics and the diffusive pro cess of random w alks is at first surprising. Diffusiv e pro cesses conserv e the total amount of the substance dif- fusing, whereas no such conserv ation law holds for epi- demics [12]. The intuition for the equiv alence of the tw o pro cesses is the following. A no de’s eigenv ector central- it y giv es the num b er of paths connecting it to all other no des in the graph [21]; hence, the pro duct of eigen v ector cen tralities of a pair of no des captures how muc h of the substance is newly created when the epidemic follo ws the edge linking the pair. By enco ding the amount of non- conserv ation in edge reweigh ting, this sc heme allows the epidemic to b e reduced to diffusion. B. Qualit y Measure for the Replicator The equiv alence pro ved ab o ve allows us to exploit the prop erties of the symmetric normalized Laplacian, along with its relationship to graph partitioning, for epi- demic diffusion. Since the replicator is simply L s of the rew eighted graph e A , sp ectral clustering using the repli- cator corresp onds to a relaxation of normalized cut on this rew eighted graph. The appropriate measure for as- sessing graph cut quality with the replicator is therefore normalized cut on the reweigh ted graph e N ( S ). 4 C. An Illustrativ e Example W e use a simple example to highlight the differences b et ween traditional graph partitioning and one based on epidemics. Consider the graph in Figure 1, which sho ws a dense cluster connected through no de 6 to a sparsely link ed cluster. Such a configuration is common in so cial net works, where a high-degree hub linking different com- m unities may obscure communit y b oundaries. W e ex- p ect a go o d partition to group no de 6 with other no des in its clique. Ho wev er, the cut ( B ) that minimizes nor- malized cut ( N ( S )) groups no de 6 with nodes 1–5 and assigns nodes 7–11 to the other cluster. Multiple cuts minimize ratio cut ( R ( S )), including one that groups to- gether no des 3–5. No de 6 has the highest eigen vector cen trality . F ur- thermore, no des that b elong to the clique hav e higher cen trality v alues than other nodes. Consequen tly , in the r eweighte d graph, the edges linking node 6 to the rest of the clique are more “exp ensive” to cut, and no des 6–11 are group ed together b y the preferred cut ( A ) that min- imizes both the ratio cut e R ( S ) and the normalized cut e N ( S ) on the reweigh ted graph. The qualit y measures of the cuts are sho wn in the table in Fig. 1. By giving edges linking central no des a higher weigh t, epidemic- based graph partitioning thus preserves dense, clique-like structures. Accordingly , deleting these edges will hav e the greatest impact on reducing the spread of an epi- demic [22]. Qualit y Cut A Cut B Original gr aph R ( S ) 1.83 1.83 N ( S ) 0.528 0.417 R eweighte d gr aph e R ( S ) 11.4 32.3 e N ( S ) 0.747 0.778 FIG. 1. (Color Online)(Left) An example graph. The p ossible cuts are sho wn b y the dotted curv es A and B. (Right) Qualit y measures of cuts A and B on the original and reweigh ted graph. D. Efficien t Sp ectral P artitioning W e now describ e an efficient metho d for sp ectral clus- tering using epidemic diffusion based on sp ectral bisec- tion [6]. First, w e create a v ector v that is the comp onen- t wise ratio of the second eigen vector ψ to the first eigen- v ector θ of the operator R and sort its v alues. Next, we examine all N − 1 cuts in this ordering (where N = | V | ) and pick one corresp onding to the partition that mini- mizes an appropriate qualit y measure. The qualit y mea- sure we use with R is normalized cut on the rew eighted graph ( e N ( S )). W e compare the resulting partition with those pro duced b y applying an analogous splitting pro- cedure to L , with qualit y measure R ( S ), and L s , with qualit y measure N ( S ) (on the original graph). The prop osed optimization pro cedure is exhaustive, since it tests all N − 1 possible cuts within the ordering pro duced by v . It may seem that there w ould be some loss in accuracy from restricting our searc h to cuts in a one-dimensional pro jection, rather than searc hing ov er the entire subspace spanned by the first tw o eigenv ectors θ and ψ . Ho wev er, it has b een observed [2, 23] that the comp onen twise ratio of the second to first eigenv ector of L s is precisely equal to the second eigenv ector of the random walk Laplacian L rw , whose first eigenv ector is a constan t v ector. Th us, our algorithm is effective b ecause it is a computationally efficien t pro cedure for finding the b est normalized cut in the tw o-dimensional eigenspace of e L rw , i.e., L rw on the reweigh ted graph. The adv antages of using L rw in sp ectral clustering are discussed in [7]. IV. EV ALUA TION ON SYNTHETIC GRAPHS W e use syn thetic graphs to gain b etter insigh t into the differences b etw een op erators L , L s , and R and the characteristics of graphs for which differen t op era- tors find b etter solutions. Lancic hinetti and F ortunato ha ve prop osed an algorithm to generate random graphs with known hierarchical comm unity structure [24]. The N nodes are divided into macro communities, which are themselv es comp osed of micro communities, and then edges b et ween no des are created using mixing parame- ters µ 1 and µ 2 . The parameter µ 1 designates the fraction of a node’s edges that will connect to no des in a differ- en t macro comm unity , and µ 2 giv es the fraction of edges that will connect to no des in a differen t micro comm u- nit y within the same macro communit y . The remaining (1 − µ 1 − µ 2 ) fraction of edges link to other nodes within the same micro and macro communities. These b enc h- mark net w orks allo w us to systematically explore th e p er- formance of different sp ectral clustering approaches. FIG. 2. Eac h pixel represen ts the mean av erage clustering co efficien t (left) and the standard deviation (right) across 1 00 runs for fixed ( µ 1 , µ 2 ). Using soft w are av ailable on [25], we generated 100 graphs for each set of parameter v alues. W e to ok N = 5 100 with t wo macro communities. W e v aried µ 1 and µ 2 b et ween 0 and 0 . 5. The a verage clustering co efficient ranged betw een 0.23 and 0.6421, suggesting that the syn- thetic graphs ha ve properties similar to those often found in real world netw orks [26]. W e partition each benchmark graph using L , L s , and R b y minimizing their respective quality measures. T o ev aluate the resulting partitions, we use the Normalized Mutual Information (NMI) measure [27], which compares the partition to the ground truth comm unities. When the v alue of this measure is 1.0, the partitioning metho d has successfully recov ered the underlying comm unity struc- ture. W e calculate the av erage and standard deviation of the NMI scores for a fixed set of parameters and display the results in Figure 3. Laplacian L : Ratio Cut Symmetric Normalized Laplacian L s : Normalized Cut Replicator R : Normalized Cut (reweigh ted) FIG. 3. NMI scores for minimizing the resp ectiv e op erators’ qualit y measure. Each pixel represents the av erage (left) or standard deviation (right) NMI score across 100 runs for fixed ( µ 1 , µ 2 ). As the prop ortion of a no de’s edges that connect to individuals in the opposite comm unity , µ 1 , increases, it b ecomes more difficult to divide the netw ork in to the correct comm unities. W e find that L and L s giv e b et- ter results when µ 1 is small (very few links betw een the t wo comm unities). As µ 1 increases, R dominates with a higher NMI score. Additionally , R has the low est stan- dard deviation of the three operators, indicating a consis- ten t performance in identifying the underlying comm u- nities. V. CONCLUSION Sp ectral partitioning traditionally uses the graph Laplacian. In this paper, w e hav e introduced a method for spectral partitioning using the replicator, an op erator describing epidemic diffusion on graphs. W e hav e sho wn that this op erator is equiv alent to the symmetric nor- malized Laplacian on a different graph, where edges are rew eighted according to the eigenv ector cen trality mea- sure. By reweigh ting the edges, a higher weigh t is placed on globally imp ortant nodes. Thus, this metho d tends to preserve cliques and other dense clusters. W e hav e in tro duced a sp ectral bisection approach based on the component wise ratio of the second to the first eigenv ector of R , choosing the partition b y splitting the sorted v ector so as to minimize an appropriate qualit y measure. Comparing the performance of different meth- o ds on synthetic graphs with known communit y struc- ture, w e hav e shown that sp ectral partitioning using the replicator is b etter able to reco ver the underlying com- m unity structure, esp ecially in cases where more edges b et ween the tw o macro communities make it more diffi- cult for the Laplacian and symmetric normalized Lapla- cian to iden tify comm unities. By reweigh ting the edges using eigenv ector cen tralit y , the replicator assigns more imp ortance to cen tral nodes. Thus, the edges that pass b et ween clusters are giv en less influence if they do not link no des of high centralit y . By limiting the cuts to influen tial edges, the metho d leads to a more accurate reconstruction of the communit y structure. Ac knowledgemen ts The authors are grateful to Arjuna Flenner, Yv es v an Gennip, and Blak e Hun ter for many instructiv e con ver- sations and suggestions. KL and RG are also greatly in- debted to Shanghua T eng, whose insights and en thusiasm con tinue to inspire them. This work has been funded by the Air F orce Office of Scientific Researc h under con tracts F A9550-10-1-0569 and F A9550-10-1-0102, D ARP A under Con tract No. W911NF-12-1-0034, the National Science F oundation under grant 1217605, and the Department of Energy Office of Science Adv anced Computing Research (ASCR) program in Applied Mathematics. 6 [1] J. Lesk ov ec, K. J. Lang, A. Dasgupta, and M. W. Ma- honey , in Pr oc e e dings of 17th International World Wide Web Confer enc e (WWW2008) (2008). [2] J. Shi and J. Malik, IEEE T ransactions on Pattern Anal- ysis and Machine Intelligence 22 (8), 888 (2000). [3] A. Bertozzi and A. Flenner, Multiscale Mo deling & Sim- ulation 10 (3), 1090 (2012). [4] F. R. K. Ch ung, Sp ectr al Gr aph The ory , vol. 92 of CBMS R e gional Confer enc e Series in Mathematics (American Mathematical So ciet y , 1996). [5] A. Y. Ng, M. I. Jordan, and Y. W eiss, in A dvanc es in Neur al Information Pr o c essing Systems (2001), v ol. 14, pp. 849–856. [6] D. A. Spielman and S.-H. T eng, Linear Algebra and its Applications 421 (2-3), 284 (Mar. 2007). [7] U. von Luxburg, Statistics and Computing 17 (4), 395 (2007). [8] M. Jerrum and A. Sinclair, in Pr o c. ACM Symp osium on The ory of Computing (1988), STOC ’88, pp. 235–244. [9] M. Rosv all and C. T. Bergstrom, Pro c. National Academ y of Sciences 105 (4), 1118 (2008). [10] R. M. Anderson and R. Ma y , Infe ctious dise ases of hu- mans: dynamics and contr ol (Oxford Univ ersit y Press, 1991). [11] E. M. Rogers, Diffusion of Innovations, 5th Edition (F ree Press, 2003). [12] K. Lerman and R. Ghosh, Phys. Rev. E 86 (026108) (2012). [13] P . Bonacich and P . Llo yd, So cial Netw orks 23 (3), 191 (2001). [14] R. Kannan and S. V empala, Sp e ctr al Algorithms , v ol. 4 of F oundations and T r ends in The or etic al Computer Science (Publishers Inc., Hanov er, MA, 2009). [15] M. E. J. Newman, Pro c. National Academy of Sciences 103 (23), 8577 (2006). [16] S. F ortunato, Physics Reports 486 , 75 (2010). [17] L. Lov´ asz, Random Walks on Gr aphs: A Survey (1993), pp. 353–397. [18] H. W. Hethcote, SIAM Review 42 (4), 599 (2000). [19] Y. W ang, D. Chakrabarti, C. W ang, and C. F aloutsos, in Pr o c. 22nd Symp osium on R eliable Distribute d Systems (IEEE, 2003), pp. 25–34. [20] F. Chung, Pro c. National Academy of Sciences 104 (50), 19735 (2007). [21] R. Ghosh and K. Lerman, Phys. Rev. E 83 (6), 066118 (2011). [22] H. T ong, B. A. Prak ash, T. Eliassi-Rad, M. F alout- sos, and C. F aloutsos, in Pr o c. 21st A CM International Confer enc e on Information and know le dge management (2012), CIKM ’12, pp. 245–254. [23] Y. W eiss, in Pr o c e e dings IEEE International Confer enc e on Computer Vision (1999), pp. 975–982. [24] A. Lancic hinetti and S. F ortunato, Ph ys. Rev. E 80 (1) (2009). [25] S. F ortunato, Benchmark gr aphs to test c ommunity dete c- tion algorithms (October 2011), https://sites.google. com/site/santofortunato/inthepress2 . [26] D. J. W atts, The American Journal of So ciology 105 (2), 493 (1999). [27] L. Danon, A. D ´ ıaz-Guilera, J. Duch, and A. Arenas, Journal of Statistical Mechanics: Theory and Exp eri- men t (2005).