{"title": "Scalable Inference of Overlapping Communities", "book": "Advances in Neural Information Processing Systems", "page_first": 2249, "page_last": 2257, "abstract": "We develop a scalable algorithm for posterior inference of overlapping communities in large networks. Our algorithm is based on stochastic variational inference in the mixed-membership stochastic blockmodel. It naturally interleaves subsampling the network with estimating its community structure. We apply our algorithm on ten large, real-world networks with up to 60,000 nodes. It converges several orders of magnitude faster than the state-of-the-art algorithm for MMSB, finds hundreds of communities in large real-world networks, and detects the true communities in 280 benchmark networks with equal or better accuracy compared to other scalable algorithms.", "full_text": "Scalable Inference of Overlapping Communities\n\nPrem Gopalan David Mimno Sean M. Gerrish Michael J. Freedman David M. Blei\n\n{pgopalan,mimno,sgerrish,mfreed,blei}@cs.princeton.edu\n\nDepartment of Computer Science\n\nPrinceton University\nPrinceton, NJ 08540\n\nAbstract\n\nWe develop a scalable algorithm for posterior inference of overlapping communi-\nties in large networks. Our algorithm is based on stochastic variational inference\nin the mixed-membership stochastic blockmodel (MMSB). It naturally interleaves\nsubsampling the network with estimating its community structure. We apply our\nalgorithm on ten large, real-world networks with up to 60,000 nodes. It converges\nseveral orders of magnitude faster than the state-of-the-art algorithm for MMSB,\n\ufb01nds hundreds of communities in large real-world networks, and detects the true\ncommunities in 280 benchmark networks with equal or better accuracy compared\nto other scalable algorithms.\n\n1\n\nIntroduction\n\nA central problem in network analysis is to identify communities, groups of related nodes with\ndense internal connections and few external connections [1, 2, 3]. Classical methods for community\ndetection assume that each node participates in a single community [4, 5, 6]. This assumption is\nlimiting, especially in large real-world networks. For example, a member of a large social network\nmight belong to overlapping communities of co-workers, neighbors, and school friends.\nTo address this problem, researchers have developed several methods for detecting overlapping com-\nmunities in observed networks. These methods include algorithmic approaches [7, 8] and probabilis-\ntic models [2, 3, 9, 10]. In this paper, we focus on the mixed-membership stochastic blockmodel\n(MMSB) [2], a probabilistic model that allows each node of a network to exhibit a mixture of\ncommunities. The MMSB casts community detection as posterior inference: Given an observed\nnetwork, we estimate the posterior community memberships of its nodes.\nThe MMSB can capture complex community structure and has been adapted in several ways [11,\n12]; however, its applications have been limited because its corresponding inference algorithms\nhave not scaled to large networks [2].\nIn this work, we develop algorithms for the MMSB that\nscale, allowing us to study networks that were previously out of reach for this model. For example,\nwe analyzed social networks with as many as 60,000 nodes. With our method, we can use the\nMMSB to analyze large networks, \ufb01nding approximate posteriors in minutes with networks for\nwhich the original algorithm takes hours. When compared to other scalable methods for overlapping\ncommunity detection, we found that the MMSB gives better predictions of new connections and\nmore closely recovers ground-truth communities. Further, we can now use the MMSB to compute\ndescriptive statistics at scale, such as which nodes bridge communities.\nThe original MMSB algorithm optimizes the variational objective by coordinate ascent, processing\nevery pair of nodes in each iteration [2]. This algorithm is inef\ufb01cient, and it quickly becomes\nintractable for large networks. In this paper, we develop stochastic optimization algorithms [13, 14]\nto \ufb01t the variational distribution, where we obtain noisy estimates of the gradient by subsampling\nthe network.\n\n1\n\n\f(a)\n\n(b)\n\nFigure 1: Figure 1(a) shows communities (see \u00a72) discovered in a co-authorship network of 1,600 re-\nsearchers [16] by an a-MMSB model with 50 communities. The color of author nodes indicates their most\nlikely posterior community membership. The size of nodes indicates bridgeness [17], a measure of participa-\ntion in multiple communities. Figure 1(b) shows a graphical model of the a-MMSB. The prior over multinomial\n\u03c0 is a symmetric Dirichlet distribution. Priors over Bernoulli \u03b2 are Beta distributions.\n\nOur algorithm alternates between subsampling from the network and adjusting its estimate of the\nunderlying communities. While this strategy has been used in topic modeling [15], the MMSB\nintroduces new challenges because the Markov blanket of each node is much larger than that of\na document. Our simple sampler usually selects unconnected nodes (due to sparse real-world net-\nworks). We develop better sampling methods that focus more on the informative data in the network,\ne.g., the observed links, and thus make inference even faster.\n\n2 Modeling overlapping communities\n\nIn this section, we introduce the assortative mixed-membership stochastic blockmodel (a-MMSB),\na statistical model of networks that models nodes participating in multiple communities. The a-\nMMSB is a subclass of the mixed-membership stochastic blockmodel (MMSB) [2].1\nLet y denote the observed links of an undirected network, where yab = 1 if nodes a and b are\nlinked and 0 otherwise. Let K denote the number of communities. Each node a is associated with\ncommunity memberships \u03c0a, a distribution over communities; each community is associated with a\ncommunity strength \u03b2k \u2208 (0, 1), which captures how tightly its members are linked. The probability\nthat two nodes are linked is governed by the similarity of their community memberships and the\nstrength of their shared communities.\nWe capture these assumptions in the following generative process of a network.\n\n1. For each community k, draw community strength \u03b2k \u223c Beta(\u03b7).\n2. For each node a, draw community memberships \u03c0a \u223c Dirichlet(\u03b1).\n3. For each pair of nodes a and b,\n\n(a) Draw interaction indicator za\u2192b \u223c \u03c0a.\n(b) Draw interaction indicator za\u2190b \u223c \u03c0b.\n(c) Draw link yab \u223c Bernoulli(r), where\n\n(cid:26)\u03b2k\n\n\u0001\n\nr =\n\nif za\u2192b,k = za\u2190b,k = 1,\nif za\u2192b (cid:54)= za\u2190b.\n\n(1)\n\n1We use a subclass of the MMSB models that is appropriate for community detection in undirected net-\nworks. In particular, we assume assortativity, i.e., that links imply that nodes are similar. We call this special\ncase the assortative MMSB or a-MMSB. In \u00a72 we argue why the a-MMSB is more appropriate for community\ndetection than the MMSB. We note that our algorithms are immediately applicable to the MMSB as well.\n\n2\n\nBARABASI, AJEONG, HNEWMAN, MKLEINBERG, JYahoo Labs!\"!#$!\"\"#$%\"#$$&\"$$&#$'$!\"#!!\"$!%#!&!%#\"&!'#&!(!)(!*!+,!-.!/0&12!(!345567&8'!595:912#&/2!%;\"$!%;!$!';$!$($$)$ 29h\n8.7h > 67h\n2.8h > 67h\n22.1h > 67h\n-\n10.3d\n-\n2.5d\n-\n5.2d\n9.5d\n-\n\nSOURCE\n[22]\n[16]\n[23]\n[23]\n[23]\n[23]\n[23],[24]\n[25]\n[26]\n[27]\n\nFigure 3: Stochastic a-MMSB (with random pair sampling) scales better and \ufb01nds communities as good as\nbatch a-MMSB on real networks (Top). Strati\ufb01ed random node sampling is an order of magnitude faster than\nother sampling methods on the hep-ph, astro-ph and hep-th2 networks (Bottom).\nsigni\ufb01cantly improves convergence speed on real networks. Third, we compare our algorithm with\nleading algorithms in terms of accuracy on benchmark graphs and ability to predict links.\nWe measure convergence by computing the link prediction accuracy on a validation set. We set\naside a validation and a test set, each having 10% of the network links and an equal number of\nnon-links (see \u00a73.2). We approximate the probability that a link exists between two nodes using\nposterior expectations of \u03b2 and \u03c0. We then calculate perplexity, which is the exponential of the\naverage predictive log likelihood of held-out node pairs.\nFor random pair and strati\ufb01ed random pair sampling, we use a mini-batch size S = N/2 for graphs\nwith N nodes. For the strati\ufb01ed random node sampling, we set the number of non-link sets m = 10.\nBased on experiments, we set the parameters \u03ba = 0.5 and \u03c40 = 1024. We set hyperparameters \u03b1 =\n1/K and {\u03b71, \u03b70} proportional to the expected number of links and non-links in each community.\nWe implemented all algorithms in C++.\nComparing scalability to batch algorithms. AM is an order of magnitude faster than standard\nbatch inference for a-MMSB [2]. Figure 2 shows the time to convergence for four networks3 of\nvarying types, node sizes N and sparsity d. Figure 3 shows test perplexity for batch vs. stochastic\ninference. For many networks, AM learns rapidly during the early iterations, achieving 90% of the\nconverged perplexity in less than 70% of the full convergence time. For all but the two smallest\nnetworks, batch inference did not converge within the allotted time. AM lets us ef\ufb01ciently \ufb01t a\nmixed-membership model to large networks.\nComparing sampling methods. Figure 3 shows that strati\ufb01ed random node sampling converges an\norder of magnitude faster than random node sampling. It is statistically more ef\ufb01cient because the\nobservations in each iteration include all the links of a node and a random sample of its non-links.\n\n3Following [1], we treat the directed citation network hep-th2 as an undirected network.\n\n7\n\ngravity 5Ktime (hours)Perplexity0102030400102030405060onlinebatchhep\u2212th 10K01020304050600102030405060hep\u2212ph 12K0102030400102030405060astro\u2212ph 19K0102030400102030405060hep\u2212ph 12Ktime (hours)Perplexity10152025\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf01234\u25cf\u25cf\u25cf\u25cfstratified nodenodestratified pairpairastro\u2212ph 19Ktime (hours)Perplexity1520253035\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf0510152025\u25cf\u25cf\u25cf\u25cfstratified nodenodestratified pairpairhep\u2212th2 27Ktime (hours)Perplexity152025303540\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf\u25cf010203040\u25cf\u25cf\u25cf\u25cfstratified nodenodestratified pairpair\f(a)\n\n(b)\n\n(c)\n\n(d)\n\nFigure 4: Figures (a) and (b) show that Stochastic a-MMSB (AM) outperforms the Poisson model (PM),\nClique Percolation (CP), and Link Clustering (LC) in accurately recovering overlapping communities in 280\nbenchmark networks [28]. Each \ufb01gure shows results on a binary partition of the 280 networks. Accuracy\nis measured using normalized mutual information (NMI) [28]; error bars denote the 95% con\ufb01dence interval\naround the mean NMI. Figures (c) and (d) show that a-MMSB generalizes to new data better than PM on the\nnetscience and us-air network, respectively. Each algorithm was run with 10 random initializations per K.\nFigure 3 also shows that strati\ufb01ed random pair sampling converges \u223c1x\u20132x faster than random pair\nsampling.\nComparing accuracy to scalable algorithms. AM can recover communities with equal or better\naccuracy than the best scalable algorithms: the Poisson model (PM) [3], Clique percolation (CP) [7]\nand Link clustering (LC) [8]. We measure the ability of algorithms to recover overlapping commu-\nnities in synthetic networks generated by the benchmark tool [28].4 Our synthetic networks re\ufb02ect\nreal-world networks by modeling noisy links and by varying community densities from sparse to\ndense. We evaluate using normalized mutual information (NMI) between discovered communities\nand the ground truth communities [28]. We ran PM and a-MMSB until validation log likelihood\nchanged by less than 0.001%. CP and LC are deterministic, but results vary between parameter\nsettings. We report the best solution for each model.5\nFigure 4 shows results for the 280 synthetic networks split in two ways. AM outperforms PM,\nLC, and CP on noisy networks and networks with sparse communities, and it matches the best\nperformance in the noiseless case and the dense case. CP performs best on networks with dense\ncommunities\u2014they tend to have more k-cliques\u2014but with a larger margin of error than AM.\nComparing predictive accuracy to PM. Stochastic a-MMSB also beats PM [3], the best scal-\nable probabilistic model of overlapping communities, in predictive accuracy. On two networks, we\nevaluated both algorithms\u2019 ability to predict held out links and non-links. We ran both PM and a-\nMMSB until their validation log likelihood changed less than 0.001%. Figures 4(c) and 4(d) show\ntraining and testing likelihood. PM over\ufb01ts, while the a-MMSB generalizes well.\nUsing the a-MMSB as an exploratory tool. AM opens the door to large-scale exploratory analysis\nof real-world networks. In addition to the co-authorship network in Figure 1(a), we analyzed the\n\u201ccond-mat\u201d collaboration network [26] with the number of communities set to 300. This network\ncontains 40,421 scientists and 175,693 links. In the supplement, we visualized the top authors in\nthe network by a measure of their participation in different communities (bridgeness [17]). Finding\nsuch bridging nodes in a network is an important task in disease prevention and marketing.\n\nAcknowledgments\n\nD.M. Blei is supported by ONR N00014-11-1-0651, NSF CAREER 0745520, AFOSR FA9550-09-\n1-0668, the Alfred P. Sloan foundation, and a grant from Google.\n\n4We\n\nof\n\nthese\n\ngenerated\n\n280\n\nK , 0.15 N\n\nK , .., 0.35 N\n\n#nodes\u2208\nnetworks\nfor\ncombinations\n{400}; #communities\u2208{5, 10}; #nodes with at\nleast 3 overlapping communities\u2208{100}; community\nsizes\u2208{equal, unequal}, when unequal,\nK ]; average node\ndegree\u2208 {0.1 N\nK }, the maximum node degree=2\u00d7average node degree; % links of a\nK , 0.4 N\nnode that are noisy\u2208 {0, 0.1}; random runs\u2208{1,..,5}.\n5CP \ufb01nds a solution per clique size; LC \ufb01nds a solution per threshold at which the dendrogram is cut [8] in\nsteps of 0.1 from 0 to 1; PM and a-MMSB \ufb01nd a solution \u2200K \u2208 {k(cid:48), k(cid:48) + 10} where k(cid:48) is the true number\nof communities\u2014increasing by 10 allows for potentially a larger number of communities to be detected; a-\nMMSB also \ufb01nds a solution for each of random pair or strati\ufb01ed random pair sampling methods with the\nhyperparameters \u03b7 set to the default or set to \ufb01t dense clusters.\n\nthe community sizes are in the range [ N\n\nparameters:\n\n2K , 2N\n\n8\n\nNMI0.00.10.20.30.40.50 noiseAMPMLCCP10% noisyAMPMLCCPNMI0.00.10.20.30.40.50.6sparseAMPMLCCPdenseAMPMLCCPheld\u2212out log likelihood\u221210\u22128\u22126\u22124\u22122K=5PMAMK=20PMAMK=40PMAMtrainingtestheld\u2212out log likelihood\u22125\u22124\u22123\u22122\u22121K=5PMAMK=20PMAMK=40PMAMtrainingtest\fReferences\n[1] Santo Fortunato. Community detection in graphs. Physics Reports, 486(35):75\u2013174, 2010.\n[2] E. Airoldi, D. Blei, S. Fienberg, and E. Xing. Mixed membership stochastic blockmodels. Journal of\n\nMachine Learning Research, 9:1981\u20132014, 2008.\n\n[3] Brian Ball, Brian Karrer, and M. E. J. Newman. Ef\ufb01cient and principled method for detecting communities\n\nin networks. Physical Review E, 84(3):036103, 2011.\n\n[4] M. E. J. Newman and M. Girvan. Finding and evaluating community structure in networks. Physical\n\nReview E, 69(2):026113, 2004.\n\n[5] K. Nowicki and T. Snijders. Estimation and prediction for stochastic blockstructures. Journal of the\n\nAmerican Statistical Association, 96(455):1077\u20131087, 2001.\n\n[6] Peter J. Bickel and Aiyou Chen. A nonparametric view of network models and Newman-Girvan and other\n\nmodularities. Proceedings of the National Academy of Sciences, 106(50):21068\u201321073, 2009.\n\n[7] Imre Dernyi, Gergely Palla, and Tams Vicsek. Clique percolation in random networks. Physical Review\n\nLetters, 94(16):160202, 2005.\n\n[8] Yong-Yeol Ahn, James P. Bagrow, and Sune Lehmann. Link communities reveal multiscale complexity\n\nin networks. Nature, 466(7307):761\u2013764, 2010.\n\n[9] M. E. J. Newman and E. A. Leicht. Mixture models and exploratory analysis in networks. Proceedings\n\nof the National Academy of Sciences, 104(23):9564\u20139569, 2007.\n\n[10] A. Goldenberg, A. Zheng, S. Fienberg, and E. Airoldi. A survey of statistical network models. Founda-\n\ntions and Trends in Machine Learning, 2:129\u2013233, 2010.\n\n[11] W. Fu, L. Song, and E. Xing. Dynamic mixed membership blockmodel for evolving networks. In ICML,\n\n2009.\n\n[12] Qirong Ho, Ankur P. Parikh, and Eric P. Xing. A multiscale community blockmodel for network explo-\n\nration. Journal of the American Statistical Association, 107(499):916\u2013934, 2012.\n\n[13] H. Robbins and S. Monro. A stochastic approximation method. The Annals of Mathematical Statistics,\n\n22(3):400\u2013407, 1951.\n\n[14] M. Hoffman, D. Blei, C. Wang, and J. Paisley. Stochastic variational inference. arXiv:1206.7051, 2012.\n[15] M. Hoffman, D. Blei, and F. Bach. Online learning for latent Dirichlet allocation. In NIPS, 2010.\n[16] M. E. J. Newman. Finding community structure in networks using the eigenvectors of matrices. Physical\n\nReview E, 74(3):036104, 2006.\n\n[17] Tams Nepusz, Andrea Petrczi, Lszl Ngyessy, and Flp Bazs. Fuzzy communities and the concept of\n\nbridgeness in complex networks. Physical Review E, 77(1):016107, 2008.\n\n[18] M. Jordan, Z. Ghahramani, T. Jaakkola, and L. Saul. Introduction to variational methods for graphical\n\nmodels. Machine Learning, 37:183\u2013233, 1999.\n\n[19] S. Amari. Differential geometry of curved exponential families-curvatures and information loss. The\n\nAnnals of Statistics, 1982.\n\n[20] M. Morup, M.N. Schmidt, and L.K. Hansen. In\ufb01nite multiple membership relational modeling for com-\n\nplex networks. In IEEE MLSP, 2011.\n\n[21] M. Kim and J. Leskovec. Modeling social networks with node attributes using the multiplicative attribute\n\ngraph model. In UAI, 2011.\n\n[22] RITA. U.S. Air Carrier Traf\ufb01c Statistics, Bur. Trans. Stats, 2010.\n[23] J. Leskovec, J. Kleinberg, and C. Faloutsos. Graph evolution: Densi\ufb01cation and shrinking diameters.\n\nACM TKDD, 2007.\n\n[24] J. Gehrke, P. Ginsparg, and J. M. Kleinberg. Overview of the 2003 KDD cup. SIGKDD Explorations,\n\n5:149\u2013151, 2003.\n\n[25] B. Klimmt and Y. Yang. Introducing the Enron corpus. In CEAS, 2004.\n[26] M. E. J. Newman. The structure of scienti\ufb01c collaboration networks. Proceedings of the National\n\nAcademy of Sciences, 98(2):404\u2013409, 2001.\n\n[27] J. Leskovec, K. J. Lang, A. Dasgupta, and M. W. Mahone. Community structure in large networks:\n\nNatural cluster sizes and the absence of large well-de\ufb01ned cluster. In Internet Mathematics, 2008.\n\n[28] Andrea Lancichinetti and Santo Fortunato. Benchmarks for testing community detection algorithms on\n\ndirected and weighted graphs with overlapping communities. Physical Review E, 80(1):016118, 2009.\n\n9\n\n\f", "award": [], "sourceid": 1114, "authors": [{"given_name": "Prem", "family_name": "Gopalan", "institution": null}, {"given_name": "Sean", "family_name": "Gerrish", "institution": null}, {"given_name": "Michael", "family_name": "Freedman", "institution": null}, {"given_name": "David", "family_name": "Blei", "institution": null}, {"given_name": "David", "family_name": "Mimno", "institution": null}]}