Detecting community structure using biased random merging

Abstract Decomposing a network into small modules or communities is beneficial for understanding the structure and dynamics of the network. One of the most prominent approaches is to repeatedly join communities together in pairs with the greatest increase in modularity so that a dendrogram that shows the order of joins is obtained. Then the community structure is acquired by cutting the dendrogram at the levels corresponding to the maximum modularity. However, there tends to be multiple pairs of communities that share the maximum modularity increment and the greedy agglomerative procedure may only merge one of them. Although the modularity function typically admits a lot of high-scoring solutions, the greedy strategy may fail to reach any of them. In this paper we propose an enhanced data structure in order to enable diverse choices of merging operations in community finding procedure. This data structure is actually a max-heap equipped with an extra array that stores the maximum modularity increments; and the corresponding community pairs is merged in the next move. By randomly sampling elements in this head array, additional diverse community structures can be efficiently extracted. The head array is designed to host the community pairs corresponding to the most significant increments in modularity so that the modularity structures obtained out of the sampling exhibit high modularity scores that are, on the average, even greater than what the CNM algorithm produces. Our method is tested on both real-world and computer-generated networks.

[1]  Ulrik Brandes,et al.  On Modularity Clustering , 2008, IEEE Transactions on Knowledge and Data Engineering.

[2]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[3]  Brian W. Kernighan,et al.  An efficient heuristic procedure for partitioning graphs , 1970, Bell Syst. Tech. J..

[4]  M E J Newman,et al.  Finding and evaluating community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  E A Leicht,et al.  Community structure in directed networks. , 2007, Physical review letters.

[6]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[7]  M E J Newman,et al.  Modularity and community structure in networks. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[8]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[9]  M. Newman.  Analysis of weighted networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  M E Newman,et al.  Scientific collaboration networks. I. Network construction and fundamental results. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Benjamin H. Good,et al.  Performance of modularity maximization in practical contexts. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Jari Saramäki,et al.  Small But Slow World: How Network Topology and Burstiness Slow Down Spreading , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[14]  K. Kaski,et al.  Limited resolution in complex network community detection with Potts model approach , 2006 .

[15]  M E J Newman,et al.  Community structure in social and biological networks , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[16]  Santo Fortunato,et al.  Community detection in graphs , 2009, ArXiv.

[17]  Luonan Chen,et al.  Quantitative function for community detection. , 2008 .

[18]  Xiang-Sun Zhang,et al.  Modularity optimization in community detection of complex networks , 2009 .

[19]  Alex Arenas,et al.  Synchronization reveals topological scales in complex networks. , 2006, Physical review letters.

[20]  Xin-She Yang,et al.  Introduction to Algorithms , 2021, Nature-Inspired Optimization Algorithms.

[21]  Leon Danon,et al.  Comparing community structure identification , 2005, cond-mat/0505245.

[22]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[23]  T. Vicsek,et al.  Uncovering the overlapping community structure of complex networks in nature and society , 2005, Nature.

[24]  Jian-Guo Liu,et al.  Detecting community structure in complex networks via node similarity , 2010 .

[25]  Christophe Andrieu,et al.  A tutorial on adaptive MCMC , 2008, Stat. Comput..

[26]  Réka Albert,et al.  Near linear time algorithm to detect community structures in large-scale networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  N. Alves.  Unveiling community structures in weighted networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[29]  M E J Newman,et al.  Fast algorithm for detecting community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Albert-László Barabási,et al.  Internet: Diameter of the World-Wide Web , 1999, Nature.

[31]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[32]  W. Zachary,et al.  An Information Flow Model for Conflict and Fission in Small Groups , 1977, Journal of Anthropological Research.

[33]  M. Newman,et al.  Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  A. Arenas,et al.  Models of social networks based on social distance attachment. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Andrea Lancichinetti,et al.  Community detection algorithms: a comparative analysis: invited presentation, extended abstract , 2009, VALUETOOLS.

[36]  S. Fortunato,et al.  Resolution limit in community detection , 2006, Proceedings of the National Academy of Sciences.

[37]  M. Newman,et al.  Finding community structure in very large networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  F. Radicchi,et al.  Benchmark graphs for testing community detection algorithms. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Karl Branting,et al.  Information Theoretic Criteria for Community Detection , 2008, SNAKDD.

[40]  M. Newman,et al.  The structure of scientific collaboration networks. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[41]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[42]  Duanbing Chen,et al.  A fast and efficient heuristic algorithm for detecting community structures in complex networks , 2009 .

[43]  M. Newman,et al.  Finding community structure in networks using the eigenvectors of matrices. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.