Finding community structures in complex networks using mixed integer optimisation

Abstract.The detection of community structure has been used to reveal the relationships between individual objects and their groupings in networks. This paper presents a mathematical programming approach to identify the optimal community structures in complex networks based on the maximisation of a network modularity metric for partitioning a network into modules. The overall problem is formulated as a mixed integer quadratic programming (MIQP) model, which can then be solved to global optimality using standard optimisation software. The solution procedure is further enhanced by developing special symmetry-breaking constraints to eliminate equivalent solutions. It is shown that additional features such as minimum/maximum module size and balancing among modules can easily be incorporated in the model. The applicability of the proposed optimisation-based approach is demonstrated by four examples. Comparative results with other approaches from the literature show that the proposed methodology has superior performance while global optimum is guaranteed.

[1]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[2]  Gary Klein,et al.  Optimal clustering: A model and method , 1991 .

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

[4]  Fang Wu,et al.  Finding communities in linear time: a physics approach , 2003, ArXiv.

[5]  Petter Holme,et al.  Subnetwork hierarchies of biochemical pathways , 2002, Bioinform..

[6]  A. Arenas,et al.  Community detection in complex networks using extremal optimization. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

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

[9]  Hawoong Jeong,et al.  Random field Ising model and community structure in complex networks , 2005, cond-mat/0502672.

[10]  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.

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

[12]  Alex Pothen,et al.  PARTITIONING SPARSE MATRICES WITH EIGENVECTORS OF GRAPHS* , 1990 .

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

[14]  Stefan Boettcher,et al.  Extremal Optimization for Graph Partitioning , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  A Díaz-Guilera,et al.  Self-similar community structure in a network of human interactions. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[17]  Alexander Rives,et al.  Modular organization of cellular networks , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[18]  A. Medus,et al.  Detection of community structures in networks via global optimization , 2005 .

[19]  D. Parisi,et al.  Self-contained algorithms to detect communities in networks , 2004 .

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

[21]  David Lusseau,et al.  The emergent properties of a dolphin social network , 2003, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[22]  D. Lusseau,et al.  The bottlenose dolphin community of Doubtful Sound features a large proportion of long-lasting associations , 2003, Behavioral Ecology and Sociobiology.

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

[24]  Donald E. Knuth,et al.  The Stanford GraphBase - a platform for combinatorial computing , 1993 .

[25]  P. Bork,et al.  Genome evolution reveals biochemical networks and functional modules , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[26]  Claudio Castellano,et al.  Defining and identifying communities in networks. , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[27]  Stefan Bornholdt,et al.  Detecting fuzzy community structures in complex networks with a Potts model. , 2004, Physical review letters.

[28]  C. Lee Giles,et al.  Self-Organization and Identification of Web Communities , 2002, Computer.

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

[31]  Douglas H. Fisher,et al.  Iterative Optimization and Simplification of Hierarchical Clusterings , 1996, J. Artif. Intell. Res..

[32]  Jean-Pierre Eckmann,et al.  Curvature of co-links uncovers hidden thematic layers in the World Wide Web , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[33]  Mark Newman,et al.  Detecting community structure in networks , 2004 .

[34]  R. Guimerà,et al.  Functional cartography of complex metabolic networks , 2005, Nature.

[35]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[36]  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.

[37]  Sophia Tsoka,et al.  Robustness of the p53 network and biological hackers , 2005, FEBS letters.

[38]  Mika Gustafsson,et al.  Comparison and validation of community structures in complex networks , 2006 .