Abstract
Community structure is one of the important characteristics of complex networks. In the recent decade, many models and algorithms have been designed to identify communities in a given network, among which there is a class of methods that globally search the best community structure by optimizing some modularity criteria. However, it has been recently revealed that these methods may either fail to find known qualified communities (a phenomenon called resolution limit) or even yield false communities (the misidentification phenomenon) in some networks. In this paper, we propose a new model which is immune to the above phenomena. The model is constructed by restating community identification as a combinatorial optimization problem. It aims to partition a network into as many qualified communities as possible. This model is formulated as a linear integer programming problem and its NP-completeness is proved. A qualified min-cut based bisecting algorithm is designed to solve this model. Numerical experiments on both artificial networks and real-life complex networks show that the combinatorial model/algorithm has promising performance and can overcome the limitations in existing algorithms.
Similar content being viewed by others
References
Agarwal G, Kempe D (2008) Modularity-maximizing graph communities via mathematical programming. Eur Phys J B 66(3):409–418
Albert R, Barabási A (2002) Statistical mechanics of complex networks. Rev Mod Phys 74(1):47–97
Alimonti P, Kann V (2000) Some APX-completeness results for cubic graphs. Theor Comput Sci 237(1–2):123–134
Arenas A, Fernandez A, Gomez S (2008) Analysis of the structure of complex networks at different resolution levels. New J Phys 10:053039
Cafieri S, Hansen P, Liberti L (2010) Edge ratio and community structure in networks. Phys Rev E 81(2):026105
Danon L, Duch J, Diaz-Guilera A, Arenas A (2005) Comparing community structure identification. J Stat Mech, P09008
Everett M, Borgatti S (1998) Analyzing clique overlap. Connections 21(1):49–61
Fortunato S (2010) Community detection in graphs. Phys Rep 486:75–174
Fortunato S, Barthelemy M (2007) Resolution limit in community detection. Proc Natl Acad Sci USA 104(1):36–41
Girvan M, Newman M (2002) Community structure in social and biological networks. Proc Natl Acad Sci USA 99(12):7821–7826
Gleiser P, Danon L (2003) Community structure in jazz. Adv Complex Syst 6(4):565–573
Guimerà R, Amaral L (2005) Functional cartography of complex metabolic networks. Nature 433(7028):895–900
Hu Y, Chen H, Zhang P, Li M, Di Z Fan Y (2008) Comparative definition of community and corresponding identifying algorithm. Phys Rev E 78(2):026121
Lancichinetti A, Fortunato S, Kertész J (2009) Detecting the overlapping and hierarchical community structure in complex networks. New J Phys 11:033015
Li Z, Zhang S, Wang RS, Zhang XS, Chen L (2008) Quantitative function for community detection. Phys Rev E 77(3):036109
Newman M (2006) Modularity and community structure in networks. Proc Natl Acad Sci USA 103(23):8577–8582
Newman M, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E 69(2):026113
Palla G, Derényi I, Farkas I, Vicsek T (2005) Uncovering the overlapping community structure of complex networks in nature and society. Nature 435(7043):814–818
Radicchi F, Castellano C, Cecconi F, Loreto V, Parisi D (2004) Defining and identifying communities in networks. Proc Natl Acad Sci USA 101(9):2658–2663
Ravasz E, Somera A, Mongru D, Oltvai Z, Barabasi A (2002) Hierarchical organization of modularity in metabolic networks. Science 297(5586):1551–1555
Reichardt J, Bornholdt S (2004) Detecting fuzzy community structures in complex networks with a Potts model. Phys Rev Lett 93(21):218701
Ronhovde P, Nussinov Z (2009) Multiresolution community detection for megascale networks by information-based replica correlations. Phys Rev E 80(1):016109
Rosvall M, Bergstrom C (2007) An information-theoretic framework for resolving community structure in complex networks. Proc Natl Acad Sci USA 104(18):7327–7331
Rosvall M, Bergstrom C (2008) Maps of random walks on complex networks reveal community structure. Proc Natl Acad Sci USA 105(4):1118–1123
Schuetz P, Caflisch A (2008) Multistep greedy algorithm identifies community structure in real-world and computer-generated networks. Phys Rev E 78(17):026112
Shi J, Malik J (2000) Normalized cuts and image segmentation. IEEE Trans Pattern Anal Mach Intell 22(8):888–905
Šíma J, Schaeffer S (2006) On the NP-completeness of some graph cluster measures. Lect Notes Comput Sci 3831:530–537
Stoer M, Wagner F (1997) A simple min-cut algorithm. J ACM 44(4):585–591
Wang RS, Zhang S, Wang Y, Zhang XS, Chen L (2008) Clustering complex networks and biological networks by nonnegative matrix factorization with various similarity measures. Neurocomputing 72:134–141
Zachary W (1977) An information flow model for conflict and fission in small groups. J Anthropol Res 33:452–473
Zhang S, Wang RS, Zhang XS (2007) Uncovering fuzzy community structure in complex networks. Phys Rev E 76(4):046103
Zhang XS, Wang RS, Wang Y, Wang J, Qiu Y, Wang L, Chen L (2009) Modularity optimization in community detection of complex networks. Europhys Lett 87(3):38002
Zhu X, Gerstein M, Snyder M (2007) Getting connected: analysis and principles of biological networks. Genes Dev 21(9):1010–1024
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, XS., Li, Z., Wang, RS. et al. A combinatorial model and algorithm for globally searching community structure in complex networks. J Comb Optim 23, 425–442 (2012). https://doi.org/10.1007/s10878-010-9356-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-010-9356-0