The use of sparsest cuts to reveal the hierarchical community structure of social networks

Weshow that good community structures can be obtained by partitioning a social network in a succession of divisive sparsest cuts. A network flow algorithm based on fundamental principles of graph theory is introduced to identify the sparsest cuts and an underlying hierarchical community structure of the network via maximum concurrent flow. Matula [Matula, DavidW., 1985. Concurrent flow and concurrent connectivity in graphs. In: Alavi, Y., et al. (Eds.), Graph Theory and its Applications to Algorithms and Computer Science.Wiley,NewYork,NY, pp. 543–559.] established themaximumconcurrentflowproblem F Multipartite cut Divisive algorithm Graph theory (MCFP), and papers on divisive vs. agglomerative average-linkage hierarchical clustering [e.g., Matula, David W., 1983. Cluster validity by concurrent chaining. In: Felsenstein, J. (Ed.), Numerical Taxonomy: Proc. of the NATO Adv. Study Inst., vol. 1. Springer-Verlag, New York, pp. 156–166 (Proceedings of NATO ASI SeriesG);Matula, DavidW., 1986.Divisive vs. agglomerative average linkagehierarchical clustering. In: Gaul, W., and Schader, M. (Eds.), Classification as a Tool of Research. Elsevier, North-Holland, Amsterdam, Byron pp. 289–301; Thompson, U N C O R R E C TE D P R O O Please cite this article in press as: Mann, C.F., et al., The use of sparsest cu Soc Netw (2008), doi:10.1016/j.socnet.2008.03.004 problemwithvariable capacitie of Engineering and Applied Sci social network by way of spars The MCFP extends the [Ford J Canadian Journal of Mathemati works. Princeton University Pr concurrent flow. The density of alently, links or bonds) betwee possible. The minimum density density is the average weight o Sparsest cuts (i.e., minimum de programming until a multipart sparsest cut. Empirical results on real-wo with embedded communities the weighted Girvan–Newman weighted networks. Physical R measure of accuracyM is define Apreliminary version of this paperwas previously presented at the International Sunbelt SocialNetworkConference (Sunbelt XXVII) and the8th European SocialNetwork Conference, the official conference of the International Network for Social Network Analysis (INSNA) held at Corfu Island, GreeceMay 1–6, 2007 (e.g., Mann et al., 2007). ∗ Corresponding author at: Department of Computer Science and Engineering, Southern Methodist University, P.O. Box 750122, Dallas, TX 75275-0122, United States. Tel.: +1 972 705 1027; fax: +1 972 705 4192. E-mail address: [email protected] (C.F. Mann). 0378-8733/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.socnet.2008.03.004 J., 1985. A flow rerouting algorithm for the maximum concurrent flow sanddemands, and its application tocluster analysis.MasterThesis. School ence, Southern Methodist University] provide the basis for partitioning a est cuts and/or maximum concurrent flow. r., Lester R., Fulkerson, Delbert R., 1956. Maximal flow through a network. cs 8, 399–404; Ford Jr., Lester R., Fulkerson, Delbert R., 1962. Flows in Netess, Princeton, NJ] source–sink max-flow problem to all-pairs maximum an (S, T)-cut is |(S, T)\/(|S|•|T|) where |(S, T)| is the number of edges (equivn communities S and T with |S|•|T| being the maximum number of edges cut in the network is the sparsest cut. When the edges are weighted, the f the (S, T)-cut edges, with absent edges treated as edges of zero weight. ts to reveal the hierarchical community structure of social networks, nsity) of the remaining components are iteratively determined via linear ite cut is identified that is more constraining to concurrent flow than any rld networks with known hierarchical structures and random networks are used to compare this sparsest-cut community structure criterion to community structure criterion [e.g., Newman, M.E.J., 2004. Analysis of eview E 70, 056131] that is based on edge betweenness centrality. A new d to evaluate the results of the graph partitionings found by these criteria. © 2008 Elsevier B.V. All rights reserved.