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Issue No.02 - February (2008 vol.20)
pp: 172-188
ABSTRACT
Modularity is a recently introduced quality measure for graph clusterings. It has immediately received considerable attention in several disciplines, and in particular in the complex systems literature, although its properties are not well understood. We study the problem of finding clusterings with maximum modularity, thus providing theoretical foundations for past and present work based on this measure. More precisely, we prove the conjectured hardness of maximizing modularity both in the general case and with the restriction to cuts, and give an Integer Linear Programming formulation. This is complemented by first insights into the behavior and performance of the commonly applied greedy agglomaration approach.
INDEX TERMS
Computations on discrete structures, Knowledge and data engineering tools and techniques, Graphs and networks, Inter programming, G.1.6.a Constrained optimization, Combinatorial algorithms, Graph Theory, Clustering, Clustering, Mathematics and statistics
CITATION
Ulrik Brandes, Daniel Delling, Marco Gaertler, Robert G?rke, Martin Hoefer, Zoran Nikoloski, Dorothea Wagner, "On Modularity Clustering", IEEE Transactions on Knowledge & Data Engineering, vol.20, no. 2, pp. 172-188, February 2008, doi:10.1109/TKDE.2007.190689
REFERENCES
 [1] M.E.J. Newman and M. Girvan, “Finding and Evaluating Community Structure in Networks,” Physical Rev. E, vol. 69, no. 026113, http://link.aps.org/abstract/PRE/v69e026113 , 2004. [2] S. Fortunato and M. Barthelemy, “Resolution Limit in Community Detection,” Proc. Nat'l Academy of Sciences, vol. 104, no. 1, pp. 36-41, 2007. [3] E. Ziv, M. Middendorf, and C. Wiggins, “Information-Theoretic Approach to Network Modularity,” Physical Rev. E, vol. 71, no. 046117, 2005. [4] S. Muff, F. Rao, and A. Caflisch, “Local Modularity Measure for Network Clusterizations,” Physical Rev. E, vol. 72, no. 056107, 2005. [5] P. Fine, E.D. Paolo, and A. Philippides, “Spatially Constrained Networks and the Evolution of Modular Control Systems,” Proc. Ninth Int'l Conf. Simulation of Adaptive Behavior (SAB '06), 2006. [6] M. Gaertler, R. Görke, and D. Wagner, “Significance-Driven Graph Clustering,” Proc. Third Int'l Conf. Algorithmic Aspects in Information and Management (AAIM '07), pp. 11-26, June 2007. [7] M.E.J. Newman, “Fast Algorithm for Detecting Community Structure in Networks,” Physical Rev. E, vol. 69, no. 066133, 2004. [8] A. Clauset, M.E.J. Newman, and C. Moore, “Finding Community Structure in Very Large Networks,” Physical Rev. E, vol. 70, no. 066111, http://link.aps.org/abstract/PRE/v70e066111 , 2004. [9] M. Newman, “Modularity and Community Structure in Networks,” Proc. Nat'l Academy of Sciences, pp. 8577-8582, 2005. [10] S. White and P. Smyth, “A Spectral Clustering Approach to Finding Communities in Graph,” Proc. SIAM Int'l Conf. Data Mining (SDM '05), 2005. [11] R. Guimerà, M. Sales-Pardo, and L.A.N. Amaral, “Modularity from Fluctuations in Random Graphs and Complex Networks,” Physical Rev. E, vol. 70, no. 025101, 2004. [12] J. Reichardt and S. Bornholdt, “Statistical Mechanics of Community Detection,” Physical Rev. E, vol. 74, no. 016110, 2006. [13] J. Duch and A. Arenas, “Community Detection in Complex Networks Using Extremal Optimization,” Physical Rev. E, vol. 72, no. 027104, 2005. [14] M. Gaertler, “Clustering,” Network Analysis: Methodological Foundations, U. Brandes and T. Erlebach, eds. pp. 178-215, Springer-Verlag, http://springerlink.metapress.comopenurl.asp?genre= article&is sn=0302-9743&volume=3418&spage=178 , Feb. 2005. [15] L. Danon, A. Díaz-Guilera, J. Duch, and A. Arenas, “Comparing Community Structure Identification,” J. Statistical Mechanics, 2005. [16] M.R. Garey and D.S. Johnson, Computers and Intractability. A Guide to the Theory of ${\cal NP}\hbox{-}{\rm Completeness}$ . W.H. Freeman and Co., 1979. [17] M. Newman, “Analysis of Weighted Networks,” technical report, Univ. of Michigan, July 2004. [18] C.J. Alpert and A.B. Kahng, “Recent Directions in Netlist Partitioning: A Survey,” Integration: The VLSI J., vol. 19, nos. 1-2, pp. 1-81, , 1995. [19] E. Hartuv and R. Shamir, “A Clustering Algorithm Based on Graph Connectivity,” Information Processing Letters, vol. 76, nos. 4-6, pp.175-181, , 2000. [20] S. Vempala, R. Kannan, and A. Vetta, “On Clusterings: Good, Bad and Spectral,” Proc. 41st Ann. IEEE Symp. Foundations of Computer Science (FOCS '00), pp. 367-378, 2000. [21] I. Giotis and V. Guruswami, “Correlation Clustering with a Fixed Number of Clusters,” Proc. 17th Ann. ACM-SIAM Symp. Discrete Algorithms (SODA '06), pp. 1167-1176, , 2006. [22] T. Bui, S. Chaudhuri, F. Leighton, and M. Sipser, “Graph Bisection Algorithms with Good Average Case Behavior,” Combinatorica, vol. 7, no. 2, pp. pp. 171-191, 1987. [23] A.K. Jain, M.N. Murty, and P.J. Flynn, “Data Clustering: A Review,” ACM Computing Surveys, vol. 31, no. 3, pp. 264-323, 1999. [24] U. Brandes, D. Delling, M. Gaertler, R. Görke, M. Hoefer, Z. Nikoloski, and D. Wagner, “On Modularity: NP-Completeness and Beyond,” Technical Report 2006-19, ITI Wagner, Faculty of Informatics, Universität Karlsruhe (TH), 2006. [25] W.W. Zachary, “An Information Flow Model for Conflict and Fission in Small Groups,” J. Anthropological Research, vol. 33, pp.452-473, 1977. [26] M.E.J. Newman and M. Girvan, “Mixing Patterns and Community Structure in Networks,” Statistical Mechanics of Complex Networks, Lecture Notes in Physics 625, R. Pastor-Satorras, M. Rubi, and A.Diaz-Guilera, eds., pp. 66-87, Springer-Verlag, 2003. [27] D. Delling, M. Gaertler, R. Görke, and D. Wagner, “Experiments on Comparing Graph Clusterings,” Technical Report 2006-16, ITI Wagner, Faculty of Informatics, Universität Karlsruhe (TH), 2006. [28] D. Lusseau, K. Schneider, O.J. Boisseau, P. Haase, E. Slooten, and S.M. Dawson, “The Bottlenose Dolphin Community of Doubtful Sound Features a Large Proportion of Long-Lasting Associations,” Behavioral Ecology and Sociobiology, vol. 54, no. 4, pp. 396-405, 2003. [29] M.E.J. Newman and M. Girvan, “Finding and Evaluating Community Structure in Networks,” http://vlsicad.cs.ucla.edu/~cheese/survey.htmlhttp:/ /citeseer.nj.nec.com/hartuv99clustering.htmlhttp:/ /portal.acm.org/ citation.cfm?id=1109557.1109686#http:// arxiv.org/abs/cond-mat0308217, Aug. 2003.