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.
This work was partially supported by the DFG under grants BR 2158/2-3, WA 654/14-3, Research Training Group 1042 ”Explorative Analysis and Visualization of Large Information Spaces” and the EU under grant DELIS (contract no. 001907).
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References
Brandes, U., Erlebach, T. (eds.): Network Analysis. LNCS, vol. 3418. Springer, Heidelberg (2005)
Newman, M.E.J., Girvan, M.: Finding and evaluating community structure in networks. Physical Review E 69 (2004)
Fortunato, S., Barthelemy, M.: Resolution Limit in Community Detection. In: Proceedings of the National Academy of Sciences, vol. 104, pp. 36–41 (2007)
Ziv, E., Middendorf, M., Wiggins, C.: Information-Theoretic Approach to Network Modularity. Physical Review E 71 (2005)
Muff, S., Rao, F., Caflisch, A.: Local Modularity Measure for Network Clusterizations. Physical Review E 72 (2005)
Gaertler, M., Görke, R., Wagner, D.: Significance-Driven Graph Clustering. In: AAIM 2007. LNCS, pp. 11–26. Springer, Heidelberg (2007)
Clauset, A., Newman, M.E.J., Moore, C.: Finding community structure in very large networks. Physical Review E 70 (2004)
Newman, M.: Modularity and Community Structure in Networks. In: Proceedings of the National Academy of Sciences, pp. 8577–8582 (2005)
White, S., Smyth, P.: A Spectral Clustering Approach to Finding Communities in Graph. In: SIAM Data Mining Conference (2005)
Reichardt, J., Bornholdt, S.: Statistical Mechanics of Community Detection. Physical Review E 74 (2006)
Duch, J., Arenas, A.: Community Detection in Complex Networks using Extremal Optimization. Physical Review E 72 (2005)
Brandes, U., Delling, D., Gaertler, M., Görke, R., Höfer, M., Nikoloski, Z., Wagner, D.: On Modularity Clustering. IEEE Transactions on Knowledge and Data Engineering (to appear, 2007)
Danon, L., Díaz-Guilera, A., Duch, J., Arenas, A.: Comparing community structure identification. Journal of Statistical Mechanics (2005)
Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of \(\mathcal{NP}\)-Completeness. W.H. Freeman and Company, New York (1979)
Newman, M.: Analysis of Weighted Networks. Technical report, Cornell University, Santa Fe Institute, University of Michigan (2004)
Alpert, C.J., Kahng, A.B.: Recent Directions in Netlist Partitioning: A Survey. Integration: The VLSI Journal 19, 1–81 (1995)
Hartuv, E., Shamir, R.: A Clustering Algorithm based on Graph Connectivity. Information Processing Letters 76, 175–181 (2000)
Vempala, S., Kannan, R., Vetta, A.: On Clusterings - Good, Bad and Spectral. In: FOCS 2000. Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, pp. 367–378 (2000)
Giotis, I., Guruswami, V.: Correlation Clustering with a Fixed Number of Clusters. In: SODA 2006. Proceedings of the 17th Annual ACM–SIAM Symposium on Discrete Algorithms, New York, NY, USA, pp. 1167–1176 (2006)
Bui, T.N., Chaudhuri, S., Leighton, F.T., Sipser, M.: Graph bisection algorithms with good average case behavior. Combinatorica 7, 171–191 (1987)
Zachary, W.W.: An Information Flow Model for Conflict and Fission in Small Groups. Journal of Anthropological Research 33, 452–473 (1977)
Brandes, U., Gaertler, M., Wagner, D.: Experiments on Graph Clustering Algorithms. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 568–579. Springer, Heidelberg (2003)
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Brandes, U. et al. (2007). On Finding Graph Clusterings with Maximum Modularity. In: Brandstädt, A., Kratsch, D., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2007. Lecture Notes in Computer Science, vol 4769. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74839-7_12
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DOI: https://doi.org/10.1007/978-3-540-74839-7_12
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