Abstract
Clustering applications dealing with perception based or biased data lead to models with non-disjunct clusters. There, objects to be clustered are allowed to belong to several clusters at the same time which results in a fuzzy clustering. It can be shown that this is equivalent to searching all maximal cliques in dynamic graphs like G t = (V,E t), where E t − 1 ⊂ E t, t = 1,...,T; E 0 = φ. In this article algorithms are provided to track all maximal cliques in a fully dynamic graph.
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References
B. Bollobás, Modern Graph Theory. Springer: New York, 1998.
I.M. Bomze, M. Budinich, P.M. Pardalos, and M. Pelillo, “The maximum clique problem,” in Handbook of Combinatorial Optimization, volume Suppl. vol. A:4, Kluwer Academic Publishers: Boston, MA, 1999.
I.M. Bomze and V. Stix, “Genetic engineering via negative fitness: Evolutionary dynamics for global optimization,” Annals of Oper. Res., vol. 89, pp. 297–318, 1999.
C. Bron and J. Kerbosch, “Algorithm 457: Finding all cliques of an undirected graph,” Commun. ACM, vol. 16, no. 9, pp. 575–577, 1973.
M. Broom, C. Cannings, and G. T. Vickers, “On the number of local maxima of a constrained quadratic form,” Proc. R. Soc. Lond., vol. 443, pp. 573–584, 1993.
E. Cambouropoulos, A. Smaill, and G. Widmer, “A clustering algorithm for melodic analysis,” in Proceedings of the Diderot'99, 1999.
E. Cambouropoulos and G. Widmer, “Melodic clustering: Motivic analysis of Schumann's Träumerei,” in Proceedings of JIM 2000, Bordeaux, 2000.
C. Cannings and G.T. Vickers, “Patterns of ESSs II,” J. Theor. Biol., vol. 132, pp. 409–420, 1988.
D. Eppstein, “Clustering for faster network simplex pivots,” in Proc. 5th ACM-SIAM Symp. Discrete Algorithms, 1994, pp. 160–166.
D. Eppstein, Z. Galil, and G.F. Italiano, Algorithms and Theory of Computation Handbook, chapter Dynamic Graph Algorithms. CRC Press: New York, 1999, pp. 8-1–8-25.
Z. Galil and G.F. Italiano, “Fully dynamic algorithms for 2-edge-connectivity,” SIAM J. Comput., vol. 21, pp. 1047–1069, 1992.
E.J. Gardiner, P.J. Artymiuk, and P. Willett, “Clique-detection algorithms for matching three-dimensional molecular structures,” Journal of Molecular Graphics and Modelling, vol. 15, no. 4, pp. 245–253, 1997.
M.A. Gluck and J.E. Corter, “Information, uncertainty, and the utility of categories,” in Proc. 7th Ann. Conf. of the Cognitive Science Society, 1985.
J. A. Hartigan, Clustering Algorithms, Wiley Series in Probability and Mathematical Statistics. Wiley: New York, 1975.
M.R. Henzinger and V. King, “Maintaining minimum spanning trees in dynamic graphs,” in Proc. 24th Int. Coll. Automata, Languages and Programming, 1997, pp. 594–604.
D.S. Johnson and M.A. Tricks (Eds.), “Cliques, coloring and satisfiability: Second dimacs implementation challenge,” DIMACS Series in Discrete Mathematics and Theoretical Computer Science 26, American Mathematical Society, Procidence, 1996.
G. Li, V. Uren, E. Motta, S.B. Shum, and J. Domingue, “Claimaker: Weaving a semantic web of research papers,” in 1st International Semantic Web Conference, 2002.
J.W. Moon and L. Moser, “On cliques in graphs,” Isr. J. Math., vol. 3, pp. 23–28, 1965.
T.S. Motzkin and E.G. Straus, “Maxima for graphs and a new proof of a theorem of Turán,” Canad. J. Math., vol. 17, no. 4, pp. 533–540, 1965.
G.T. Vickers and C. Cannings, “Patterns of ESSs I,” J. Theor. Biol., vol. 132, pp. 387–408, 1988.
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Stix, V. Finding All Maximal Cliques in Dynamic Graphs. Computational Optimization and Applications 27, 173–186 (2004). https://doi.org/10.1023/B:COAP.0000008651.28952.b6
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DOI: https://doi.org/10.1023/B:COAP.0000008651.28952.b6