Abstract
A planted partition graph is an Erdős-Rényi type random graph, where, based on a given partition of the vertex set, vertices in the same part are linked with a higher probability than vertices in different parts. Graphs of this type are frequently used to evaluate graph clustering algorithms, i.e., algorithms that seek to partition the vertex set of a graph into densely connected clusters. We propose a self-evident modification of this model to generate sequences of random graphs that are obtained by atomic updates, i.e., the deletion or insertion of an edge or vertex. The random process follows a dynamically changing ground-truth clustering that can be used to evaluate dynamic graph clustering algorithms. We give a theoretical justification of our model and show how the corresponding random process can be implemented efficiently.
This work was partially supported by the DFG under grant WA 654/19-1 and WA 654/15-1.
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References
Fortunato, S.: Community detection in graphs. Physics Reports 486(3-5), 75–174 (2010)
Schaeffer, S.E.: Graph Clustering. Computer Science Review 1(1), 27–64 (2007)
Bollobás, B.: Random Graphs. Cambridge University Press (2001)
Gilbert, H.: Random Graphs. The Annals of Mathematical Statistics 30(4), 1141–1144 (1959)
Condon, A., Karp, R.M.: Algorithms for Graph Partitioning on the Planted Partition Model. Randoms Structures and Algorithms 18(2), 116–140 (2001)
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)
Gaertler, M., Görke, R., Wagner, D.: Significance-Driven Graph Clustering. In: Kao, M.-Y., Li, X.-Y. (eds.) AAIM 2007. LNCS, vol. 4508, pp. 11–26. Springer, Heidelberg (2007)
Girvan, M., Newman, M.E.J.: Community structure in social and biological networks. Proceedings of the National Academy of Science of the United States of America 99(12), 7821–7826 (2002)
Watts, D.J.: Small worlds: The dynamics of networks between order and randomness. Princeton University Press (1999)
Watts, D.J., Strogatz, S.H.: Collective dynamics of ’small-world’ networks. Nature 393(6684), 440–442 (1998)
Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)
Leskovec, J., Kleinberg, J.M., Faloutsos, C.: Graphs Over Time: Densification Laws, Shrinking Diameters and Possible Explanations. In: Proceedings of the 11th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 177–187. ACM Press (2005)
Vázquez, A.: Growing network with local rules: Preferential attachment, clustering hierarchy, and degree correlations. Physical Review E 67, 056104 (2003)
Bagrow, J.: Evaluating local community methods in networks. Journal of Statistical Mechanics: Theory and Experiment, P05001 (2008), doi:10.1088/1742-5468/2008/05/P05001
Lancichinetti, A., Fortunato, S.: Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities. Physical Review E 80(1), 016118 (2009)
Fan, Y., Li, M., Zhang, P., Wu, J., Di, Z.: Accuracy and precision of methods for community identification in weighted networks. Physica A 377(1), 363–372 (2007)
Guimerà, R., Sales-Pardo, M., Amaral, L.A.N.: Module identification in bipartite and directed networks. Physical Review E 76, 036102 (2007)
Zhou, H.: Network landscape from a Brownian particle’s perspective. Physical Review E 67, 041908 (2003)
Sawardecker, E.N., Sales-Pardo, M., Amaral, L.A.N.: Detection of node group membership in networks with group overlap. The European Physical Journal B 67, 277–284 (2009)
Aldecoa, R., Marín, I.: Closed benchmarks for network community structure characterization. Physical Review E 85, 026109 (2012)
Brandes, U., Mader, M.: A Quantitative Comparison of Stress-Minimization Approaches for Offline Dynamic Graph Drawing. In: van Kreveld, M., Speckmann, B. (eds.) GD 2011. LNCS, vol. 7034, pp. 99–110. Springer, Heidelberg (2011)
Robins, G., Pattison, P., Kalish, Y., Lusher, D.: An introduction to exponential random graph (p*) models for social networks. Social Networks 29(2), 173–191 (2007)
Snijders, T.A.: The Statistical Evaluation of Social Network Dynamics. Sociological Methodology 31(1), 361–395 (2001)
Clementi, A.E.F., Macci, C., Monti, A., Pasquale, F., Silvestri, R.: Flooding time in edge-Markovian dynamic graphs. SIAM Journal on Discrete Mathematics 24(4), 1694–1712 (2010)
Baumann, H., Crescenzi, P., Fraigniaud, P.: Parsimonious flooding in dynamic graphs. In: Proceedings of the 28th ACM Symposium on Principles of Distributed Computing, pp. 260–269. ACM Press (2009)
Görke, R., Staudt, C.: A Generator for Dynamic Clustered Random Graphs. Technical report, Informatik, Uni Karlsruhe, TR 2009-7 (2009)
Görke, R.: An Algorithmic Walk from Static to Dynamic Graph Clustering. PhD thesis, Fakultät für Informatik (February 2010)
Görke, R., Maillard, P., Staudt, C., Wagner, D.: Modularity-Driven Clustering of Dynamic Graphs. In: Festa, P. (ed.) SEA 2010. LNCS, vol. 6049, pp. 436–448. Springer, Heidelberg (2010)
Görke, R., Kluge, R., Schumm, A., Staudt, C., Wagner, D.: An Efficient Generator for Clustered Dynamic Random Networks. Technical report, Karlsruhe Reports in Informatics 2012, 17 (2012)
Behrends, E.: Introduction to Markov Chains With Special Emphasis on Rapid Mixing. Friedrick Vieweg & Son (October 2002)
Batagelj, V., Brandes, U.: Efficient Generation of Large Random Networks. Physical Review E 036113 (2005)
Fisher, R.A., Yates, F.: Statistical Tables for Biological, Agricultural and Medical Research. Oliver and Boyd, London (1948)
Fan, C.T., Muller, M.E., Rezucha, I.: Development of Sampling Plans by Using Sequential (Item by Item) Selection Techniques and Digital-Computers. Journal of the American Statistical Association 57(298), 387–402 (1962)
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Görke, R., Kluge, R., Schumm, A., Staudt, C., Wagner, D. (2012). An Efficient Generator for Clustered Dynamic Random Networks. In: Even, G., Rawitz, D. (eds) Design and Analysis of Algorithms. MedAlg 2012. Lecture Notes in Computer Science, vol 7659. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34862-4_16
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DOI: https://doi.org/10.1007/978-3-642-34862-4_16
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