Abstract
The clustering coefficient of an unweighted network has been extensively used to quantify how tightly connected is the neighbor around a node and it has been widely adopted for assessing the quality of nodes in a social network. The computation of the clustering coefficient is challenging since it requires to count the number of triangles in the graph. Several recent works proposed efficient sampling, streaming and MapReduce algorithms that allow to overcome this computational bottleneck. As a matter of fact, the intensity of the interaction between nodes, that is usually represented with weights on the edges of the graph, is also an important measure of the statistical cohesiveness of a network. Recently various notions of weighted clustering coefficient have been proposed but all those techniques are hard to implement on large-scale graphs.
In this work we show how standard sampling techniques can be used to obtain efficient estimators for the most commonly used measures of weighted clustering coefficient. Furthermore we also propose a novel graph-theoretic notion of clustering coefficient in weighted networks.
Work partially done while visiting scientist at Google Research NY. Partially supported from Google Focused Award “Algorithms for Large-scale Data Analysis”, EU FET project MULTIPLEX 317532, EU ERC project PAAI 259515.
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References
Applying social network analysis to the information in cvs repositories. In: 1st International Workshop on Mining Software Repositories (MSR)
Barrat, A., Barthlemy, M., Pastor-Satorras, R., Vespignani, A.: The architecture of complex weighted networks. Proceedings of the National Academy of Sciences of the United States of America
Becchetti, L., Boldi, P., Castillo, C., Gionis, A.: Efficient semi-streaming algorithms for local triangle counting in massive graphs. In: KDD 2008 (2008)
Bollobs, B.: Mathematical results on scale-free random graphs. In: Handbook of Graphs and Networks
Budak, C., Agrawal, D., El Abbadi, A.: Structural trend analysis for online social networks. In: VLDB 2011 (2011)
Buriol, L., Frahling, G., Leonardi, S., Marchetti-Spaccamela, A., Sohler, C.: Counting triangles in data streams. In: PODS 2006 (2006)
Castillo, C., Donato, D., Becchetti, L., Boldi, P., Leonardi, S., Santini, M., Vigna, S.: A reference collection for web spam. SIGIR 2006 (2006)
Dean, J., Ghemawat, S.: Mapreduce: Simplified data processing on large clusters. In: OSDI 2004 (2004)
Fagiolo, G.: Clustering in complex directed networks. Phys. Rev. E.
Hardiman, S.J., Katzir, L.: Estimating clustering coefficients and size of social networks via random walk. In: WWW 2013 (2013)
Hintsanen, P., Toivonen, H.: Finding reliable subgraphs from large probabilistic graphs. Data Min. Knowl. Discov.
Hintsanen, P., Toivonen, H.: Finding reliable subgraphs from large probabilistic graphs. Data Min. Knowl. Discov. (2008)
Jha, M., Seshadhri, C., Pinar, A.: A space efficient streaming algorithm for triangle counting using the birthday paradox. In: KDD 2013 (2013)
Kalna, G., Higham, D.J.: Clustering coefficients for weighted networks. In: Symposium on Network Analysis in Natural Sciences and Engineering
Kwak, H., Lee, C., Park, H., Moon, S.: What is twitter, a social network or a news media?. In: WWW 2010 (2010)
Latapy, M.: Main-memory triangle computations for very large (sparse(power-law)) graphs. Theoretical Computer Science
Leskovec, J., Horvitz, E.: Planetary-scale views on a large instant-messaging network. In: WWW 2008 (2008)
Liberty, E.: Simple and deterministic matrix sketches. In: KDD 2014 (2014)
Newman, M.E.J.: Analysis of weighted networks. Phys. Rev. E 70, 056131 (2004)
Newman, M.E.J., Watts, D.J., Strogatz, S.H.: Random graph models of social networks. Proc. Natl. Acad. Sci. USA 99, 2566–2572 (2002)
Onnela, J.-P., Saramäki, J., Kertész, J., Kaski, K.: Intensity and coherence of motifs in weighted complex networks. Physical Review E
Opsahl, T., Panzarasa, P.: Clustering in weighted networks. Social Networks
Pagh, R., Tsourakakis, C.E.: Colorful triangle counting and a mapreduce implementation
Saramäki, J., Kivelä, M., Onnela, J.-P., Kaski, K., Kertesz, J.: Generalizations of the clustering coefficient to weighted complex networks. Physical Review E
Schank, T., Wagner, D.: Approximating clustering coefficient and transitivity. Journal of Graph Algorithms and Applications
Schank, Thomas, Wagner, Dorothea: Finding, Counting and Listing All Triangles in Large Graphs, an Experimental Study. In: Nikoletseas, Sotiris E. (ed.) WEA 2005. LNCS, vol. 3503, pp. 606–609. Springer, Heidelberg (2005)
Suri, S., Vassilvitskii, S.: Counting triangles and the curse of the last reducer. In: WWW 2011 (2011)
Tsourakakis, C.E., Kang, U., Miller, G.L., Faloutsos, C.: Doulion: counting triangles in massive graphs with a coin. In: KDD 2009 (2009)
Tsourakakis, C.E., Kolountzakis, M.N., Miller, G.L.: Triangle sparsifiers. J. Graph Algorithms Appl.
Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature
Zhang, B., Horvath, S., et al.: A general framework for weighted gene co-expression network analysis. Statistical Applications in Genetics and Molecular Biology
Zhang, Y., Zhang, Z., Guan, J., Zhou, S.: Analytic solution to clustering coefficients on weighted networks. arXiv preprint arXiv:0911.0476 (2009)
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Lattanzi, S., Leonardi, S. (2014). Efficient Computation of the Weighted Clustering Coefficient. In: Bonato, A., Graham, F., Prałat, P. (eds) Algorithms and Models for the Web Graph. WAW 2014. Lecture Notes in Computer Science(), vol 8882. Springer, Cham. https://doi.org/10.1007/978-3-319-13123-8_4
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DOI: https://doi.org/10.1007/978-3-319-13123-8_4
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