Abstract
Dynamic graphs can capture changing relationships in many real datasets that evolve over time. One of the most basic questions about networks is the identification of the “most important” vertices in a network. Measures of vertex importance called centrality measures are used to rank vertices in a graph. In this work, we focus on Katz Centrality. Typically, scores are calculated through linear algebra but in this paper we present an new alternative, agglomerative method of calculating Katz scores and extend it for dynamic graphs. We show that our static algorithm is several orders of magnitude faster than the typical linear algebra approach while maintaining good quality of the scores. Furthermore, our dynamic graph algorithm is faster than pure static recomputation every time the graph changes and maintains high recall of the highly ranked vertices on both synthetic and real graphs.
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References
Caldarelli, G.: Scale-Free Networks: Complex Webs in Nature and Technology. Oxford University Press, Oxford (2007)
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.-U.: Complex networks: structure and dynamics. Phys. Rep. 424(4), 175–308 (2006)
Katz, L.: A new status index derived from sociometric analysis. Psychometrika 18(1), 39–43 (1953)
Freeman, L.C.: A set of measures of centrality based on betweenness. Sociometry 40(1), 35–41 (1977)
Green, O., McColl, R., Bader, D.A.: A fast algorithm for streaming betweenness centrality. In: 2012 International Conference on and 2012 International Conference on Social Computing (SocialCom), Privacy, Security, Risk and Trust (PASSAT), pp. 11–20. IEEE (2012)
Page, L., Brin, S., Motwani, R., Winograd, T.: The PageRank citation ranking: bringing order to the web (1999)
Bahmani, B., Chowdhury, A., Goel, A.: Fast incremental and personalized pagerank. Proc. VLDB Endow. 4(3), 173–184 (2010)
Langville, A.N., Meyer, C.D.: Updating pagerank with iterative aggregation. In: Proceedings of the 13th International World Wide Web Conference on Alternate Track Papers and Posters, pp. 392–393. ACM (2004)
Riedy, J.: Updating pagerank for streaming graphs. In: 2016 IEEE International Parallel and Distributed Processing Symposium Workshops, pp. 877–884. IEEE (2016)
Foster, K.C., Muth, S.Q., Potterat, J.J., Rothenberg, R.B.: A faster katz status score algorithm. Comput. Math. Organ. Theory 7(4), 275–285 (2001)
Liben-Nowell, D., Kleinberg, J.: The link-prediction problem for social networks. J. Assoc. Inf. Sci. Technol. 58(7), 1019–1031 (2007)
Newman, M.: Networks: An Introduction. Oxford University Press, Oxford (2010)
Benzi, M., Klymko, C.: Total communicability as a centrality measure. J. Complex Netw. 1(2), 124–149 (2013)
Benzi, M., Klymko, C.: A matrix analysis of different centrality measures. arXiv preprint arXiv:1312.6722 (2014)
Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)
Erdös, P., Rényi, A.: On random graphs, i. Publicationes Mathematicae (Debrecen) 6, 290–297 (1959)
Chakrabarti, D., Zhan, Y., Faloutsos, C.: R-mat: a recursive model for graph mining. In: SDM, vol. 4, pp. 442–446. SIAM (2004)
Kunegis, J.: KONECT: the koblenz network collection. In: Proceedings of the 22nd International Conference on World Wide Web, pp. 1343–1350. ACM (2013)
Albert, R., Jeong, H., Barabási, A.-L.: Internet: diameter of the world-wide web. Nature 401(6749), 130–131 (1999)
Hawick, K., James, H.: Node importance ranking and scaling properties of some complex road networks (2007)
Acknowledgments
Eisha Nathan is in part supported by the National Physical Science Consortium Graduate Fellowship. The work depicted in this paper was sponsored in part by the National Science Foundation under award #1339745. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation.
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Nathan, E., Bader, D.A. (2018). Approximating Personalized Katz Centrality in Dynamic Graphs. In: Wyrzykowski, R., Dongarra, J., Deelman, E., Karczewski, K. (eds) Parallel Processing and Applied Mathematics. PPAM 2017. Lecture Notes in Computer Science(), vol 10777. Springer, Cham. https://doi.org/10.1007/978-3-319-78024-5_26
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DOI: https://doi.org/10.1007/978-3-319-78024-5_26
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