Abstract
We address the problem of determining the natural neighbourhood of a given node i in a large nonunifom network G in a way that uses only local computations, i.e. without recourse to the full adjacency matrix of G. We view the problem as that of computing potential values in a diffusive system, where node i is fixed at zero potential, and the potentials at the other nodes are then induced by the adjacency relation of G. This point of view leads to a constrained spectral clustering approach. We observe that a gradient method for computing the respective Fiedler vector values at each node can be implemented in a local manner, leading to our eventual algorithm. The algorithm is evaluated experimentally using three types of nonuniform networks: randomised “caveman graphs”, a scientific collaboration network, and a small social interaction network.
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References
Brandes, U., Gaertler, M., Wagner, D.: Experiments on graph clustering algorithms. In: Di Battista, G., Zwick, U. (eds.) ESA 2003, vol. 2832, pp. 568–579. Springer, Heidelberg (2003)
Brémaud, P.: Markov Chains, Gibbs Fields, Monte Carlo Simulation, and Queues. Springer, New York (1999)
Chung, F.R.K.: Spectral Graph Theory, American Mathematical Society, Providence, RI (1997)
Chung, F.R.K., Ellis, R.B.: A chip-firing game and Dirichlet eigenvalues. Discrete Mathematics 257, 341–355 (2002)
Doyle, P.G., Snell, J.L.: Random Walks and Electric Networks. In: Mathematical Association of America, Washington, DC (1984)
Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification. Wiley, New York (2001)
Fiedler, M.: Algebraic connectivity of graphs. Czechoslovak Mathematical Journal 23, 298–305 (1973)
Fiedler, M.: A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czechoslovak Mathematical Journal 25, 619–633 (1975)
Flake, G.W., Lawrence, S., Giles, C.L., Coetzee, F.M.: Self-organization and identification of Web communities. IEEE Computer 35(3), 66–71 (2002)
Gkantsidis, C., Mihail, M., Zegura, E.: Spectral analysis of Internet topologies. In: Proceedings of the 22nd Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM 2003), pp. 364–374. IEEE, New York (2003)
Guattery, S., Miller, G.L.: On the quality of spectral separators. SIAM Journal on Matrix Analysis and Applications 19(3), 701–719 (1998)
He, X., Zha, H., Ding, C.H.Q., Simon, H.: Web document clustering using hyperlink structures. Computational Statistics & Data Analysis 41(1), 19–45 (2002)
Hopcroft, J., Khan, O., Kulis, B., Selman, B.: Natural communities in large linked networks. In: Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 541–546. ACM, New York (2003)
Kannan, R., Vempala, S., Vetta, A.: On clusterings: Good, bad and spectral. Journal of the ACM 51(3), 497–515 (2004)
Kempe, D., McSherry, F.: A decentralized algorithm for spectral analysis. In: Proceedings of the 36th ACM Symposium on Theory of Computing (STOC 2004). ACM, New York (2004)
Newman, M.E.J.: The structure and function of complex networks. SIAM Review 45, 167–256 (2003)
Newman, M.E.J.: Fast algorithm for detecting community structure in networks. Physical Review E 69, 066113 (2004)
Newman, M.E.J., Girvan, M.: Finding and evaluating community structure in networks. Physical Review E 69, 026113 (2004)
Pothen, A., Simon, H.D., Liou, K.P.: Partitioning sparse matrices with eigenvectors of graphs. SIAM Journal of Matrix Analysis and Applications 11, 430–452 (1990)
Schaeffer, S.E.: Stochastic local clustering for massive graphs. In: Ho, T.-B., Cheung, D., Liu, H. (eds.) PAKDD 2005. LNCS, vol. 3518, pp. 354–360. Springer, Heidelberg (2005)
Spielman, D.A., Teng, S.-H.: Spectral partitioning works: planar graphs and finite element meshes. In: Proceedings of the 37th IEEE Symposium on Foundations of Computing (FOCS 1996), pp. 96–105. IEEE Computer Society, Los Alamitos (1996)
Virtanen, S.E.: Clustering the Chilean Web. In: Proceedings of the First Latin American Web Congress, pp. 229–231. IEEE Computer Society, Los Alamitos (2003)
Virtanen, S.E.: Properties of nonuniform random graph models. Research Report A77, Helsinki University of Technology, Laboratory for Theoretical Computer Science, Espoo, Finland (May 2003), http://www.tcs.hut.fi/Publications/info/bibdb.HUT-TCS-A77.shtml
Watts, D.J.: Small Worlds: The Dynamics of Networks between Ordeer and Randomness. Princeton University Press, Princeton (1999)
Wu, F., Huberman, B.A.: Finding communities in linear time: a physics approach. The European Physics Journal B 38, 331–338 (2004)
Zachary, W.W.: An information flow model for conflict and fission in small groups. Journal of Anthropological Research 33, 452–473 (1977)
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Orponen, P., Schaeffer, S.E. (2005). Local Clustering of Large Graphs by Approximate Fiedler Vectors. In: Nikoletseas, S.E. (eds) Experimental and Efficient Algorithms. WEA 2005. Lecture Notes in Computer Science, vol 3503. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11427186_45
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DOI: https://doi.org/10.1007/11427186_45
Publisher Name: Springer, Berlin, Heidelberg
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