Bounded Arboricity to Determine the Local Structure of Sparse Graphs

  • Gaurav Goel
  • Jens Gustedt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4271)


A known approach of detecting dense subgraphs (communities) in large sparse graphs involves first computing the probability vectors for short random walks on the graph, and then using these probability vectors to detect the communities, see Latapy and Pons [2005]. In this paper we focus on the first part of such an approach i.e. the computation of the probability vectors for the random walks, and propose a more efficient algorithm for computing these vectors in time complexity that is linear in the size of the output, in case the input graphs are restricted to a family of graphs of bounded arboricity. Such classes of graphs cover a large number of cases of interest, e.g all minor closed graph classes (planar graphs, graphs of bounded treewidth etc) and random graphs within the preferential attachment model, see Barabási and Albert [1999]. Our approach is extensible to other models of computation (PRAM, BSP or out-of-core computation) and also w.h.p. stays within the same complexity bounds for Erdős Renyi graphs.


Random Walk Random Graph Probability Vector Input Graph Sparse Graph 


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  1. Aiello, W., Chung, F., Lu, L.: Random evolution in massive graphs. In: Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, pp. 510–519 (2001), URL:
  2. Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)CrossRefMathSciNetGoogle Scholar
  3. Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)MATHCrossRefMathSciNetGoogle Scholar
  4. Cheriton, D., Tarjan, R.E.: Finding minimum spanning trees. SIAM J. Computing 5, 724–742 (1976)MATHCrossRefMathSciNetGoogle Scholar
  5. Dehne, F.K.H.A., Dittrich, W., Hutchinson, D.: Efficient external memory algorithms by simulating coarse-grained parallel algorithms. In: ACM Symposium on Parallel Algorithms and Architectures, pp. 106–115 (1997)Google Scholar
  6. Fiorio, C., Gustedt, J.: Two linear time union-find strategies for image processing. Theoretical Computer Science 154, 165–181 (1996)MATHCrossRefMathSciNetGoogle Scholar
  7. Fortune, S., Wyllie, J.: Parallelism in random access machines. In: 10th ACM Symposium on Theory of Computing, May 1978, pp. 114–118 (1978)Google Scholar
  8. Gebremedhin, A.H., Lassous, I.G., Gustedt, J., Telle, J.A.: PRO: a model for parallel resource-optimal computation. In: 16th Annual International Symposium on High Performance Computing Systems and Applications, pp. 106–113. IEEE (The Institute of Electrical and Electronics Engineers), Los Alamitos (2002)Google Scholar
  9. Gustedt, J.: Minimum spanning trees for minor-closed graph classes in parallel. In: Symposium on Theoretical Aspects of Computer Science, pp. 421–431 (1998)Google Scholar
  10. Gustedt, J.: Towards realistic implementations of external memory algorithms using a coarse grained paradigm. In: Kumar, V., Gavrilova, M.L., Tan, C.J.K., L’Ecuyer, P. (eds.) ICCSA 2003. LNCS, vol. 2668, pp. 269–278. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. Kannan, S., Naor, M., Rudich, S.: Implicit representation of graphs. SIAM Journal On Discrete Mathematics 5, 596–603 (1992), URL: MATHCrossRefMathSciNetGoogle Scholar
  12. Latapy, M., Pons, P.: Computing communities in large networks using random walks. In: Yolum, p., Güngör, T., Gürgen, F., Özturan, C. (eds.) ISCIS 2005. LNCS, vol. 3733, pp. 284–293. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. Mader, W.: Homomorphieeigenschaften und mittlere Kantendichte von Graphen. Math. Ann. 174, 265–268 (1967)MATHCrossRefMathSciNetGoogle Scholar
  14. Nash-Williams, C.S.J.A.: Edge-disjoint spanning trees of finite graphs. J. London Math. Soc (2) 36, 445–450 (1961)MATHCrossRefMathSciNetGoogle Scholar
  15. Valiant, L.G.: A bridging model for parallel computation. Communications of the ACM 33(8), 103–111 (1990)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Gaurav Goel
    • 1
    • 2
  • Jens Gustedt
    • 1
    • 3
  1. 1.INRIA LorraineFrance
  2. 2.IIT DelhiIndia
  3. 3.LORIAFrance

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