External-Memory Network Analysis Algorithms for Naturally Sparse Graphs

  • Michael T. Goodrich
  • Paweł Pszona
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6942)


In this paper, we present a number of network-analysis algorithms in the external-memory model. We focus on methods for large naturally sparse graphs, that is, n-vertex graphs that have O(n) edges and are structured so that this sparsity property holds for any subgraph of such a graph. We give efficient external-memory algorithms for the following problems for such graphs:

  1. 1

    Finding an approximate d-degeneracy ordering.

  2. 2

    Finding a cycle of length exactly c.

  3. 3

    Enumerating all maximal cliques.

Such problems are of interest, for example, in the analysis of social networks, where they are used to study network cohesion.


Random Graph Undirected Graph Maximal Clique External Memory Recursive Call 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Alon, N., Gutner, S.: Linear time algorithms for finding a dominating set of fixed size in degenerated graphs. Algorithmica 54(4), 544–556 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alon, N., Kahn, J., Seymour, P.D.: Large induced degenerate subgraphs. Graphs and Combinatorics 3, 203–211 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alon, N., Yuster, R., Zwick, U.: Finding and counting given length cycles. Algorithmica 17(3), 209–223 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Batagelj, V., Zaveršnik, M.: An O(m) algorithm for cores decomposition of networks (2003),
  6. 6.
    Bauer, R., Krug, M., Wagner, D.: Enumerating and generating labeled k-degenerate graphs. In: 7th Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pp. 90–98. SIAM, Philadelphia (2010)Google Scholar
  7. 7.
    Bollobás, B.: On generalized graphs. Acta Mathematica Hungarica 16, 447–452 (1904), doi:10.1007/BF01904851CrossRefzbMATHGoogle Scholar
  8. 8.
    Bron, C., Kerbosch, J.: Algorithm 457: finding all cliques of an undirected graph. Commun. ACM 16(9), 575–577 (1973)CrossRefzbMATHGoogle Scholar
  9. 9.
    Cheng, J., Ke, Y., Chu, S., Ozsu, T.: Efficient core decomposition in massive networks. In: IEEE Int. Conf. on Data Engineering, ICDE (2011)Google Scholar
  10. 10.
    Chrobak, M., Eppstein, D.: Planar orientations with low out-degree and compaction of adjacency matrices. Theor. Comput. Sci. 86(2), 243–266 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Doreian, P., Woodard, K.L.: Defining and locating cores and boundaries of social networks. Social Networks 16(4), 267–293 (1994)CrossRefGoogle Scholar
  12. 12.
    Eppstein, D., Löffler, M., Strash, D.: Listing all maximal cliques in sparse graphs in near-optimal time. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010. LNCS, vol. 6506, pp. 403–414. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Eppstein, D., Strash, D.: Listing all maximal cliques in large sparse real-world graphs. arXiv eprint, 1103.0318 (2011)Google Scholar
  14. 14.
    Erdős, P., Hajnal, A.: On chromatic number of graphs and set-systems. Acta Mathematica Hungarica 17(1-2), 61–99 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fernholz, D., Ramachandran, V.: The giant k-core of a random graph with a specified degree sequence (2003) (manuscript)Google Scholar
  16. 16.
    Freuder, E.C.: A sufficient condition for backtrack-free search. J. ACM 29, 24–32 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Golovach, P.A., Villanger, Y.: Parameterized complexity for domination problems on degenerate graphs. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 195–205. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  18. 18.
    Goodrich, M.T., Pszona, P.: External-memory network analysis algorithms for naturally sparse graphs. arXiv eprint, 1106.6336 (2011)Google Scholar
  19. 19.
    Irani, S.: Coloring inductive graphs on-line. Algorithmica 11, 53–72 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kirousis, L.M., Thilikos, D.M.: The linkage of a graph. SIAM Journal on Computing 25(3), 626–647 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kleinberg, J.: The small-world phenomenon: an algorithm perspective. In: 32nd ACM Symp. on Theory of Computing (STOC), pp. 163–170 (2000)Google Scholar
  22. 22.
    Lick, D.R., White, A.T.: k-degenerate graphs. Canadian Journal of Mathematics 22, 1082–1096 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Monien, B.: How to find long paths efficiently. Annals of Discrete Mathematics 25, 239–254 (1985)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Pittel, B., Spencer, J., Wormald, N.: Sudden emergence of a giant k-core in a random graph. Journal of Combinatorial Theory, Series B 67(1), 111–151 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Riordan, O.: The k-core and branching processes. Probability And Computing 17, 111 (2008)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Seidman, S.B.: Network structure and minimum degree. Social Networks 5(3), 269–287 (1983)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Tomita, E., Tanaka, A., Takahashi, H.: The worst-case time complexity for generating all maximal cliques and computational experiments. Theor. Comput. Sci. 363(1), 28–42 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Vitter, J.S.: External memory algorithms and data structures: dealing with massive data. ACM Comput. Surv. 33, 209–271 (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Michael T. Goodrich
    • 1
  • Paweł Pszona
    • 1
  1. 1.Dept. of Computer ScienceUniversity of CaliforniaIrvineUSA

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