Finding Induced Subgraphs in Scale-Free Inhomogeneous Random Graphs

  • Ellen Cardinaels
  • Johan S. H. van Leeuwaarden
  • Clara StegehuisEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10836)


We study the induced subgraph isomorphism problem on inhomogeneous random graphs with infinite variance power-law degrees. We provide a fast algorithm that determines for any connected graph H on k vertices if it exists as induced subgraph in a random graph with n vertices. By exploiting the scale-free graph structure, the algorithm runs in O(nk) time for small values of k. We test our algorithm on several real-world data sets.



The work of JvL and CS was supported by NWO TOP grant 613.001.451. The work of JvL was further supported by the NWO Gravitation Networks grant 024.002.003, an NWO TOP-GO grant and by an ERC Starting Grant.


  1. 1.
    Albert, R., Jeong, H., Barabási, A.L.: Internet: diameter of the world-wide web. Nature 401(6749), 130–131 (1999)CrossRefGoogle Scholar
  2. 2.
    Boguñá, M., Pastor-Satorras, R.: Class of correlated random networks with hidden variables. Phys. Rev. E 68, 036112 (2003)CrossRefGoogle Scholar
  3. 3.
    Bollobás, B., Janson, S., Riordan, O.: The phase transition in inhomogeneous random graphs. Random Struct. Algorithms 31(1), 3–122 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brach, P., Cygan, M., Łacki, J., Sankowski, P.: Algorithmic complexity of power law networks. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, pp. 1306–1325. Society for Industrial and Applied Mathematics, Philadelphia (2016)Google Scholar
  5. 5.
    Britton, T., Deijfen, M., Martin-Löf, A.: Generating simple random graphs with prescribed degree distribution. J. Stat. Phys. 124(6), 1377–1397 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chung, F., Lu, L.: The average distances in random graphs with given expected degrees. Proc. Natl. Acad. Sci. USA 99(25), 15879–15882 (2002) (electronic)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Clauset, A., Shalizi, C.R., Newman, M.E.J.: Power-law distributions in empirical data. SIAM Rev. 51(4), 661–703 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Faloutsos, M., Faloutsos, P., Faloutsos, C.: On power-law relationships of the internet topology. ACM SIGCOMM Comput. Commun. Rev. 29, 251–262 (1999)CrossRefGoogle Scholar
  9. 9.
    Fountoulakis, N., Friedrich, T., Hermelin, D.: On the average-case complexity of parameterized clique. arXiv:1410.6400v1 (2014)
  10. 10.
    Fountoulakis, N., Friedrich, T., Hermelin, D.: On the average-case complexity of parameterized clique. Theor. Comput. Sci. 576, 18–29 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Friedrich, T., Krohmer, A.: Cliques in hyperbolic random graphs. In: INFOCOM Proceedings 2015, pp. 1544–1552. IEEE (2015)Google Scholar
  12. 12.
    Friedrich, T., Krohmer, A.: Parameterized clique on inhomogeneous random graphs. Disc. Appl. Math. 184, 130–138 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Garey, M.R., Johnson, D.S., Garey, M.R.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W H FREEMAN & CO (2011)Google Scholar
  14. 14.
    Grochow, J.A., Kellis, M.: Network motif discovery using subgraph enumeration and symmetry-breaking. In. RECOMB, pp. 92–106 (2007)Google Scholar
  15. 15.
    Heydari, H., Taheri, S.M.: Distributed maximal independent set on inhomogeneous random graphs. In: 2017 2nd Conference on Swarm Intelligence and Evolutionary Computation (CSIEC). IEEE, March 2017Google Scholar
  16. 16.
    van der Hofstad, R.: Random Graphs and Complex Networks, vol. 1. Cambridge University Press, Cambridge (2017)CrossRefGoogle Scholar
  17. 17.
    van der Hofstad, R., van Leeuwaarden, J.S.H., Stegehuis, C.: Optimal subgraph structures in scale-free networks. arXiv:1709.03466 (2017)
  18. 18.
    Janson, S., Łuczak, T., Norros, I.: Large cliques in a power-law random graph. J. Appl. Probab. 47(04), 1124–1135 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Jeong, H., Tombor, B., Albert, R., Oltvai, Z.N., Barabási, A.L.: The large-scale organization of metabolic networks. Nature 407(6804), 651–654 (2000)CrossRefGoogle Scholar
  20. 20.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations. The IBM Research Symposia Series, pp. 85–103. Springer, Boston (1972). Scholar
  21. 21.
    Kashtan, N., Itzkovitz, S., Milo, R., Alon, U.: Efficient sampling algorithm for estimating subgraph concentrations and detecting network motifs. Bioinformatics 20(11), 1746–1758 (2004)CrossRefGoogle Scholar
  22. 22.
    Leskovec, J., Krevl, A.: SNAP Datasets: Stanford large network dataset collection (2014). Accessed 14 Mar 2017
  23. 23.
    Niu, X., Sun, X., Wang, H., Rong, S., Qi, G., Yu, Y.: - weaving chinese linking open data. In: Aroyo, L., Welty, C., Alani, H., Taylor, J., Bernstein, A., Kagal, L., Noy, N., Blomqvist, E. (eds.) ISWC 2011. LNCS, vol. 7032, pp. 205–220. Springer, Heidelberg (2011). Scholar
  24. 24.
    Norros, I., Reittu, H.: On a conditionally poissonian graph process. Adv. Appl. Probab. 38(01), 59–75 (2006)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Omidi, S., Schreiber, F., Masoudi-Nejad, A.: MODA: an efficient algorithm for network motif discovery in biological networks. Genes Genetic Syst. 84(5), 385–395 (2009)CrossRefGoogle Scholar
  26. 26.
    Park, J., Newman, M.E.J.: Statistical mechanics of networks. Phys. Rev. E 70, 066117 (2004)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Schreiber, F., Schwobbermeyer, H.: MAVisto: a tool for the exploration of network motifs. Bioinformatics 21(17), 3572–3574 (2005)CrossRefGoogle Scholar
  28. 28.
    Vázquez, A., Pastor-Satorras, R., Vespignani, A.: Large-scale topological and dynamical properties of the internet. Phys. Rev. E 65, 066130 (2002)CrossRefGoogle Scholar
  29. 29.
    Williams, V.V., Wang, J.R., Williams, R., Yu, H.: Finding four-node subgraphs in triangle time. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, pp. 1671–1680. Society for Industrial and Applied Mathematics, Philadelphia (2015)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Ellen Cardinaels
    • 1
  • Johan S. H. van Leeuwaarden
    • 1
  • Clara Stegehuis
    • 1
    Email author
  1. 1.Eindhoven University of TechnologyEindhovenThe Netherlands

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