Quad Census Computation: Simple, Efficient, and Orbit-Aware

  • Mark Ortmann
  • Ulrik Brandes
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9564)


The prevalence of select substructures is an indicator of network effects in applications such as social network analysis and systems biology. Moreover, subgraph statistics are pervasive in stochastic network models, and they need to be assessed repeatedly in MCMC sampling and estimation algorithms. We present a new approach to count all induced and non-induced 4-node subgraphs (the quad census) on a per-node and per-edge basis, complete with a separation into their non-automorphic roles in these subgraphs. It is the first approach to do so in a unified manner, and is based on only a clique-listing subroutine. Computational experiments indicate that, despite its simplicity, the approach outperforms previous, less general approaches.


Social Network Analysis Preferential Attachment Subgraph Frequency Diagonal Edge Exponential Random Graph Model 
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.


  1. 1.
    Auber, D., Chiricota, Y., Jourdan, F., Melançon, G.: Multiscale visualization of small world networks. In: 9th IEEE Symposium on Information Visualization (InfoVis 2003), 20–21 October 2003, Seattle, WA, USA (2003)Google Scholar
  2. 2.
    Batagelj, V., Mrvar, A.: A subquadratic triad census algorithm for large sparse networks with small maximum degree. Soc. Netw. 23(3), 237–243 (2001)CrossRefGoogle Scholar
  3. 3.
    Chiba, N., Nishizeki, T.: Arboricity and subgraph listing algorithms. SIAM J. Comput. 14(1), 210–223 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Eppstein, D., Goodrich, M.T., Strash, D., Trott, L.: Extended dynamic subgraph statistics using h-index parameterized data structures. Theoret. Comput. Sci. 447, 44–52 (2012). doi: 10.1016/j.tcs.2011.11.034 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Eppstein, D., Spiro, E.S.: The h-index of a graph and its application to dynamic subgraph statistics. In: Dehne, F., Gavrilova, M., Sack, J.-R., Tóth, C.D. (eds.) WADS 2009. LNCS, vol. 5664, pp. 278–289. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  6. 6.
    Hočevar, T., Demšar, J.: A combinatorial approach to graphlet counting. Bioinformatics 30(4), 559–565 (2014)CrossRefGoogle Scholar
  7. 7.
    Holland, P.W., Leinhardt, S.: A method for detecting structure in sociometric data. Am. J. Sociol. 76(3), 492–513 (1970)CrossRefGoogle Scholar
  8. 8.
    Holland, P.W., Leinhardt, S.: Local structure in social networks. Sociol. Methodol. 7, 1–45 (1976)CrossRefGoogle Scholar
  9. 9.
    Kloks, T., Kratsch, D., Müller, H.: Finding and counting small induced subgraphs efficiently. Inf. Process. Lett. 74(3–4), 115–121 (2000)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Kowaluk, M., Lingas, A., Lundell, E.: Counting and detecting small subgraphs via equations and matrix multiplication. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, 23–25 January 2011, pp. 1468–1476 (2011)Google Scholar
  11. 11.
    Leskovec, J., Krevl, A.: SNAP Datasets: stanford large network dataset collection, June 2014.
  12. 12.
    Lin, M.C., Soulignac, F.J., Szwarcfiter, J.L.: Arboricity, h-index, and dynamic algorithms. Theor. Comput. Sci. 426, 75–90 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Marcus, D., Shavitt, Y.: RAGE - A rapid graphlet enumerator for large networks. Comput. Netw. 56(2), 810–819 (2012)CrossRefGoogle Scholar
  14. 14.
    Melançon, G., Sallaberry, A.: Edge metrics for visual graph analytics: a comparative study. In: 12th International Conference on Information Visualisation, IV 2008, 8–11 July 2008, London, UK, pp. 610–615 (2008)Google Scholar
  15. 15.
    Milenković, T., Lai, J., Pržulj, N.: GraphCrunch: a tool for large network analyses. BMC Bioinformatics 9(70), 1–11 (2008)Google Scholar
  16. 16.
    Milenković, T., Pržulj, N.: Uncovering biological network function via graphlet degree signatures. Cancer Inf. 6, 257–273 (2008)Google Scholar
  17. 17.
    Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., Alon, U.: Network motifs: simple building blocks of complex networks. Science 298(5594), 824–827 (2002)CrossRefGoogle Scholar
  18. 18.
    Ortmann, M., Brandes, U.: Triangle listing algorithms: back from the diversion. In: 2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments, ALENEX 2014, Portland, Oregon, USA, 5 January 2014, pp. 1–8 (2014)Google Scholar
  19. 19.
    Pržulj, N., Corneil, D.G., Jurisica, I.: Modeling interactome: scale-free or geometric? Bioinformatics 20(18), 3508–3515 (2004)CrossRefGoogle Scholar
  20. 20.
    Robins, G., Pattison, P., Kalish, Y., Lusher, D.: An introduction to exponential random graph (\(p^*\)) models for social networks. Soc. Netw. 29(2), 173–191 (2007)CrossRefGoogle Scholar
  21. 21.
    Solava, R.W., Michaels, R.P., Milenković, T.: Graphlet-based edge clustering reveals pathogen-interacting proteins. Bioinformatics 28(18), 480–486 (2012)CrossRefGoogle Scholar
  22. 22.
    Traud, A.L., Kelsic, E.D., Mucha, P.J., Porter, M.A.: Comparing community structure to characteristics in online collegiate social networks. SIAM Rev. 53(3), 526–543 (2011)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wernicke, S., Rasche, F.: FANMOD: a tool for fast network motif detection. Bioinformatics 22(9), 1152–1153 (2006)CrossRefGoogle Scholar
  24. 24.
    Zhou, X., Nishizeki, T.: Edge-coloring and \(f\)-coloring for various classes of graphs. In: Du, D.Z., Zhang, X.S. (eds.) Algorithms and Computation. LNCS, vol. 834, pp. 199–207. Springer, Heidelberg (1994)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Computer and Information ScienceUniversity of KonstanzKonstanzGermany

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