Triad-Based Comparison and Signatures of Directed Networks

  • Xiaochuan Xu
  • Gesine ReinertEmail author
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 812)


We introduce two methods for comparing directed networks based on triad counts, called TriadEuclid and TriadEMD. TriadEuclid clusters the Euclidean distance between triad counts, whereas TriadEMD is an adaptation of NetEMD for directed networks. We apply both methods to cluster synthetic networks, a set of web networks including google, twitter, peer-to-peer, amazon, slashdot and citation networks, as well as world trade networks from 1962-2000. Furthermore, we find signature triads and signature orbits for each type of networks in our data, which show the main triad and orbit contributions of the networks when comparing them to the other networks in the respective data set.


Network comparison Triad Signature triad Signature orbit 



This work was partially supported by the Alan Turing Institute. GR acknowledges the COST Action CA15109. The authors would like to thank Luis Ospina-Forero and Martin O’Reilly for helpful discussions, and Andrew Elliott for helpful discussions as well as computing support. They would also like to thank the anonymous referees for many helpful comments.

Supplementary material (254 kb)
Supplementary material 1 (zip 254 KB)


  1. 1.
    Ali, W., Rito, T., Reinert, G., Sun, F., Deane, C.M.: Alignment-free protein interaction network comparison. Bioinformatics 30(17), i430–i437 (2014)Google Scholar
  2. 2.
    Aparicio, D., Ribeiro, P., Silva, F.: Extending the applicability of graphlets to directed networks. IEEE/ACM Trans. Comput. Biol. Bioinform. (TCBB) 14(6), 1302–1315 (2017)Google Scholar
  3. 3.
    Borgwardt, K.M., Kriegel, H.P., Vishwanathan, S., Schraudolph, N.N.: Graph kernels for disease outcome prediction from protein-protein interaction networks. Pacific Symp. Biocomput. 12, 4–15 (2007)Google Scholar
  4. 4.
    Dwass, M.: Modified randomization tests for nonparametric hypotheses. Ann. Math. Stat. 181–187 (1957)Google Scholar
  5. 5.
    Faust, K.: A puzzle concerning triads in social networks: graph constraints and the triad census. Soc. Netw. 32(3), 221–233 (2010)Google Scholar
  6. 6.
    Feenstra, R.C., Lipsey, R.E., Deng, H., Ma, A.C., Mo, H.: World trade flows: 1962–2000. Technical report, National Bureau of Economic Research (2005)Google Scholar
  7. 7.
    Holland, P.W., Leinhardt, S.: Local structure in social networks. Sociol. Methodol. 7, 1–45 (1976)Google Scholar
  8. 8.
    Hubert, L., Arabie, P.: Comparing partitions. J. Classif. 2(1), 193–218 (1985)Google Scholar
  9. 9.
    Kuchaiev, O., Pržulj, N.: Integrative network alignment reveals large regions of global network similarity in yeast and human. Bioinformatics 27(10), 1390–1396 (2011)Google Scholar
  10. 10.
    Leskovec, J., Krevl, A.: SNAP Datasets: Stanford large network dataset collection. (2014). Accessed 21 May 2017
  11. 11.
    Mamano, N., Hayes, W.B.: Sana: Simulated annealing far outperforms many other search algorithms for biological network alignment. Bioinformatic (2017)Google Scholar
  12. 12.
    Milo, R., Itzkovitz, S., Kashtan, N., Levitt, R., Shen-Orr, S., Ayzenshtat, I., Sheffer, M., Alon, U.: Superfamilies of evolved and designed networks. Science 303(5663), 1538–1542 (2004)Google Scholar
  13. 13.
    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)Google Scholar
  14. 14.
    Newman, M.: Networks:An Introduction. Oxford University Press (2010).
  15. 15.
    Neyshabur, B., Khadem, A., Hashemifar, S., Arab, S.S.: Netal: a new graph-based method for global alignment of protein-protein interaction networks. Bioinformatics 29(13), 1654–1662 (2013)Google Scholar
  16. 16.
    Picard, F., Daudin, J.J., Koskas, M., Schbath, S., Robin, S.: Assessing the exceptionality of network motifs. J. Comput. Biol. 15(1), 1–20 (2008)Google Scholar
  17. 17.
    Pržulj, N.: Biological network comparison using graphlet degree distribution. Bioinformatics 23(2), e177–e183 (2007)Google Scholar
  18. 18.
    Rito, T., Wang, Z., Deane, C.M., Reinert, G.: How threshold behaviour affects the use of subgraphs for network comparison. Bioinformatics 26(18), i611–i617 (2010)Google Scholar
  19. 19.
    Rubner, Y., Tomasi, C., Guibas, L.J.: A metric for distributions with applications to image databases. In: Sixth International Conference on Computer Vision, pp. 59–66. IEEE (1998)Google Scholar
  20. 20.
    Sarajlić, A., Malod-Dognin, N., Yaveroğlu, Ö.N., Pržulj, N.: Graphlet-based characterization of directed networks. Sci. Rep. 6, 35,098 (2016)Google Scholar
  21. 21.
    Villani, C.: Optimal Transport: Old and New, vol. 338. Springer Science & Business Media (2008)Google Scholar
  22. 22.
    Wale, N., Watson, I.A., Karypis, G.: Comparison of descriptor spaces for chemical compound retrieval and classification. Knowl. Inf. Syst. 14(3), 347–375 (2008)Google Scholar
  23. 23.
    Wasserman, S., Faust, K.: Social Network Analysis: Methods and Applications, vol. 8. Cambridge University Press (1994)Google Scholar
  24. 24.
    Wegner, A.E., Ospina-Forero, L., Gaunt, R.E., Deane, C.M., Reinert, G.: Identifying networks with common organizational principles. J. Complex Netw. (2017)Google Scholar
  25. 25.
    Wilson, R.C., Zhu, P.: A study of graph spectra for comparing graphs and trees. Pattern Recognit. 41(9), 2833–2841 (2008)Google Scholar
  26. 26.
    Yaveroğlu, Ö.N., et al.: Revealing the hidden language of complex networks. Sci. Rep. 4, 4547 (2014)Google Scholar

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Authors and Affiliations

  1. 1.Department of StatisticsUniversity of OxfordOxfordUK

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