The Case for Kendall’s Assortativity
Abstract
Since the seminal work of Litvak and van der Hofstad [12], it has been known that Newman’s assortativity [14, 15], being based on Pearson’s correlation, is subject to a pernicious size effect which makes large networks with heavy-tailed degree distributions always unassortative. Usage of Spearman’s \(\rho \), or even Kendall’s \(\tau \) was suggested as a replacement [6], but the treatment of ties was problematic for both measures. In this paper we first argue analytically that the tie-aware version of \(\tau \) solves the problems observed in [6], and we show that Newman’s assortativity is heavily influenced by tightly knit communities. Then, we perform for the first time a set of large-scale computational experiments on a variety of networks, comparing assortativity based on Kendall’s \(\tau \) and assortativity based on Pearson’s correlation, showing that the pernicious effect of size is indeed very strong on real-world large networks, whereas the tie-aware Kendall’s \(\tau \) can be a practical, principled alternative.
References
- 1.Boldi, P., Codenotti, B., Santini, M., Vigna, S.: UbiCrawler: a scalable fully distributed web crawler. Softw. Pract. Exp. 34(8), 711–726 (2004)CrossRefGoogle Scholar
- 2.Boldi, P., Marino, A., Santini, M., Vigna, S.: BUbiNG: Massive crawling for the masses. ACM Trans. Web 12(2), 12:1–12:26 (2019)Google Scholar
- 3.Chierichetti, F., Kumar, R., Lattanzi, S., Mitzenmacher, M., Panconesi, A., Raghavan, P.: On compressing social networks. In: KDD 2009: Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 219–228. ACM, New York (2009)Google Scholar
- 4.Daniels, H.E.: The relation between measures of correlation in the universe of sample permutations. Biometrika 33(2), 129–135 (1943)MathSciNetCrossRefGoogle Scholar
- 5.Goodman, L.A., Kruskal, W.H.: Measures of association for cross classifications. J. Am. Stat. Assoc. 49(268), 732–764 (1954)zbMATHGoogle Scholar
- 6.van der Hoorn, P., Litvak, N.: Degree-degree dependencies in directed networks with heavy-tailed degrees. Internet Math. 11(2), 155–179 (2015)MathSciNetCrossRefGoogle Scholar
- 7.Kendall, M.G.: The treatment of ties in ranking problems. Biometrika 33(3), 239–251 (1945)MathSciNetCrossRefGoogle Scholar
- 8.Kleinberg, J.M.: Authoritative sources in a hyperlinked environment. J. ACM 46(5), 604–632 (1999)MathSciNetCrossRefGoogle Scholar
- 9.Knight, W.R.: A computer method for calculating Kendall’s tau with ungrouped data. J. Am. Stat. Assoc. 61(314), 436–439 (1966)CrossRefGoogle Scholar
- 10.Kurucz, M., Benczur, A., Csalogany, K., Lukacs, L.: Spectral clustering in telephone call graphs. In: Proceedings of the 9th WebKDD and 1st SNA-KDD 2007 Workshop on Web Mining and Social Network Analysis, WebKDD/SNA-KDD 2007, pp. 82–91. ACM (2007)Google Scholar
- 11.Kwak, H., Lee, C., Park, H., Moon, S.: What is Twitter, a social network or a news media? In: Proceedings of the 19th International Conference on World Wide Web, WWW 2010, pp. 591–600. ACM (2010)Google Scholar
- 12.Litvak, N., van der Hofstad, R.: Uncovering disassortativity in large scale-free networks. Phys. Rev. E 87, 022801 (2013)CrossRefGoogle Scholar
- 13.Mislove, A., Marcon, M., Gummadi, K.P., Druschel, P., Bhattacharjee, B.: Measurement and analysis of online social networks. In: Proceedings of the 7th ACM SIGCOMM Conference on Internet Measurement, pp. 29–42. ACM (2007)Google Scholar
- 14.Newman, M.E.J.: Assortative mixing in networks. Phys. Rev. Lett. 89, 208701 (2002)CrossRefGoogle Scholar
- 15.Newman, M.E.J.: Mixing patterns in networks. Phys. Rev. E 67, 026126 (2003)MathSciNetCrossRefGoogle Scholar
- 16.Spearman, C.: The proof and measurement of association between two things. Am. J. Psychol. 15(1), 72–101 (1904)CrossRefGoogle Scholar
- 17.Van Mieghem, P., Wang, H., Ge, X., Tang, S., Kuipers, F.A.: Influence of assortativity and degree-preserving rewiring on the spectra of networks. Eur. Phys. J. B 76(4), 643–652 (2010)CrossRefGoogle Scholar
- 18.Vigna, S.: A weighted correlation index for rankings with ties. In: Srinivasan, S., Ramamritham, K., Kumar, A., Ravindra, M.P., Bertino, E., Kumar, R. (eds.) Proceedings of the 24th International Conference on World Wide Web, pp. 1166–1176. ACM (2015)Google Scholar