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.
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Notes
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We use “unassortative” for networks with a correlation close to 0, and “disassortative” for networks with a correlation close to \(-1\).
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We mention that also Goodman–Kruskal’s \(\gamma \) [5], defined as the difference between concordances and discordances divided by their sum, provides a principled treatment of ties. However, Kendall’s \(\tau \) has some advantages, and in particular the possibility of defining tie-aware weighted versions [18].
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Boldi, P., Vigna, S. (2020). The Case for Kendall’s Assortativity. In: Cherifi, H., Gaito, S., Mendes, J., Moro, E., Rocha, L. (eds) Complex Networks and Their Applications VIII. COMPLEX NETWORKS 2019. Studies in Computational Intelligence, vol 882. Springer, Cham. https://doi.org/10.1007/978-3-030-36683-4_24
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