Degree-Degree Dependencies in Directed Networks with Heavy-Tailed Degrees

Abstract

In network theory, Pearson's correlation coefficients are most commonly used to measure the degree assortativity of a network. We investigate the behavior of these coefficients in the setting of directed networks with heavy-tailed degree sequences. We prove that for graphs where the in-and out-degree sequences satisfy a power law with realistic parameters, Pearson's correlation coefficients converge to a nonnegative number in the infinite network size limit. We propose alternative measures for degree-degree dependencies in directed networks based on Spearman's rho and Kendall's tau. Using examples and calculations on the Wikipedia graphs for nine different languages, we show why these rank correlation measures are more suited for measuring degree assortativity in directed graphs with heavy-tailed degrees.

Phase transitions for scaling of structural correlations in directed networks

“…However, Pearson's correlation coefficients are unable to measure strong negative degree-degree dependencies in large networks where the variance of the degrees is infinite, as was shown for undirected networks in [25,26] and for directed networks in [23]. Since our interest is mainly in networks in the infinite variance domain, i.e., 1 < γ ± 2, we need different measures.…”

“…Exact formulas for these three measures, in terms of the degrees, are given in [23]. In [21] formulas are given in terms of the empirical distributions of D α and D β and their joint distribution, evaluated at (D α i ,D β j ) for an edge i → j selected uniformly at random.…”

“…1. For the remainder of this paper we denote by E the number of edges and adopt the notation style from [22,23] to index the degree types by α,β ∈ {+,−}.…”

“…Since this data will contain many ties, one needs to use ranking schemes that deal with these ties. In [23] two such schemes are considered, resolving ties at random and assigning an average rank to tied values, which give two correlation measures denoted by ρ β α and ρ β α , respectively. Here, the subscript index denotes the degree type of the source, while the superscript index denotes the degree type of the target of a directed edge.…”

“…Since our interest is mainly in networks in the infinite variance domain, i.e., 1 < γ ± 2, we need different measures. In [23] it was suggested to use rank correlations, related to Spearman's rho [27] and Kendall's tau [28], to measure degree-degree dependencies.…”