Degree-degree correlations in directed networks with heavy-tailed degrees

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, Pearson's correlation coefficients converge to a non-negative number in the infinite network size limit. We propose alternative measures for degree-degree correlations in directed networks based on Spearman's rho and Kendall's tau. Using examples and calculations on the Wikipedia graphs for nine dfferent languages, we show why these rank correlation measures are more suited for measuring degree assortativity in directed graphs with heavy-tailed degrees.