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

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