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

TL;DR

This paper examines how traditional correlation measures behave in large directed networks with heavy-tailed degrees and proposes rank-based alternatives that better capture degree dependencies.

Contribution

It demonstrates the limitations of Pearson's correlation in heavy-tailed directed networks and introduces Spearman's rho and Kendall's tau as more suitable measures.

Findings

Pearson's correlation converges to a non-negative number in large heavy-tailed directed networks.

Rank-based measures like Spearman's rho and Kendall's tau are more effective for heavy-tailed degree dependencies.

Empirical analysis on Wikipedia graphs supports the proposed measures.

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 non-negative 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.