Degree-degree correlations in random graphs with heavy-tailed degrees

Abstract: Mixing patterns in large self-organizing networks, such as the Internet, the World Wide Web, social and biological networks are often characterized by degree-degree {dependencies} between neighbouring nodes. One of the problems with the commonly used Pearson's correlation coefficient (termed as the assortativity coefficient) is that {in disassortative networks its magnitude decreases} with the network size. This makes it impossible to compare mixing patterns, for example, in two web crawls of different size. We start with a simple model of two heavy-tailed highly correlated random variable $X$ and $Y$, and show that the sample correlation coefficient converges in distribution either to a proper random variable on $[-1,1]$, or to zero, and if $X,Y\ge 0$ then the limit is non-negative. We next show that it is non-negative in the large graph limit when the degree distribution has an infinite third moment. We consider the alternative degree-degree dependency measure, based on the Spearman's rho, and prove that it converges to an appropriate limit under very general conditions. We verify that these conditions hold in common network models, such as configuration model and Preferential Attachment model. We conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns.