Disassortativity of percolating clusters in random networks

TL;DR

This paper analyzes the degree-degree correlations in giant components of uncorrelated random networks during percolation, showing that disassortativity naturally emerges near the percolation threshold and confirming findings with simulations.

Contribution

It derives a general expression for the assortativity of giant components and proves disassortativity occurs near the percolation threshold in uncorrelated networks.

Findings

Assortativity $r$ is non-positive for $p \,\geq\, p_c$.

Average degree of neighbors scales as $k^{-1}$ at the threshold.

Disassortativity emerges in giant components near percolation threshold.

Abstract

We provide arguments for the property of the degree-degree correlations of giant components formed by the percolation process on uncorrelated random networks. Using the generating functions, we derive a general expression for the assortativity of a giant component, $r$, which is defined as Pearson's correlation coefficient for degrees of directly connected nodes. For uncorrelated random networks in which the third moment for the degree distribution is finite, we prove the following two points. (1) Assortativity $r$ satisfies the relation $r\le 0$ for $p\ge p_{\rm c}$. (2) The average degree of nodes adjacent to degree-$k$ nodes at the percolation threshold is proportional to $k^{-1}$ independently of the degree distribution function. These results claim that disassortativity emerges in giant components near the percolation threshold. The accuracy of the analytical treatment is confirmed by extensive Monte Carlo simulations.