Percolation and Loop Statistics in Complex Networks

TL;DR

This paper investigates how structural correlations in complex networks influence percolation transitions and loop statistics, revealing that degree distribution alone is insufficient to characterize critical phenomena.

Contribution

It demonstrates that genuine structural correlations are crucial for understanding percolation behavior and classifies networks based on loop scaling behaviors.

Findings

Networks with similar degree distributions can exhibit different percolation critical phenomena.

Two classes of networks are identified based on the scaling of finite loops with network size.

Percolation critical phenomena correlate with loop statistics and structural correlations.

Abstract

Complex networks display various types of percolation transitions. We show that the degree distribution and the degree-degree correlation alone are not sufficient to describe diverse percolation critical phenomena. This suggests that a genuine structural correlation is an essential ingredient in characterizing networks. As a signature of the correlation we investigate a scaling behavior in $M_N(h)$, the number of finite loops of size $h$, with respect to a network size $N$. We find that networks, whose degree distributions are not too broad, fall into two classes exhibiting $M_N(h)\sim ({constant})$ and $M_N(h) \sim (\ln N)^\psi$, respectively. This classification coincides with the one according to the percolation critical phenomena.