Local clustering in scale-free networks with hidden variables

TL;DR

This paper studies how local clustering behaves in scale-free networks with hidden variables, revealing how clustering scales with network size and degree distribution, especially for exponents between 2 and 3.

Contribution

It provides a detailed analysis of clustering decay in correlated random graphs with hidden variables, including explicit scaling laws for different degree exponents.

Findings

Clustering decreases with hidden variable or degree.

Average clustering coefficient scales as $C o N^{2- au}\, ext{ln}N$ for certain parameters.

Clustering decay is extremely slow near $ au=2$, remaining significant in very large networks.

Abstract

We investigate the presence of triangles in a class of correlated random graphs in which hidden variables determine the pairwise connections between vertices. The class rules out self-loops and multiple edges and allows for negative degree correlations (disassortative mixing) due to infinite-variance degrees controlled by a structural cutoff $h_s$ and natural cutoff $h_c$. We show that local clustering decreases with the hidden variable (or degree). We also determine how the average clustering coefficient $C$ scales with the network size $N$, as a function of $h_s$ and $h_c$. For scale-free networks with exponent $2<\tau<3$ and the default choices $h_s\sim N^{1/2}$ and $h_c\sim N^{1/(\tau-1)}$ this gives $C\sim N^{2-\tau}\ln N$ for the universality class at hand. We characterize the extremely slow decay of $C$ when $\tau\approx 2$ and show that for $\tau=2.1$, say, clustering only starts to vanish for networks as large as $N=10^{11}$.