Local clustering in scale-free networks with hidden variables

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 $N=10^{11}$.