Large deviation and anomalous fluctuations scaling in degree assortativity on configuration networks

Abstract

By constructing a multicanonical Monte Carlo simulation, we obtain the full probability distribution $ρ_N(r)$ of the degree assortativity coefficient $r$ on configuration networks of size $N$ by using the multiple histogram reweighting method. We suggest that $ρ_N(r)$ obeys a large deviation principle, $ρ_N \left(r-r_N^* \right) \asymp {e^{ - {N^ξ}I\left( {r- r_N^* } \right)}}$, where the rate function $I$ is convex and possesses its unique minimum at $r=r_N^*$, and $ξ$ is an exponent that scales $ρ_N$'s with $N$. We show that $ξ=1$ for Poisson random graphs, and $ξ\geq1$ for scale-free networks in which $ξ$ is a decreasing function of the degree distribution exponent $γ$. Our results reveal that the fluctuations of $r$ exhibits an anomalous scaling with $N$ in highly heterogeneous networks.