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

By constructing a multicanonical Monte Carlo simulation, we obtain the full probability distribution $rho_N(r)$ of the degree assortativity coefficient $r$ on configuration networks of size $N$ by using the multiple histogram reweighting method. We suggest that $rho_N(r)$ obeys a large deviation principle, $rho_N left( r- r_N^* right) asymp {e^{ - {N^xi }Ileft( {r- r_N^* } right)}}$, where the rate function $I$ is convex and possesses its unique minimum at $r=r_N^*$, and $xi$ is an exponent that scales $rho_N$s with $N$. We show that $xi=1$ for Poisson random graphs, and $xigeq1$ for scale-free networks in which $xi$ is a decreasing function of the degree distribution exponent $gamma$. Our results reveal that the fluctuations of $r$ exhibits an anomalous scaling with $N$ in highly heterogeneous networks.