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

By constructing a multicanonical Monte Carlo simulation and using the multiple histogram reweighting method, we obtain the full probability distribution $\rho_N(r)$ of the degree assortativity coefficient $r$ on configuration networks of size $N$. We suggest that $\rho_N(r)$ obeys a large deviation principle, $\rho_N \left( r-r_N^* \right) \asymp {e^{-{N^\xi } I\left( {r-r_N^* } \right)}}$, 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 $\xi\geq1$ 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.