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, $\rho_N \left(r-r_N^* \right) \asymp {e^{ - {N^\xi }I\left( {r- r_N^* } \right)}}$,wheretheratefunction$I$ is convex and possesses its unique minimum at $r=r_N^{\ast }$, and $\xi$ is an exponent that scales ${\rho }_N$’s with $N$. We show that $\xi =1$ for Poisson random graphs, and $\xi \ge 1$ 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.