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 ${\rho }_N(r)$ 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, ${\rho }_N\left(r-r_N^{\ast }\right)\asymp e^{-N^{\xi }I\left(r-r_N^{\ast }\right)}$, where the rate function $I$ 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.

9 pages, 9 figures

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• Generalized optimal paths and weight distributions revealed through the large deviations of random walks on networks
• Analytic solution of the two-star model with correlated degrees