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

## TL;DR

This paper investigates the probability distribution of degree assortativity in networks, revealing large deviation principles and anomalous fluctuation scaling, especially in heterogeneous scale-free networks.

## Contribution

It introduces a multicanonical Monte Carlo method to analyze the full distribution of degree assortativity and establishes a large deviation principle with a novel scaling exponent.

## Key findings

- The distribution obeys a large deviation principle with a convex rate function.
- The scaling exponent $\xi$ equals 1 for Poisson graphs and varies for scale-free networks.
- Fluctuations exhibit anomalous scaling in highly heterogeneous 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, $\rho_N \left(r-r_N^* \right) \asymp {e^{ - {N^\xi }I\left( {r- r_N^* } \right)}}$, 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.