Correlations in interacting systems with a network topology

TL;DR

This paper investigates how pair correlations decay in cooperative systems on complex networks, revealing that correlations weaken faster than expected, especially in networks with fat-tailed degree distributions, limiting observable correlations to nearest neighbors.

Contribution

It provides a theoretical analysis of correlation decay in complex networks and derives the pair correlation function for the Ising model on such networks.

Findings

Correlations decay faster than 1/(ll z_ll) on average.

In fat-tailed networks, only nearest neighbor correlations are significant.

Derived the pair correlation function for the Ising model on complex networks.

Abstract

We study pair correlations in cooperative systems placed on complex networks. We show that usually in these systems, the correlations between two interacting objects (e.g., spins), separated by a distance $\ell$, decay, on average, faster than $1/(\ell z_\ell)$. Here $z_\ell$ is the mean number of the $\ell$-th nearest neighbors of a vertex in a network. This behavior, in particular, leads to a dramatic weakening of correlations between second and more distant neighbors on networks with fat-tailed degree distributions, which have a divergent number $z_2$ in the infinite network limit. In this case, only the pair correlations between the nearest neighbors are observable. We obtain the pair correlation function of the Ising model on a complex network and also derive our results in the framework of a phenomenological approach.