Correlated fluctuations in strongly-coupled binary networks beyond equilibrium

TL;DR

This paper introduces a systematic cumulant expansion method for analyzing activity and correlations in finite-sized, strongly-coupled binary networks, extending beyond equilibrium and mean-field approximations.

Contribution

It develops a novel non-linear approximation framework for single realizations of asymmetric couplings in finite networks, surpassing traditional mean-field and equilibrium theories.

Findings

Accurately predicts activity and pairwise covariances in networks with hundreds of units.

Provides an efficient algorithm for inverse problems in network connectivity.

Shows correlations are invariant under scaling of interaction strengths.

Abstract

Randomly coupled Ising spins constitute the classical model of collective phenomena in disordered systems, with applications covering ferromagnetism, combinatorial optimization, protein folding, stock market dynamics, and social dynamics. The phase diagram of these systems is obtained in the thermodynamic limit by averaging over the quenched randomness of the couplings. However, many applications require the statistics of activity for a single realization of the possibly asymmetric couplings in finite-sized networks. Examples include reconstruction of couplings from the observed dynamics, learning in the central nervous system by correlation-sensitive synaptic plasticity, and representation of probability distributions for sampling-based inference. The systematic cumulant expansion for kinetic binary (Ising) threshold units with strong, random and asymmetric couplings presented here goes beyond mean-field theory and is applicable outside thermodynamic equilibrium; a system of approximate non-linear equations predicts average activities and pairwise covariances in quantitative agreement with full simulations down to hundreds of units. The linearized theory yields an expansion of the correlation- and response functions in collective eigenmodes, leads to an efficient algorithm solving the inverse problem, and shows that correlations are invariant under scaling of the interaction strengths.