Analogue neural networks on correlated random graphs

TL;DR

This paper analyzes a generalized Hopfield neural network model with Gaussian, diluted patterns, revealing how tuning dilution affects network topology, thermodynamics, and critical behavior, including phase transitions and ergodicity breakdown.

Contribution

It provides an analytical study of the topological and thermodynamic properties of a diluted, Gaussian-pattern Hopfield model, including free energy and phase transition analysis.

Findings

Different topological regimes can be achieved by tuning dilution.

The network exhibits high cliquishness even when sparse.

Dilution rescaled the critical noise level for ergodicity breakdown.

Abstract

We consider a generalization of the Hopfield model, where the entries of patterns are Gaussian and diluted. We focus on the high-storage regime and we investigate analytically the topological properties of the emergent network, as well as the thermodynamic properties of the model. We find that, by properly tuning the dilution in the pattern entries, the network can recover different topological regimes characterized by peculiar scalings of the average coordination number with respect to the system size. The structure is also shown to exhibit a large degree of cliquishness, even when very sparse. Moreover, we obtain explicitly the replica symmetric free energy and the self-consistency equations for the overlaps (order parameters of the theory), which turn out to be classical weighted sums of 'sub-overlaps' defined on all possible sub-graphs. Finally, a study of criticality is performed through a small-overlap expansion of the self-consistencies and through a whole fluctuation theory developed for their rescaled correlations: Both approaches show that the net effect of dilution in pattern entries is to rescale the critical noise level at which ergodicity breaks down.