Analytic solution of attractor neural networks on scale-free graphs

TL;DR

This paper analyzes how the topology of scale-free networks affects the retrieval capabilities of neural networks, revealing enhanced performance compared to random graphs through theoretical and simulation methods.

Contribution

It provides an analytic framework for neural networks on scale-free graphs using replica techniques and explores phase transitions and retrieval properties.

Findings

Enhanced retrieval in scale-free networks compared to Poissonian graphs

Phase diagram with paramagnetic, retrieval, and spin glass phases

Agreement between theoretical predictions and simulations

Abstract

We study the influence of network topology on retrieval properties of recurrent neural networks, using replica techniques for diluted systems. The theory is presented for a network with an arbitrary degree distribution $p(k)$ and applied to power law distributions $p(k) \sim k^{-\gamma}$, i.e. to neural networks on scale-free graphs. A bifurcation analysis identifies phase boundaries between the paramagnetic phase and either a retrieval phase or a spin glass phase. Using a population dynamics algorithm, the retrieval overlap and spin glass order parameters may be calculated throughout the phase diagram. It is shown that there is an enhancement of the retrieval properties compared with a Poissonian random graph. We compare our findings with simulations.