The Little-Hopfield model on a Random Graph

TL;DR

This paper analyzes the Little-Hopfield neural network model on a sparse random graph, revealing phase transition lines that match those of sequential dynamics and confirming findings through numerical simulations.

Contribution

It provides an analytical solution for the Little-Hopfield model on a random graph using replica symmetry and bifurcation analysis, extending understanding of neural network phase transitions.

Findings

Phase transition lines match sequential dynamics

First-order retrieval/spin-glass transition line identified

Numerical simulations confirm analytical predictions

Abstract

We study the Hopfield model on a random graph in scaling regimes where the average number of connections per neuron is a finite number and where the spin dynamics is governed by a synchronous execution of the microscopic update rule (Little-Hopfield model).We solve this model within replica symmetry and by using bifurcation analysis we prove that the spin-glass/paramagnetic and the retrieval/paramagnetictransition lines of our phase diagram are identical to those of sequential dynamics.The first-order retrieval/spin-glass transition line follows by direct evaluation of our observables using population dynamics. Within the accuracy of numerical precision and for sufficiently small values of the connectivity parameter we find that this line coincides with the corresponding sequential one. Comparison with simulation experiments shows excellent agreement.