# Transient Dynamics of Sparsely Connected Hopfield Neural Networks with Arbitrary Degree Distributions

## TL;DR

This paper analyzes the transient behavior of sparsely connected Hopfield neural networks with various degree distributions using a probabilistic approach, providing explicit dynamics calculations and insights into optimal network configurations.

## Contribution

It introduces a recursive scheme to determine the time evolution of overlap parameters for arbitrary degree distributions in Hopfield networks.

## Key findings

- Explicit dynamics calculations for binomial, power-law, and uniform degree distributions.
- Good agreement between theoretical results and numerical simulations.
- Optimal network performance is achieved with a delta function degree distribution.

## Abstract

Using probabilistic approach, the transient dynamics of sparsely connected Hopfield neural networks is studied for arbitrary degree distributions. A recursive scheme is developed to determine the time evolution of overlap parameters. As illustrative examples, the explicit calculations of dynamics for networks with binomial, power-law, and uniform degree distribution are performed. The results are good agreement with the extensive numerical simulations. It indicates that with the same average degree, there is a gradual improvement of network performance with increasing sharpness of its degree distribution, and the most efficient degree distribution for global storage of patterns is the delta function.