Efficient Hopfield pattern recognition on a scale-free neural network

TL;DR

This paper demonstrates that implementing Hopfield neural networks on scale-free networks significantly reduces memory and computational requirements while maintaining effective pattern recognition, especially for large networks.

Contribution

The study introduces a novel approach of using scale-free networks for Hopfield models, improving efficiency in pattern recognition tasks compared to traditional fully connected networks.

Findings

Good associative memory achieved for intermediate connectivity levels

Memory and computational efficiency scale with N/m

Retrieval quality decreases with very small m

Abstract

Neural networks are supposed to recognise blurred images (or patterns) of $N$ pixels (bits) each. Application of the network to an initial blurred version of one of $P$ pre-assigned patterns should converge to the correct pattern. In the "standard" Hopfield model, the $N$ "neurons'' are connected to each other via $N^2$ bonds which contain the information on the stored patterns. Thus computer time and memory in general grow with $N^2$. The Hebb rule assigns synaptic coupling strengths proportional to the overlap of the stored patterns at the two coupled neurons. Here we simulate the Hopfield model on the Barabasi-Albert scale-free network, in which each newly added neuron is connected to only $m$ other neurons, and at the end the number of neurons with $q$ neighbours decays as $1/q^3$. Although the quality of retrieval decreases for small $m$, we find good associative memory for $1 \ll m \ll N$. Hence, these networks gain a factor $N/m \gg 1$ in the computer memory and time.