Parallel dynamics of extremely diluted symmetric Q-Ising neural networks

TL;DR

This paper investigates the parallel dynamics of extremely diluted symmetric Q-Ising neural networks using a probabilistic approach, revealing new insights especially for Q>2 and deriving equilibrium fixed-point equations.

Contribution

It introduces a recursive scheme to determine the full time evolution of order parameters in symmetric Q-Ising networks, extending analysis to cases with Q>2.

Findings

Explicit analysis for Q=2 and Q=3 models.

Results extend previous Q=2 findings.

Derived equilibrium fixed-point equations and capacity-gain diagram.

Abstract

The parallel dynamics of extremely diluted symmetric Q-Ising neural networks is studied for arbitrary Q using a probabilistic approach. In spite of the extremely diluted architecture the feedback correlations arising from the symmetry prevent a closed-form solution in contrast with the extremely diluted asymmetric model. A recursive scheme is found determining the complete time evolution of the order parameters taking into account all feedback. It is based upon the evolution of the distribution of the local field, as in the fully connected model. As an illustrative example an explicit analysis is carried out for the Q=2 and Q=3 model. These results agree with and extend the partial results existing for Q=2. For Q>2 the analysis is entirely new. Finally, equilibrium fixed-point equations are derived and a capacity-gain function diagram is obtained.