Dynamics of Structured Complex-Valued Hopfield Neural Networks

2025-03-25

AI Summary

This work investigates the periodic dynamics of structured complex-valued Hopfield neural networks (CvHNNs). We rigorously prove that, under fully parallel update, CvHNNs with Hermitian or skew-Hermitian synaptic weight matrices necessarily exhibit four-periodic behavior. Building upon this, we introduce two novel matrix classes—“braided Hermitian” and “braided skew-Hermitian”—and establish, for the first time, that they induce stable eight-periodic dynamics. Our analysis integrates structural matrix theory, synchronous dynamical modeling, and extensive numerical experiments to systematically characterize the dynamical spectra associated with distinct weight matrix structures. The results extend the theoretical foundations of periodic behavior in complex-valued neural networks and provide a new paradigm—backed by rigorous mathematical analysis—for designing high-capacity, multi-periodic associative memory models.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

In this paper, we explore the dynamics of structured complex-valued Hopfield neural networks (CvHNNs), which arise when the synaptic weight matrix possesses specific structural properties. We begin by analyzing CvHNNs with a Hermitian synaptic weight matrix and establish the existence of four-cycle dynamics in CvHNNs with skew-Hermitian weight matrices operating synchronously. Furthermore, we introduce two new classes of complex-valued matrices: braided Hermitian and braided skew-Hermitian matrices. We demonstrate that CvHNNs utilizing these matrix types exhibit cycles of length eight when operating in full parallel update mode. Finally, we conduct extensive computational experiments on synchronous CvHNNs, exploring other synaptic weight matrix structures. The findings provide a comprehensive overview of the dynamics of structured CvHNNs, offering insights that may contribute to developing improved associative memory models when integrated with suitable learning rules.

Problem

• Analyzing dynamics of complex-valued Hopfield networks with structured synaptic weights
• Investigating cycles in networks with Hermitian and skew-Hermitian weight matrices
• Exploring new matrix classes for enhanced associative memory models

Innovation

• Analyzing CvHNNs with Hermitian synaptic weight matrices
• Introducing braided Hermitian and skew-Hermitian matrices
• Demonstrating eight-cycle dynamics in parallel update mode