Dynamical Stability of Threshold Networks over Undirected Signed Graphs

TL;DR

This paper investigates how the structure of undirected signed graphs influences the stability and long-term behavior of threshold networks, introducing a new stability index linked to graph antibalance.

Contribution

It introduces the stability index for signed graphs, connecting graph structure to network dynamics, and characterizes conditions for stability and oscillations in signed threshold networks.

Findings

Graphs with negative stability index on all subgraphs lead to stable fixed points.

Presence of a non-negative stability index in some subgraph can cause oscillations.

The analysis extends to periodic update schemes in network dynamics.

Abstract

This paper, we explore the dynamics of threshold networks on undirected signed graphs. Much attention has been dedicated to understanding the convergence and long-term behavior of this model. Yet, an open question persists: How does the underlying graph structure impact network dynamics? Similar studies have been carried out for threshold networks and other types of Boolean networks, but the latter primarily focus on unsigned networks. Here, we address this question in the context of signed threshold networks. We introduce the stability index of a signed graph, related to the concepts of antibalance in signed graphs. Our index establishes a connection between the structure and the dynamics of signed threshold networks. We show that signed graphs having a negative stability index on every induced subgraph exhibit stable dynamics, i.e., the dynamics converge to fixed points regardless of their threshold parameters. Conversely, if at least one induced subgraph has a non-negative stability index, oscillations in long-term behavior may appear. Furthermore, we generalize the analysis to network dynamics under periodic update schemes.