Delay-Induced Stability Transitions in Directed Signed Consensus Networks

TL;DR

This paper investigates how time delays in signed network consensus models can cause stability transitions, leading from consensus to instability, with theoretical analysis and numerical validation.

Contribution

It introduces a delay differential equation model for signed networks with heterogeneous delays and analyzes how delays influence stability and induce transitions.

Findings

Increasing delays can destabilize the consensus state.

Delay-induced transitions include non-convergent behavior and instability.

A phase diagram maps different dynamical regimes based on delays.

Abstract

We study delay-induced transitions in consensus dynamics on signed networks with a ring topology. The proposed model is formulated as a system of delay differential equations incorporating both cooperative and antagonistic interactions, as well as heterogeneous time delays. We perform a stability analysis by deriving the associated characteristic equation and examining the real parts of its eigenvalues. It is shown that the stability of the consensus state depends critically on the magnitude of the delays. In particular, increasing time delays may destabilize the system and induce transitions from consensus to bounded non-convergent behavior or instability. A phase diagram in the parameter space is constructed to identify different dynamical regimes. Numerical simulations validate the theoretical results and illustrate the delay-induced transitions. Such delay-induced transitions have also been reported in various biological and engineered systems, highlighting the universal role of time delays in shaping collective dynamics.