Sufficient Conditions on Bipartite Consensus of Weakly Connected Matrix-weighted Networks

TL;DR

This paper establishes new sufficient conditions for bipartite consensus in weakly connected matrix-weighted networks, relaxing traditional connectivity assumptions and broadening practical applicability.

Contribution

It introduces algebraic conditions based on semidefinite matrices that guarantee bipartite consensus without requiring positive-negative spanning trees.

Findings

Derived algebraic conditions for bipartite consensus

Validated conditions through numerical simulations

Extended bipartite consensus theory to weakly connected networks

Abstract

Recent advancements in bipartite consensus, a scenario where agents are divided into two disjoint sets with agents in the same set agreeing on a certain value and those in different sets agreeing on opposite or specifically related values, have highlighted its potential applications across various fields. Traditional research typically relies on the presence of a positive-negative spanning tree, which limits the practical applicability of bipartite consensus. This study relaxes that assumption by allowing for weak connectivity within the network, where paths can be weighted by semidefinite matrices. By exploring the algebraic constraints imposed by positive-negative trees and semidefinite paths, we derive sufficient conditions for achieving bipartite consensus. Our theoretical findings are validated through numerical results.