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

TL;DR

This paper establishes new sufficient conditions for bipartite consensus in matrix-weighted networks with weak connectivity, relaxing previous assumptions and broadening potential engineering applications.

Contribution

It introduces relaxed connectivity conditions using semidefinite matrices, extending bipartite consensus theory beyond positive-negative spanning trees.

Findings

Derived new algebraic conditions for bipartite consensus

Validated conditions through numerical simulations

Broadened applicability of bipartite consensus in networked systems

Abstract

Recent advances in bipartite consensus on matrix-weighted networks, where agents are divided into two disjoint sets with those in the same set agreeing on a certain value and those in different sets converging to opposite values, have highlighted its potential applications across various fields. Traditional approaches often depend on the existence of a positive-negative spanning tree in matrix-weighted networks to achieve bipartite consensus, which greatly restricts the use of these approaches in engineering applications. This study relaxes that assumption by allowing weak connectivity within the network, where paths can be weighted by semidefinite matrices. By analyzing the algebraic constraints imposed by positive-negative trees and semidefinite paths, we derive new sufficient conditions for achieving bipartite consensus. Our findings are validated by numerical simulations.