Non-trivial consensus on directed signed matrix-weighted networks with compound measurement noises and time-varying topologies

2026-03-14

AI Summary

This work proposes a novel stochastic consensus protocol for directed, matrix-weighted signed networks subject to antagonistic interactions, compound measurement noise, and time-varying topologies. By constructing a stochastic differential equation model that integrates inter-dimensional cooperation and antagonism, composite noise, and dynamic topology, and leveraging the properties of the grounded matrix-weighted Laplacian together with a carefully designed control gain function, the protocol achieves convergence to a nontrivial consensus state both in mean square and almost surely. Notably, this result requires only boundedness of the entries of the edge weight matrices and does not rely on structural balance. The approach overcomes the traditional limitation of consensus protocols to purely cooperative networks and guarantees convergence to a prescribed non-zero consensus state under relatively weak connectivity conditions, offering both theoretical rigor and practical feasibility.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Agreement, Argumentation & Negotiation
• Game Theory and Economic Paradigms: Cooperative Game Theory

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Computational social science
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This paper studies non-trivial consensus--a relatively novel and unexplored convergence behavior--on directed signed matrix-weighted networks subject to both additive and multiplicative measurement noises under time-varying topologies. Building upon grounded matrix-weighted Laplacian properties, a stochastic dynamic model is established that simultaneously captures inter-dimensional cooperative and antagonistic interactions, compound measurement noises and time-varying network structures. Based on stochastic differential equations theory, protocols that guarantee mean square and almost sure non-trivial consensus are proposed. Specifically, for any predetermined non-trivial consensus state, all agents are proven to converge toward this non-zero value in the mean-square and almost-sure senses. The design of control gain function in our protocols highlights a balanced consideration of the cumulative effect over time, the asymptotic decay property and the finite energy corresponding to measurement noises. Notably, the conditions on time-varying topologies in our protocols only require boundedness of elements in edge weight matrices, which facilitate the practicality of concept "time-varying topology" in matrix-weighted network consensus algorithms. Furthermore, the proposed protocols operate under milder connectivity conditions and no requirements on structural (un)balance properties. The work in this paper demonstrates that groups with both cooperative and antagonistic inter-dimensional interactions can achieve consensus even in the presence of compound measurement noises and time-varying topologies, challenging the conventional belief that consensus is attainable only in fully cooperative settings.

Problem

• non-trivial consensus
• signed matrix-weighted networks
• compound measurement noises
• time-varying topologies
• directed networks

Innovation

• non-trivial consensus
• matrix-weighted networks
• compound measurement noises
• time-varying topologies
• stochastic differential equations