Non-Trivial Consensus on Directed Matrix-Weighted Networks with Cooperative and Antagonistic Interactions

2026-02-12

AI Summary

This study addresses the nontrivial consensus problem in directed signed matrix-weighted networks, moving beyond the conventional restriction to purely cooperative groups. By introducing informed agents, designing external input signals, and establishing a lower bound on coupling strength, the authors ensure that the system state globally asymptotically converges to any prescribed consensus state lying in the null space of the Laplacian matrix. They provide the first proof that the eigenvalues of a directed grounded Laplacian matrix have positive real parts without requiring structural balance, thereby achieving consensus under weaker connectivity conditions. For undirected networks, they derive relaxed coupling conditions and further extend the results to switching topologies, establishing necessary conditions for convergence in such dynamic settings.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization

Abstract

This paper investigates the non-trivial consensus problem on directed signed matrix-weighted networks\textemdash a novel convergence state that has remained largely unexplored despite prior studies on bipartite consensus and trivial consensus. Notably, we first prove that for directed signed matrix-weighted networks, every eigenvalue of the grounded Laplacians has positive real part under certain conditions. This key finding ensures the global asymptotic convergence of systems states to the null spaces of signed matrix-weighted Laplacians, providing a foundational tool for analyzing dynamics on rooted signed matrix-weighted networks. To achieve non-trivial consensus, we propose a systematic approach involving the strategic selection of informed agents, careful design of external signals, and precise determination of coupling terms. Crucially, we derive the lower bounds of the coupling coefficients. Our consensus algorithm operates under milder connectivity conditions, and does not impose restrictions on whether the network is structurally balanced or unbalanced. Moreover, the non-trivial consensus state can be preset arbitrarily as needed. We also carry out the above analysis for undirected networks, with more relaxed conditions on the coupling coefficients comparing to the directed case. This paper further studies non-trivial consensus with switching topologies, and propose the necessary condition for the convergence of switching networks. The work in this paper demonstrates that groups with both cooperative and antagonistic multi-dimensional interactions can achieve consensus, which was previously deemed exclusive to fully cooperative groups.

Problem

• non-trivial consensus
• directed matrix-weighted networks
• cooperative and antagonistic interactions
• signed Laplacian
• multi-dimensional interactions

Innovation

• non-trivial consensus
• signed matrix-weighted networks
• grounded Laplacian
• cooperative and antagonistic interactions
• switching topologies