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

2026-02-13 · Source: cs.MA updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Robotics & Autonomous Systems · Depth: Expert, extended

Summary

This paper introduces a novel approach to achieve non-trivial consensus in directed signed matrix-weighted networks, a state where multi-dimensional agents with both cooperative and antagonistic interactions converge to an arbitrarily preset, non-zero shared state. Unlike prior research focused on bipartite or trivial consensus in scalar-weighted networks, this work extends the concept to matrix-weighted networks, which better characterize inter-dimensional communication. The authors prove that for directed signed matrix-weighted networks, every eigenvalue of the grounded Laplacians has a positive real part under specific conditions, ensuring global asymptotic convergence. They derive lower bounds for coupling coefficients and propose a systematic algorithm involving informed agent selection, external signal design, and precise coupling term determination. The algorithm operates under milder connectivity conditions and is applicable to both structurally balanced and unbalanced networks, including those with switching topologies, where parameters dynamically adjust.

Key takeaway

For AI Scientists working on multi-agent system control or opinion dynamics, this research provides a robust framework for achieving non-trivial consensus in complex, matrix-weighted networks. You should consider implementing the proposed algorithm, which offers relaxed connectivity conditions and broad applicability to both balanced and unbalanced network structures. This enables steering diverse agent groups, even those with antagonistic interactions, towards a specific, non-zero target state, a capability previously limited to fully cooperative systems.

Key insights

Non-trivial consensus is achievable in complex, multi-dimensional networks with both cooperative and antagonistic interactions.

Principles

• Positive real part eigenvalues of grounded Laplacians ensure global asymptotic convergence.
• Network decomposition via positive-negative paths and "in-degree-dominated vertex" is crucial.
• Dynamic parameter adjustment is key for switching topologies.

Method

The method involves selecting informed agents, designing external signals, and determining coupling terms, with derived lower bounds for coupling coefficients, to steer multi-dimensional agents to a preset non-zero consensus state.

In practice

• Apply to UAV network formation control.
• Utilize for opinion manipulation in social networks.
• Design external inputs to achieve desired system states.

Topics

• Non-Trivial Consensus
• Matrix-Weighted Networks
• Signed Networks
• Grounded Laplacians
• Switching Topologies

Best for: AI Scientist, AI Researcher, Research Scientist, Robotics Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.MA updates on arXiv.org.