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    This study introduces a new bipartite consensus protocol for multiagent systems with matrix coupling, enabling consensus in complex antagonistic networks. The research provides algebraic conditions for achieving consensus and cluster consensus, crucial for understanding network dynamics.

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    Area of Science:

    • Control Theory
    • Network Science
    • Multiagent Systems

    Background:

    • Scalar-weighted consensus in multiagent systems is well-studied.
    • Antagonistic interaction networks require matrix coupling to describe agent state interdependencies.
    • Existing models often overlook the impact of matrix coupling in such networks.

    Purpose of the Study:

    • To investigate the second-order matrix-weighted bipartite consensus problem on undirected structurally balanced signed networks.
    • To highlight the influence of matrix coupling in antagonistic interaction networks.
    • To develop conditions for achieving bipartite consensus and cluster bipartite consensus.

    Main Methods:

    • Development of a bipartite consensus protocol.
    • Utilization of matrix-valued Gauge transformation and stability theory.
    • Derivation of algebraic conditions and algebraic graph conditions.

    Main Results:

    • An algebraic condition for achieving second-order bipartite consensus was obtained.
    • A direct algebraic graph condition for bipartite consensus was established.
    • An algebraic graph condition for cluster bipartite consensus was provided, demonstrating the critical role of matrix weights.

    Conclusions:

    • The proposed protocol effectively achieves second-order bipartite consensus in networks with matrix coupling.
    • Matrix weights are critical for consensus and can be designed to achieve a desired number of clusters.
    • Theoretical results are validated through five simulation examples.