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    This study analyzes edge dynamics in signed networks with cooperative and antagonistic interactions. It demonstrates that edge states converge regardless of network balance, offering a novel approach for bipartite consensus problems.

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    Area of Science:

    • Network Science
    • Control Theory
    • Graph Theory

    Background:

    • Signed networks exhibit complex dynamics due to cooperative and antagonistic interactions.
    • Understanding edge dynamics is crucial for analyzing network behavior and achieving consensus.
    • Existing node-based protocols often require restrictive assumptions like digon sign-symmetry.

    Purpose of the Study:

    • To characterize edge dynamics in signed networks with mixed interactions.
    • To develop an edge-based distributed protocol for state convergence.
    • To investigate the relationship between edge Laplacian matrices and network structural balance.

    Main Methods:

    • Adoption of signed digraphs to represent competing interactions.
    • Introduction of edge Laplacian matrices for network analysis.
    • Development of an edge-based distributed protocol for state convergence.

    Main Results:

    • The state of the edge system converges to a constant vector for both balanced and unbalanced signed digraphs.
    • A novel relationship between the edge Laplacian matrix and structural balance is established using undirected cycle properties.
    • The proposed edge-based method bypasses the need for digon sign-symmetry, unlike traditional node-based protocols.

    Conclusions:

    • The edge-based analysis provides a robust method for studying signed network dynamics.
    • State convergence of edge systems can be leveraged to solve bipartite consensus problems.
    • The findings offer a more general framework for signed network analysis and control.