Sufficient Conditions on Bipartite Consensus of Weakly Connected Matrix-weighted Networks

The positive/negative definite matrices are strong in the multi-agent protocol in dictating the agents' final states as opposed to the semidefinite matrices. Previous sufficient conditions on the bipartite consensus of the matrix-weighted network are heavily based on the positive-negative spanning tree whereby the strong connections permeate the network. To establish sufficient conditions for the weakly connected matrix-weighted network where such a spanning tree does not exist, we first identify a basic unit in the graph that is naturally bipartite in structure and in convergence, referred to as a continent. We then derive sufficient conditions for when several of these units are connected through paths or edges that are endowed with semidefinite matricial weights. Lastly, we discuss how consensus and bipartite consensus, unsigned and signed matrix-weighted networks should be unified, thus generalizing the obtained results to the consensus study of the matrix-weighted networks.