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The positive/negative definite matrices are strong in the multi-agentprotocol in dictating the agents' final states as opposed to the semidefinitematrices. Previous sufficient conditions on the bipartite consensus of thematrix-weighted network are heavily based on the positive-negative spanningtree whereby the strong connections permeate the network. To establishsufficient conditions for the weakly connected matrix-weighted network wheresuch a spanning tree does not exist, we first identify a basic unit in thegraph that is naturally bipartite in structure and in convergence, referred toas a continent. We then derive sufficient conditions for when several of theseunits are connected through paths or edges that are endowed with semidefinitematricial weights. Lastly, we discuss how consensus and bipartite consensus,unsigned and signed matrix-weighted networks should be unified, thusgeneralizing the obtained results to the consensus study of the matrix-weightednetworks.

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