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.

Although the consensus problem has been extensively investigated, the ties among agents are assumed to be characterized by scalar-weighted networks, which fail in characterizing interdependencies among higher-dimensional states of neighboring agents. Recently, a broader category of networks termed matrix-weighted networks has been introduced which is an immediate generalization of scalar-weighted networks Sun and Yu [27], Pan et al. [23, 20], Trinh et al. [28], Pan et al. [21, 22], Wang et al. [30], Pan et al. [19]. In fact, matrix-weighted networks naturally become relevant in scenarios such as graph effective resistance based distributed control and estimation Barooah and Hespanha [2], logical inter-dependency of multiple topics in opinion evolution Friedkin et al. [8], bearing-based formation control Zhao and Zelazo [37], array of coupled LC oscillators Tuna [29] as well as consensus and synchronization on matrix-weighted networks Trinh et al. [28], Pan et al. [20]. As opposed to scalar-weighted networks, connectivity alone does not translate to achieving consensus for matrixweighted networks. To this end, properties of weight matrices play an important role in characterizing consensus. For instance, positive definiteness and positive semi-definiteness of weight matrices have been employed to provide consensus conditions in Trinh et al. [28]; negative definiteness and negative semi-definiteness of weight matrices