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.

翻译：在多智能体协议中，正定/负定矩阵相较于半定矩阵对智能体的最终状态具有更强的调控作用。现有矩阵加权网络二部一致性的充分条件主要依赖于正负生成树，该类生成树要求强连接贯穿整个网络。为建立不存在此类生成树的弱连通矩阵加权网络的充分条件，我们首先识别图中一种兼具二部结构与收敛特性的基本单元，称之为"大陆"；继而推导出当多个此类单元通过具有半定矩阵权重的路径或边连接时的充分条件。最后，我们探讨如何统一无符号与有符号矩阵加权网络中的一致性与二部一致性研究，从而将所得结果推广至矩阵加权网络的一致性分析领域。