Cluster Consensus of the Matrix-Weighted Network on A Negative Circle

Despite the simplicity of its protocol, the matrix-weighted multi-agent system naturally admits cluster consensus under a wide range of network parameters, with mechanisms that have yet to be fully understood. Here we examine the system on a negative circle and study its clustering effect with the edge set named as the Nontrivial Balancing Set (NBS). A single NBS, upon the negation of its edges, implies a bipartition of the agents both structurally and dynamically. When the cyclic network contains more than one NBS, clusters of the agents' asymptotic values develop as a result of different bipartitions on the graph and in the Laplacian null space. We prove that the vectors associated with the NBSs span the null space of the Laplacian matrix for such networks. Furthermore, we show that agents partitioned into the same group by all of the NBSs achieve consensus, and that the clusters are separated dynamically under an appropriate initial value condition.