Bipartite Consensus on Matrix-Valued Weighted Networks

This brief examines bipartite consensus problem on matrix-valued weighted networks. First, it is shown that such networks achieve bipartite consensus if and only if the null space of the matrix-valued weighted Laplacian is spanned by a matrix-valued Gauge transformation, extending results for scalar-valued weighted networks. Second, it is shown that if a structurally balanced matrix-valued weighted network has a "positive-negative spanning tree," then the bipartite consensus can be achieved. Lastly, we show that in the case where edges are weighted by either positive definite or negative definite matrices, bipartite consensus is achieved if and only if the network is structurally balanced. Simulation results are provided to demonstrate these theoretical observations.