Interval Bipartite Consensus

In Chap. 3, it is shown that the bipartite consensus and stability can be achieved for strongly connected signed networks, where the structural balance and unbalance of their associated signed digraphs play a key role, respectively. However, two questions naturally arise: (1) what disagreement behaviors will emerge for signed networks without the strong connectivity property, and (2) can they still be tied closely to the structural balance or unbalance properties of their related signed digraphs? In this chapter, we contribute to providing answers to these two questions by proceeding further with the discussions in Chap. 3 for the disagreement behavior analysis of signed networks subject to quasi-strongly connected topologies. Thanks to the M-matrix-based analysis approach, the equivalent relation is revealed between the semi-positive stability (respectively, positive stability) for the Laplacian matrix of a quasi-strongly connected signed digraph and the structural balance (respectively, unbalance) of its rooted subgraph. Note also that the digon sign-symmetry assumption upon signed digraphs is no longer imposed. In addition, it is disclosed that the interval bipartite consensus (respectively, stability) is exhibited for a quasi-strongly connected signed network if and only if its rooted subgraph is structurally balanced (respectively, unbalanced). Simulation examples are given to validate the disagreement behavior analysis results for quasi-strongly connected signed networks.