Edge Convergence Problems on Signed Networks

This paper focuses on characterizing edge dynamics of signed networks subject to both cooperative and antagonistic interactions and copes with the state convergence problems of the resulting edge systems. To represent the two competitive classes of interactions that emerge in signed networks, signed digraphs are adopted and the relevant edge Laplacian matrices are introduced, with which an edge-based distributed protocol is presented. The relation between the edge Laplacian matrix and the structural balance (or unbalance) of a signed digraph is disclosed by taking advantage of properties of undirected cycles. Further, it is shown that for a signed network, the state of its edge system converges to a constant vector, regardless of whether its associated signed digraph is structurally balanced or unbalanced. This result does not need to impose the assumption upon the digon sign-symmetry of the signed digraph that is generally required by the node-based distributed protocols. In particular, the state convergence results of edges can be exploited to handle traditional bipartite consensus problems for the nodes of signed networks. Simulation examples are given to illustrate the effectiveness of the edge-based analysis method proposed for signed networks.