A Homomorphic Mapping Design for Multigroup Consensus With Signed Networks

This article addresses the multigroup consensus problem with signed networks, where a set of agents with positive-weight edges is treated as a group, and the edges among groups are negative weights. Based on group homomorphism theory, each group can be mapped as a virtual node, called the group representative node. Consequently, the signed networks can be represented by the homomorphic mapping graph with only negative weight edges. The definition of structural balance has been extended, eliminating the requirement for a structurally balanced network to have an even number of negative edges. As a result, a signed network with negative cycles is also considered structurally balanced. Then, multigroup symmetry consensus conditions are derived based on the characteristics of the Laplacian matrix associated with the homomorphic mapping graph. It can be concluded that average consensus and bipartite consensus are special cases of multigroup symmetric consensus. Each group in the signed networks is assigned different consensus parameters, thereby achieving asymmetric consensus, which together with symmetric consensus forms multigroup consensus control.