Bipartite consensus for multi-agent systems on directed signed networks

Collective dynamics is a complex emergence phenomenon yielded by local interactions within multi-agent systems. When agents cooperate or compete in the community, a collective behavior, such as consensus, polarization or diversity, may emerge. In this paper, we investigate a bipartite consensus process, in which all the agents converge to a final state characterized by identical modulus but opposite sign. Firstly, the interaction network of the agents is represented by a directed signed graph. A neighbor-based interaction rule is proposed for each agent with a single integrator dynamics. Then, we classify the signed network into heterogeneous networks and homogeneous networks according to the sign of edges. Under a weak connectivity assumption that the signed network has a spanning tree, some sufficient conditions are derived for bipartite consensus of multi-agent systems with the help of a structural balance theory. At the same time, signless Laplacian matrix and signed Laplacian matrix are introduced to analyze the bipartite consensus of multi-agent systems on homogeneous networks and heterogenous networks, respectively. Finally, simulation results are provided to demonstrate the bipartite consensus formation.