Structural Balance of Complex Weighted Graphs and Multi-Partite Consensus

The structural balance of a signed graph is known to be necessary and sufficient to obtain a bipartite consensus among agents with friend-foe relationships. In the real world, relationships are multifarious, and the coexistence of different opinions is ubiquitous. We are therefore motivated to study the multi-partite consensus problem of multi-agent systems, for which we extend the concept of structural balance to graphs with complex edge weights. It is shown that the generalized structural balance property is necessary and sufficient for achieving multi-partite consensus. Examples of multi-opinion social networks are illustrated as a real-world application of the proposed theoretical results.