On a New Class of Structurally Balanced Graphs for Scaled Group Consensus

Abstract

A distributed protocol that can achieve a scaled group consensus, or equivalently a multi-parition of the agents, over weighted, signed, and directed graphs is investigated. It is said that the scaled group consensus is achieved if a consensus value for a certain agent (group) can be described as the multiplication of one for another agent by a scaling factor for all agents. In this paper, it is assumed that each agent need not know all of the scaling factors and which group it belongs to. Then, the definition of n-structurally balanced graphs is proposed. Necessary and sufficient conditions are provided to achieve the multi-parition of the agents over the n-structurally balanced graphs. It is also provided that the distributed protocol can achieve in fact the scaled group consensus over the graphs. The results are illustrated through a numerical example.