Cluster consensus with point group symmetries

A cluster consensus system is a multiagent system in which autonomous agents communicate to form groups, and agents within the same group converge to the same point, called the clustering point. We introduce in this paper a class of cluster consensus dynamics, termed G- clustering dynamics for G a point group, in which the autonomous agents can form as many as jGj clusters and, moreover, the associated jGj clustering points exhibit a geometric symmetry induced by the point group. The definition of a G-clustering dynamics relies on the use of the so-called voltage graph: A G-voltage graph is a directed graph (digraph) together with a map assigning elements of a group G to the edges of the digraph. For example, in the case when G = (-1; 1), i.e., the cyclic group of order 2, a voltage graph is nothing but a signed graph. G-clustering dynamics can then be viewed as a generalization of the so-called Altafini's model, which was originally defined over a signed graph, by defining the dynamics over a voltage graph. One of the main contributions of this paper is to identify a necessary and suficient condition for the exponential convergence of a G-clustering dynamics. Various properties of voltage graphs that are necessary for establishing the convergence result are also investigated, some of which might be of independent interest in topological graph theory.