Stability of the continuous-time Altafini model

This paper considers the continuous-time Altafini model for opinion dynamics in which the interaction among a group of agents is described by a piecewise-constant switching signed digraph (or directed graph). Building on an idea proposed in [1], stability of the system described by the model is studied using a graphical approach. It is shown that for any sequence of repeatedly jointly strongly connected digraphs, without any assumption on the sign structure of the graphs, the system asymptotically reaches a consensus in absolute value, including convergence to zero and different types of bipartite consensus (or two-clustering). Necessary and sufficient conditions for exponential stability with respect to each possible type of limit states are provided. Specifically, under the assumption of repeatedly jointly strong connectivity, it is shown that (1) a certain type of two-clustering will be reached exponentially fast for almost all initial conditions if, and only if, the sequence of signed digraphs is repeatedly jointly structurally balanced corresponding to that type of two-clustering; (2) the system will converge to zero exponentially fast for all initial conditions if, and only if, the sequence of signed digraphs is repeatedly jointly structurally unbalanced.