  This paper considers the discrete-time version of Altafini's model for
opinion dynamics in which the interaction among a group of agents is described
by a time-varying signed digraph. Prompted by an idea from [1], exponential
convergence of the system is studied using a graphical approach. Necessary and
sufficient conditions for exponential convergence 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. An upper bound on the convergence rate is also
provided.
