A Generalized Discrete-Time Altafini Model

A discrete-time modulus consensus model is considered in which the interactions between the members of a networked family of n agents is described by a time-dependent gain graph whose vertices correspond to agents and whose arcs are assigned complex numbers from a prescribed cyclic group. Limiting behavior of the model's state is studied using a graphical approach. It is shown that a certain type of clustering of agents' 'opinions' or states will be reached exponentially fast for almost all initial conditions if and only if the sequence of gain graphs is 'repeatedly jointly structurally balanced' corresponding to the type of clustering being reached, where the number of clusters is at most the order of the prescribed cyclic group. It is also shown that the agents' states will all converge to zero asymptotically if the sequence of gain graphs is repeatedly jointly strongly connected and structurally unbalanced. In the special case when the cyclic group is of order two, the model simplifies to the so-called Altafini model whose gain graph is simply a signed graph.