A Generalized Discrete-Time Altafini Model
This work extends the theoretical foundations of consensus and clustering in multi-agent systems to a broader class of interaction gains, but is incremental as it generalizes an existing model.
The paper generalizes the discrete-time Altafini model to complex-valued gains from a cyclic group, proving that exponential clustering occurs under repeated joint structural balance, and consensus at zero under repeated joint strong connectivity and structural imbalance.
A discrete-time modulus consensus model is considered in which the interaction among a family of networked agents is described by a time-dependent gain graph whose vertices correspond to agents and whose arcs are assigned complex numbers from a cyclic group. Limiting behavior of the model is studied using a graphical approach. It is shown that, under appropriate connectedness, a certain type of clustering 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 that type of clustering, where the number of clusters is at most the order of a cyclic group. It is also shown that the model will reach a consensus asymptotically at zero 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.