A Generalized Discrete-Time Altafini Model

TL;DR

This paper analyzes a generalized discrete-time consensus model with complex interactions, showing conditions for clustering or consensus based on the structure of the gain graphs and their balance properties.

Contribution

It extends the Altafini model to complex gains, providing a graphical approach to characterize the conditions for clustering and consensus in networked agents.

Findings

Clustering occurs exponentially fast under repeatedly jointly structurally balanced graphs.

Consensus at zero is achieved if the gain graphs are repeatedly jointly strongly connected and unbalanced.

The model generalizes the Altafini model to complex cyclic groups.

Abstract

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.