A Combinatorial Necessary and Sufficient Condition for Cluster Consensus

TL;DR

This paper establishes a precise combinatorial condition that determines when a set of stochastic matrices guarantees cluster consensus in multi-agent systems, extending existing theories and providing practical checks.

Contribution

It introduces a new combinatorial necessary and sufficient condition for cluster consensus, extending the avoiding set condition and clarifying the role of cluster-spanning trees.

Findings

The condition is easy to verify with elementary routines.

Cluster-spanning trees are both necessary and sufficient under certain conditions.

The results unify and extend existing cluster consensus criteria.

Abstract

In this technical note, cluster consensus of discrete-time linear multi-agent systems is investigated. A set of stochastic matrices $\mathcal{P}$ is said to be a cluster consensus set if the system achieves cluster consensus for any initial state and any sequence of matrices taken from $\mathcal{P}$. By introducing a cluster ergodicity coefficient, we present an equivalence relation between a range of characterization of cluster consensus set under some mild conditions including the widely adopted inter-cluster common influence. We obtain a combinatorial necessary and sufficient condition for a compact set $\mathcal{P}$ to be a cluster consensus set. This combinatorial condition is an extension of the avoiding set condition for global consensus, and can be easily checked by an elementary routine. As a byproduct, our result unveils that the cluster-spanning trees condition is not only sufficient but necessary in some sense for cluster consensus problems.