Graph Distances and Controllability of Networks

TL;DR

This paper establishes a graph-theoretic lower bound on the controllability of diffusively coupled networks, linking controllability to graph distances, and provides methods for leader selection even with unknown weights.

Contribution

It introduces a tight, topology-based lower bound on controllability rank applicable to arbitrary networks and offers an algorithm for leader selection without knowing coupling weights.

Findings

Derived a graph distance-based lower bound on controllability

Provided an algorithm for computing the lower bound

Demonstrated leader selection method for controllability with unknown weights

Abstract

In this technical note, we study the controllability of diffusively coupled networks from a graph theoretic perspective. We consider leader-follower networks, where the external control inputs are injected to only some of the agents, namely the leaders. Our main result relates the controllability of such systems to the graph distances between the agents. More specifically, we present a graph topological lower bound on the rank of the controllability matrix. This lower bound is tight, and it is applicable to systems with arbitrary network topologies, coupling weights, and number of leaders. An algorithm for computing the lower bound is also provided. Furthermore, as a prominent application, we present how the proposed bound can be utilized to select a minimal set of leaders for achieving controllability, even when the coupling weights are unknown.