# Graph Distances and Controllability of Networks

arXiv: https://arxiv.org/abs/1608.04309 · HTML: https://mlsanity.com/paper/1608.04309/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Aug 2016.

## Built on by (most cited first, 2 of 2)

- [Control-based Graph Embeddings with Data Augmentation for Contrastive Learning](https://mlsanity.com/paper/2403.04923/) (2024)
- [Learning Backbones: Sparsifying Graphs through Zero Forcing for Effective Graph-Based Learning](https://mlsanity.com/paper/2502.17713/) (2025)

## Abstract (arXiv)

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.