Edge Augmentation with Controllability Constraints in Directed Laplacian Networks

TL;DR

This paper develops algorithms for adding edges to directed Laplacian networks to increase density and robustness while maintaining strong structural controllability bounds based on zero forcing and node distances.

Contribution

It introduces novel edge augmentation algorithms that maximize edge addition without violating controllability bounds, including a closed-form solution for zero forcing bounds.

Findings

The zero forcing-based augmentation algorithm guarantees maximum edges added.

The distance-based augmentation algorithm provides a high-probability approximate solution.

Numerical evaluations compare the effectiveness of both augmentation methods.

Abstract

In this paper, we study the maximum edge augmentation problem in directed Laplacian networks to improve their robustness while preserving lower bounds on their strong structural controllability (SSC). Since adding edges could adversely impact network controllability, the main objective is to maximally densify a given network by selectively adding missing edges while ensuring that SSC of the network does not deteriorate beyond certain levels specified by the SSC bounds. We consider two widely used bounds: first is based on the notion of zero forcing (ZF), and the second relies on the distances between nodes in a graph. We provide an edge augmentation algorithm that adds the maximum number of edges in a graph while preserving the ZF-based SSC bound, and also derive a closed-form expression for the exact number of edges added to the graph. Then, we examine the edge augmentation problem while preserving the distance-based bound and present a randomized algorithm that guarantees an approximate solution with high probability. Finally, we numerically evaluate and compare these edge augmentation solutions.