# Strong Structural Controllability of Networks under Time-Invariant and Time-Varying Topological Perturbations

## TL;DR

This paper explores the robustness of strong structural controllability in directed networks under structural changes, providing bounds, characterizations, and conditions for both static and dynamic networks.

## Contribution

It introduces a perfect graph construct, derives bounds on edge modifications, characterizes critical edge-sets, and proposes controllability conditions for time-varying networks.

## Key findings

- Derived tight bounds on edge additions and deletions preserving controllability.
- Characterized maximal edge-sets that maintain strong structural controllability.
- Proposed controllability conditions for networks with time-varying structures and weights.

## Abstract

This paper investigates the robustness of strong structural controllability for linear time-invariant and linear time-varying directed networks with respect to structural perturbations, including edge deletions and additions. In this direction, we introduce a new construct referred to as a perfect graph associated with a network with a given set of control nodes. The tight upper bounds on the number of edges that can be added to, or removed from a network, while ensuring strong structural controllability, are then derived. Moreover, we obtain a characterization of critical edge-sets, the maximal sets of edges whose any subset can be respectively added to, or removed from a network, while preserving strong structural controllability. In addition, procedures for combining networks to obtain strongly structurally controllable network-of-networks are proposed. Finally, controllability conditions are proposed for networks whose edge weights, as well as their structures, can vary over time.