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In this paper, we study the maximum edge augmentation problem in directedLaplacian networks to improve their robustness while preserving lower bounds ontheir strong structural controllability (SSC). Since adding edges couldadversely impact network controllability, the main objective is to maximallydensify a given network by selectively adding missing edges while ensuring thatSSC of the network does not deteriorate beyond certain levels specified by theSSC bounds. We consider two widely used bounds: first is based on the notion ofzero forcing (ZF), and the second relies on the distances between nodes in agraph. We provide an edge augmentation algorithm that adds the maximum numberof edges in a graph while preserving the ZF-based SSC bound, and also derive aclosed-form expression for the exact number of edges added to the graph. Then,we examine the edge augmentation problem while preserving the distance-basedbound and present a randomized algorithm that guarantees an approximatesolution with high probability. Finally, we numerically evaluate and comparethese edge augmentation solutions.

Edge Augmentation with Controllability Constraints in Directed Laplacian Networks