A submodular optimization approach to leader-follower consensus in networks with negative edges

Networked systems often contain both positive and negative interactions between nodes, with the latter often represented as negative edge weights in a graph. Such negative edges may prevent a network from performing cooperative tasks such as achieving consensus, creating a need for new control-theoretic techniques that guarantee performance in the presence of negative edges. This paper considers the problem of selecting leader nodes to maintain a fixed state in order to steer the remaining nodes to a desired state value in networks with negative edges. We present two sufficient conditions that are equivalent to submodular constraints on the set of leader nodes. The first constraint is based on the graph spectrum, while the second is formulated in terms of the determinant of the Laplacian matrix. We prove that both conditions can be formulated as submodular constraints on the set of leader nodes, leading to polynomial-time algorithms with provable approximation guarantees for selecting a minimum-size set of leader nodes to satisfy these conditions. Furthermore, we introduce necessary conditions for consensus and prove that a set of leader nodes satisfying these conditions can be selected in polynomial time. We characterize the requirements for leader selection in order to ensure consensus in line networks. The results are illustrated through numerical study.