Optimality of Fast-Matching Algorithms for Random Networks With Applications to Structural Controllability

Abstract

Network control refers to a very large and diverse set of problems including controllability of linear time-invariant dynamical systems, where the objective is to select an appropriate input to steer the network to a desired state. There are many notions of controllability, one of them being structural controllability , which is intimately connected to finding maximum matchings on the underlying network topology. In this work, we study fast, scalable algorithms for finding maximum matchings for a large class of random networks . First, we illustrate that degree distribution random networks are realistic models for real networks in terms of structural controllability. Subsequently, we analyze a popular, fast, and practical heuristic due to Karp and Sipser as well as a simplification of it. For both heuristics, we establish asymptotic optimality and provide results concerning the asymptotic size of maximum matchings for an extensive class of random networks.