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

TL;DR

This paper investigates fast algorithms for maximum matchings in random networks, demonstrating their asymptotic optimality and relevance to structural controllability in large-scale network control problems.

Contribution

It introduces and analyzes the asymptotic optimality of scalable heuristics for maximum matchings in random networks, linking degree distributions to structural controllability.

Findings

Degree distribution networks model real networks for controllability.

Karp-Sipser heuristic is asymptotically optimal for these networks.

Results provide asymptotic sizes of maximum matchings in large random networks.

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.