Controllability and maximum matchings of complex networks

TL;DR

This paper introduces a simplified core percolation theory approach to estimate maximum matchings in complex networks, providing insights into controllability linked to network structure.

Contribution

The authors develop an alternative, more interpretable theory based on core percolation to estimate maximum matchings, applicable to various random graph types.

Findings

The new theory accurately estimates maximum matching sizes.

It applies to random graphs with or without degree symmetry.

The approach clarifies the link between network structure and controllability.

Abstract

Previously, the controllability problem of a linear time-invariant dynamical system was mapped to the maximum matching (MM) problem on the bipartite representation of the underlying directed graph, and the sizes of MMs on random bipartite graphs were calculated analytically with the cavity method at zero temperature limit. Here we present an alternative theory to estimate MM sizes based on the core percolation theory and the perfect matching of cores. Our theory is much more simplified and easily interpreted, and can estimate MM sizes on random graphs with or without symmetry between out- and in-degree distributions. Our result helps to illuminate the fundamental connection between the controllability problem and the underlying structure of complex systems.