Controllability, matching ratio and graph convergence

TL;DR

This paper proves key properties of the directed matching ratio in random and scale-free networks, including concentration, convergence, and almost sure convergence, extending prior heuristic and numerical results.

Contribution

It provides rigorous proofs for the concentration and convergence of directed matching ratios in various network models, generalizing previous heuristic findings.

Findings

Directed matching ratio concentrates around its mean.

Matching ratio converges in the local weak sense.

Almost sure convergence for scale-free networks.

Abstract

There is an important parameter in control theory which is closely related to the directed matching ratio of the network, as shown by Liu, Slotine and Barab\'asi (2011). We give proofs on two main statements of that paper on the directed matching ratio, which were based on numerical results and heuristics from statistical physics. First, we show that the directed matching ratio of directed random networks given by a fix sequence of degrees is concentrated around its mean. We also examine the convergence of the (directed) matching ratio of a random (directed) graph sequence that converges in the local weak sense, and generalize the result of Elek and Lippner (2009). We prove that the mean of the directed matching ratio converges to the properly defined matching ratio parameter of the limiting graph. We further show the almost sure convergence of the matching ratios for the most widely used families of scale-free networks, which was the main motivation of Liu, Slotine and Barab\'asi.