Abstract
There is an important parameter in control theory which is closely related to the directed matching ratio of the network, as shown in the paper of Liu et al. (Nature 473:167–173, 2011). We give proofs of two main statements of Liu et al. (2011) 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 (Proc Am Math Soc 138(8):2939–2947, 2010). 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 et al. (2011).
Highlights
Introduction and ResultsLiu et al [10] examined the controllability of both real networks and network models
For the most widely used families of scale-free networks, the directed matching ratio converges to a constant. These two latter statements were based on numerical results, and for the last one methods from statistical physics were used
In Sect. 3.2.1 we prove the results on the matching ratio that imply part (2)
Summary
Liu et al [10] examined the controllability of both real networks and network models. We prove that if a sequence of random directed graphs is obtained from a convergent deterministic graph sequence by orienting each edge independently, it converges almost surely in the local weak sense, see Definition 4. This is our Lemma 5 which is similar to Proposition 2.2 in [7]. We get that for directed graphs obtained from almost sure convergent undirected graph sequences the matching ratios converge almost surely This result applies for sequences given by the random configuration model or Erdos– Rényi random graphs.
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