AbstractThere 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).

中文翻译：

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可控性、匹配率和图收敛

摘要 控制理论中有一个重要参数与网络的有向匹配率密切相关，如 Liu 等人的论文所示。（自然 473：167-173，2011 年）。我们给出了刘等人的两个主要陈述的证明。(2011) 关于定向匹配率，这是基于统计物理学的数值结果和启发式。首先，我们表明由固定度数序列给出的有向随机网络的有向匹配率集中在其均值附近。我们还检查了在局部弱意义上收敛的随机（有向）图序列的（有向）匹配比的收敛性，并概括了 Elek 和 Lippner（Proc Am Math Soc 138(8):2939–2947， 2010）。我们证明了有向匹配率的均值收敛于限制图的正确定义的匹配率参数。我们进一步展示了最广泛使用的无标度网络系列的匹配率几乎肯定收敛，这是 Liu 等人的主要动机。(2011)。