  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.
