Leírás
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, Slotine and Barabási (2011).
We gave proofs of two main statements of that paper on the directed
matching ratio, which were based on numerical results and heuristics
from statistical physics. The first result is that the directed
matching ratio of directed random networks given by a fix sequence of
degrees is concentrated around its mean. The second result is about
the convergence of the (directed) matching ratio of a random
(directed) graph sequence that converges in the local weak sense. This
generalizes the result of Elek and Lippner (2010). We proved that the
mean of the directed matching ratio converges to the properly defined
matching ratio parameter of the limiting graph. We further showed 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 Barabási.