Dorottya Beringer: Controllability, matching ratio and graph convergence

Speaker: Dorottya Beringer

Title: Controllability, matching ratio and graph convergence

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 Barabási (2011). We gave proofs on 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. We also examined the convergence of the (directed) matching ratio of a random (directed) graph sequence that converges in the local weak sense, and generalized the result of Elek and Lippner (2009). This second result is that the mean of the directed matching ratio converges to the properly defined matching ratio parameter of the limiting graph. We further show that the matching ratios of the most widely used families of scale-free networks converge almost surely. Joint work with Ádám Timár.