Tight Bounds for Connectivity of Random K-out Graphs
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Random K-out graphs are used in several applications including modeling by sensor networks secured by the random pairwise key predistribution scheme, and payment channel networks. The random K-out graph with n nodes is constructed as follows. Each node draws an edge towards K distinct nodes selected uniformly at random. The orientation of the edges is then ignored, yielding an undirected graph. An interesting property of random K-out graphs is that they are connected almost surely in the limit of large n for any K ≥2. This means that they attain the property of being connected very easily, i.e., with far fewer edges (O(n)) as compared to classical random graph models including Erdős-Rényi graphs (O(n log n)). This work aims to reveal to what extent the asymptotic behavior of random K-out graphs being connected easily extends to cases where the number n of nodes is small. We establish upper and lower bounds on the probability of connectivity when n is finite. Our lower bounds improve significantly upon the existing results, and indicate that random K-out graphs can attain a given probability of connectivity at much smaller network sizes than previously known. We also show that the established upper and lower bounds match order-wise; i.e., further improvement on the order of n in the lower bound is not possible. In particular, we prove that the probability of connectivity is 1-Θ(1/n^K^2-1) for all K ≥ 2. Through numerical simulations, we show that our bounds closely mirror the empirically observed probability of connectivity.
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