Tight Bounds for the Probability of Connectivity in Random K-out Graphs

Random K-out graphs are receiving increasing attention as a model to construct sparse, yet well connected topologies in a fully distributed fashion with applications in modeling sensor networks secured by random pairwise key predistribution schemes, decentralized learning, and cryptocurrency networks. A random K-out graph over a set of 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. Existing results on connectivity of random K-out graphs focus on an asymptotic setting where the number of nodes is infinite. In the asymptotic setting, it is known that random K-out graphs get connected for any K ≥ 2, and thus achieve connectivity easily, i.e., with far fewer edges (O(n)) as compared to classical random graph models including Erdős-Rényi graphs (O(n log n)). However, practical deployment of random K-out graphs as a tool for topology design requires an understanding of the connectivity of these graphs when the number of nodes is finite.In this work, we derive upper and lower bounds for probability of connectivity in random K-out graphs when the number of nodes is finite. Our matching upper and lower bounds prove that the probability of connectivity is $1 - \Theta \left( {1/{n^{{K^2} - 1}}} \right)$ for all K ≥ 2. Our work is the first to provide an upper bound on the probability of connectivity, which shows that further improvement on the order of n in the lower bound is not possible. Our corresponding lower bound significantly improves the existing ones. Together, our bounds provide a more precise characterization of the probability of connectivity, and reveal that random K-out graphs are an efficient way to construct a connected network topology when the number of nodes is finite. Through numerical simulations, we show that our bounds closely mirror the empirically observed probability of connectivity.