Tight Bounds for Connectivity of Random Kout Graphs
Abstract
Random Kout 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 Kout 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 Kout graphs is that they are connected almost surely in the limit of large $n$ for any $K \geq2$. 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 ErdősRényi graphs ($O(n \log n)$). This work aims to reveal to what extent the asymptotic behavior of random Kout 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 Kout 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 orderwise; 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 $1\Theta({1}/{n^{K^21}})$ for all $K \geq 2$. Through numerical simulations, we show that our bounds closely mirror the empirically observed probability of connectivity.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.10638
 Bibcode:
 2020arXiv200610638S
 Keywords:

 Computer Science  Information Theory;
 Mathematics  Combinatorics;
 Mathematics  Probability