Tight Bounds for Connectivity of Random K-out Graphs
Abstract
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 $K \geq2$. 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 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 $1-\Theta({1}/{n^{K^2-1}})$ for all $K \geq 2$. Through numerical simulations, we show that our bounds closely mirror the empirically observed probability of connectivity.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2020
- DOI:
- 10.48550/arXiv.2006.10638
- arXiv:
- arXiv:2006.10638
- Bibcode:
- 2020arXiv200610638S
- Keywords:
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- Computer Science - Information Theory;
- Mathematics - Combinatorics;
- Mathematics - Probability
- E-Print:
- doi:10.1109/ICC42927.2021.9500694