# CONNECTIVITY OF SOFT RANDOM GEOMETRIC GRAPHS

@article{Penrose2016CONNECTIVITYOS, title={CONNECTIVITY OF SOFT RANDOM GEOMETRIC GRAPHS}, author={Mathew D. Penrose}, journal={Annals of Applied Probability}, year={2016}, volume={26}, pages={986-1028} }

Consider a graph on $n$ uniform random points in the unit square, each pair being connected by an edge with probability $p$ if the inter-point distance is at most $r$. We show that as $n \to \infty$ the probability of full connectivity is governed by that of having no isolated vertices, itself governed by a Poisson approximation for the number of isolated vertices, uniformly over all choices of $p,r$. We determine the asymptotic probability of connectivity for all $(p_n,r_n)$ subject to$r_n = O…

## 109 Citations

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The regime in which the number of isolated nodes stabilizes is obtained, a precursor to establishing a threshold for connectivity, and the Poisson approximation result is derived using the Stein's method.

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### Random geometric graphs with general connection functions.

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Here the connection probability of a dense network in a convex domain in two or three dimensions is expressed in terms of contributions from boundary components for a very general class of connection functions.

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### On the Distribution of Random Geometric Graphs

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A series of upper bounds on the graph entropy are derived; in particular, the bound involving the entropy of a three-node graph is tighter than the existing bound which assumes distances are independent.

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