Abstract
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$r_n = O( n^{-\eps}),$ some $\eps >0$. We generalize the first result to higher dimensions,and to a larger class of connection probability functions.
Original language | English |
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Pages (from-to) | 986-1028 |
Number of pages | 43 |
Journal | Annals of Applied Probability |
Volume | 26 |
Issue number | 2 |
DOIs | |
Publication status | Published - Apr 2016 |
Keywords
- Random graph
- stochastic geometry
- random connection model
- connectivity
- isolated points
- continuum percolation