CONNECTIVITY OF SOFT RANDOM GEOMETRIC GRAPHS
By Mathew D. Penrose. 2013
142 citations recorded by the source corpus.
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.
Subfields: Computer Graphics and Computer-Aided Design, Condensed Matter Physics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/5fcbd78f8791714d672807ba96193edd6406d778
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In Computer Graphics and Computer-Aided Design
In Condensed Matter Physics
In Mathematical Physics
Citations in the corpus
- Recurrence versus transience for weight-dependent random connection models (2019, background): “ature. Vertices Profile Kernel Name and reference Poisson indicator plain Random geometric graph, Gilbert disc model [24] Poisson general plain Random connection model [18] Soft random geometric graph [23] lattice polynomial plain Long-range percolation [1] Poisson indicator sum Boolean model [9, 19]”