# 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}
}
• M. Penrose
• Published 2016
• Mathematics
• Annals of Applied Probability
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$r_n = O… Expand 96 Citations The random walk on the random connection model We study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices$x$and$y$are connected with probability that asymptotically behavesExpand Minimum spanning trees of random geometric graphs with location dependent weights Consider n nodes {Xi}1≤i≤n independently distributed in the unit square S, each according to a distribution f. Nodes Xi and Xj are joined by an edge if the Euclidean distance d(Xi,Xj) is less thanExpand Inhomogeneous random graphs, isolated vertices, and Poisson approximation • M. Penrose • Mathematics, Computer Science • Journal of Applied Probability • 2018 A general result on Poisson approximation by Stein's method is used for a set of points selected from a Poisson point process and this method also gives a goodPoisson approximation for U-statistics of aPoisson process. Expand Random geometric graphs with general connection functions. • Mathematics, Medicine • Physical review. E • 2016 Here the connection probability of a dense network in a convex domain in two or three dimensions is expressed in terms of contributions from boundary components for a very general class of connection functions. Expand Isolation and Connectivity in Random Geometric Graphs with Self-similar Intensity Measures • C. Dettmann • Mathematics, Physics • Journal of statistical physics • 2018 It is shown that nonuniformity can break the Poisson distribution property, but it strengthens the link between isolation and connectivity, and stretches out the connectivity transition. Expand On the Distribution of Random Geometric Graphs • Mathematics, Computer Science • 2018 IEEE International Symposium on Information Theory (ISIT) • 2018 A series of upper bounds on the graph entropy are derived; in particular, the bound involving the entropy of a three-node graph is tighter than the existing bound which assumes distances are independent. Expand Asymptotic adaptive threshold for connectivity in a random geometric social network. • Mathematics • 2018 Consider a dynamic random geometric social network identified by$s_t$independent points$x_t^1,\ldots,x_t^{s_t}$in the unit square$[0,1]^2$that interact in continuous time$t\geq 0$. TheExpand Recurrent random graphs in high dimensions: random connection model and long-range percolation We study the behavior of the random walk in both a contiuum and discrete independent long-range percolation model, in which two given vertices$x$and$y$are connected with probability thatExpand The random connection model: Connectivity, edge lengths, and degree distributions This paper proves some useful results on the asymptotic behavior of the length of the edges and the degree distribution in the connectivity regime and works for connection functions g that are not necessarily compactly supported but satisfy g(r)=o(r-c). Expand Connectivity of networks with general connection functions • Computer Science, Physics • ArXiv • 2014 Here, the full connection probability of a dense network in a convex polygonal or polyhedral domain is expressed in terms of contributions from boundary components, for a very general class of connection functions. Expand #### References SHOWING 1-10 OF 21 REFERENCES On Connectivity Thresholds in the Intersection of Random Key Graphs on Random Geometric Graphs • Mathematics, Computer Science • 2013 A random graph in which the RKG is superposed on the familiar random geometric graph (RGG). Expand Connectivity threshold of Bluetooth graphs • Computer Science, Mathematics • Random Struct. Algorithms • 2014 It is proved that no connectivity can take place with high probability for a range of parameters$r, c$and completely characterize the connectivity threshold (in$c\$) for values of r close the critical value for connectivity in the underlying random geometric graph. Expand
On connectivity thresholds in superposition of random key graphs on random geometric graphs
• Mathematics, Computer Science
• 2013 IEEE International Symposium on Information Theory
• 2013
A random graph in which the RKG is superposed on the familiar random geometric graph (RGG) for the graph to be asymptotically almost surely connected. Expand
The longest edge of the random minimal spanning tree
For n points placed uniformly at random on the unit square, suppose Mn (respectively,Mn) denotes the longest edge-length of the nearest neighbor graph (respectively, the minimal spanning tree) onExpand
Faulty Random Geometric Networks
• Mathematics, Computer Science
• Parallel Process. Lett.
• 2000
This paper first analyzes how to emulate an original random geometric network G on a faulty network F and shows that, with high probability, random geometric networks with (edge or node) faults do have a Hamiltonian cycle, provided the failure probability is constant. Expand
On a continuum percolation model
Consider particles placed in space by a Poisson process. Pairs of particles are bonded together, independently of other pairs, with a probability that depends on their separation, leading to theExpand
Full Connectivity: Corners, Edges and Faces
• Physics, Computer Science
• ArXiv
• 2012
A cluster expansion for the probability of full connectivity of high density random networks in confined geometries is developed and general analytical formulas that show a persistence of universality in a different form to percolation theory are derived. Expand
Random Geometric Graphs
This chapter discusses probabilistic ingredients, the largest component for a binomial process, and connectedivity and the number of components in a graph-like model. Expand
Asymptotic Distribution of The Number of Isolated Nodes in Wireless Ad Hoc Networks with Unreliable Nodes and Links
• Computer Science
• GLOBECOM
• 2006
The connectivity of a wireless ad hoc network that is composed of unreliable nodes and links is studied by investigating the distribution of the number of isolated nodes in the network and it is shown that if all nodes have a maximum transmission radius r n=radiclnn+xi/pipn for some constant xi, then the total number ofisolated nodes is asymptotically Poisson with mean e -xi. Expand
HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS
• Mathematics
• 2011
We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2-connected. This answers a question of Penrose. We also showExpand