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 15 November 2013
  • 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… 

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References

SHOWING 1-10 OF 21 REFERENCES

On Connectivity Thresholds in the Intersection of Random Key Graphs on Random Geometric Graphs

A random graph in which the RKG is superposed on the familiar random geometric graph (RGG).

On k-connectivity for a geometric random graph

  • M. Penrose
  • Mathematics
    Random Struct. Algorithms
  • 1999
For n points uniformly randomly distributed on the unit cube in d dimensions, with d≥2, let ρn (respectively, σn) denote the minimum r at which the graph, obtained by adding an edge between each pair

On connectivity thresholds in superposition of random key graphs on random geometric graphs

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.

Connectivity threshold of Bluetooth graphs

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.

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) on

Random geometric graphs.

An analytical expression for the cluster coefficient is derived, which shows that the graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality are distinctly different from standard random graphs, even for infinite dimensionality.

Faulty Random Geometric Networks

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.

On a continuum percolation model

  • M. Penrose
  • Mathematics
    Advances in Applied Probability
  • 1991
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 the

Asymptotic Distribution of The Number of Isolated Nodes in Wireless Ad Hoc Networks with Unreliable Nodes and Links

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.

HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS

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 show