26 research outputs found
Continuum Percolation in the Relative Neighborhood Graph
In the present study, we establish the existence of nontrivial site
percolation threshold in the Relative Neighborhood Graph (RNG) for Poisson
stationary point process with unit intensity in the plane
Continuum percolation with steps in an annulus
Let A be the annulus in R^2 centered at the origin with inner and outer radii
r(1-\epsilon) and r, respectively. Place points {x_i} in R^2 according to a
Poisson process with intensity 1 and let G_A be the random graph with vertex
set {x_i} and edges x_ix_j whenever x_i-x_j\in A. We show that if the area of A
is large, then G_A almost surely has an infinite component. Moreover, if we fix
\epsilon, increase r and let n_c=n_c(\epsilon) be the area of A when this
infinite component appears, then n_c\to1 as \epsilon \to 0. This is in contrast
to the case of a ``square'' annulus where we show that n_c is bounded away from
1.Comment: Published at http://dx.doi.org/10.1214/105051604000000891 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Continuum Percolation in the Intrinsically Secure Communications Graph
The intrinsically secure communications graph (iS-graph) is a random graph
which captures the connections that can be securely established over a
large-scale network, in the presence of eavesdroppers. It is based on
principles of information-theoretic security, widely accepted as the strictest
notion of security. In this paper, we are interested in characterizing the
global properties of the iS-graph in terms of percolation on the infinite
plane. We prove the existence of a phase transition in the Poisson iS-graph,
whereby an unbounded component of securely connected nodes suddenly arises as
we increase the density of legitimate nodes. Our work shows that long-range
communication in a wireless network is still possible when a secrecy constraint
is present.Comment: Accepted in the IEEE International Symposium on Information Theory
and its Applications (ISITA'10), Taichung, Taiwan, Oct. 201
Percolation in invariant Poisson graphs with i.i.d. degrees
Let each point of a homogeneous Poisson process in R^d independently be
equipped with a random number of stubs (half-edges) according to a given
probability distribution mu on the positive integers. We consider
translation-invariant schemes for perfectly matching the stubs to obtain a
simple graph with degree distribution mu. Leaving aside degenerate cases, we
prove that for any mu there exist schemes that give only finite components as
well as schemes that give infinite components. For a particular matching scheme
that is a natural extension of Gale-Shapley stable marriage, we give sufficient
conditions on mu for the absence and presence of infinite components
Analyzing Interference from Static Cellular Cooperation using the Nearest Neighbour Model
The problem of base station cooperation has recently been set within the
framework of Stochastic Geometry. Existing works consider that a user
dynamically chooses the set of stations that cooperate for his/her service.
However, this assumption often does not hold. Cooperation groups could be
predefined and static, with nodes connected by fixed infrastructure. To analyse
such a potential network, in this work we propose a grouping method based on
proximity. It is a variation of the so called Nearest Neighbour Model. We
restrict ourselves to the simplest case where only singles and pairs of base
stations are allowed to be formed. For this, two new point processes are
defined from the dependent thinning of a Poisson Point Process, one for the
singles and one for the pairs. Structural characteristics for the two are
provided, including their density, Voronoi surface, nearest neighbour, empty
space and J-function. We further make use of these results to analyse their
interference fields and give explicit formulas to their expected value and
their Laplace transform. The results constitute a novel toolbox towards the
performance evaluation of networks with static cooperation.Comment: 10 pages, 6 figures, 12 total subfigures, WIOPT-SPASWIN 201
Poisson trees, succession lines and coalescing random walks
We give a deterministic algorithm to construct a graph with no loops (a tree
or a forest) whose vertices are the points of a d-dimensional stationary
Poisson process S, subset of R^d. The algorithm is independent of the origin of
coordinates. We show that (1) the graph has one topological end --that is, from
any point there is exactly one infinite self-avoiding path; (2) the graph has a
unique connected component if d=2 and d=3 (a tree) and it has infinitely many
components if d\ge 4 (a forest); (3) in d=2 and d=3 we construct a bijection
between the points of the Poisson process and Z using the preorder-traversal
algorithm. To construct the graph we interpret each point in S as a space-time
point (x,r)\in\R^{d-1}\times R. Then a (d-1) dimensional random walk in
continuous time continuous space starts at site x at time r. The first jump of
the walk is to point x', at time r'>r, (x',r')\in S, where r' is the minimal
time after r such that |x-x'|<1. All the walks jumping to x' at time r'
coalesce with the one starting at (x',r'). Calling (x',r') = \alpha(x,r), the
graph has vertex set S and edges {(s,\alpha(s)), s\in S}. This enables us to
shift the origin of S^o = S + \delta_0 (the Palm version of S) to another point
in such a way that the distribution of S^o does not change (to any point if d =
2 and d = 3; point-stationarity).Comment: 15 pages. Second version with minor correction
Percolation in the Secrecy Graph
The secrecy graph is a random geometric graph which is intended to model the
connectivity of wireless networks under secrecy constraints. Directed edges in
the graph are present whenever a node can talk to another node securely in the
presence of eavesdroppers, which, in the model, is determined solely by the
locations of the nodes and eavesdroppers. In the case of infinite networks, a
critical parameter is the maximum density of eavesdroppers that can be
accommodated while still guaranteeing an infinite component in the network,
i.e., the percolation threshold. We focus on the case where the locations of
the nodes and eavesdroppers are given by Poisson point processes, and present
bounds for different types of percolation, including in-, out- and undirected
percolation.Comment: 22 pages, 3 figure