Poisson point processes applied to sensors networks

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Poisson point processes applied to sensors networks

Laurent Decreusefond, Eduardo Ferraz, Hugues Randriambololona

To cite this version:

Laurent Decreusefond, Eduardo Ferraz, Hugues Randriambololona. Poisson point processes applied to sensors networks. Spatial Networks Models for Wirelles Communications, Apr 2010, Cambridge, United Kingdom. �hal-00472487�

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Poisson Point Processes Applied to Sensor Networks

{eduardo.ferraz, laurent.decreusefond, randriam}@telecom-paristech.fr

Motivation

The development of wireless ad hoc network-ing capability together with the decreasnetwork-ing costs and sizes of the electronical circuits al-low an increasing using of sensor networks in building, utilities, industrial, home, agricul-ture, defense and many other contexts. The topology of such networks, particularly the connectiveness and coverage area, are impor-tant and, sometimes, critical factors. Recently, many works dealing with topology give inter-pretations and techniques capable to be applied on sensor networks. Besides, it is not possible to control the positions and number of sensors in many of those networks, which lead us to study the topology of random sensor networks by using the Poisson point processes.

Problem Formulation

The principal idea of the problem is that sen-sors {S1, S2, ..., Sn} have a power suply

allow-ing them to transmit theirs ID’s and maybe some environmental information (such as tem-perature, pression, presence/absence of an ele-ment etc). At the same time, the sensors have receivers which can identify the transmitted ID’s of other sensors above a threshold power. The sensors, knowing theirs ID’s and the ID’s of the close neighbors, create an information network.

Physical Features of the System

• The sensors lie over a d-torus Tda and the

dimensions of the sensor are considered too reduced compared to the system, so the position of the sensor Si is given by

xi ∈ Td_a = (ui,1, ..., ud,i), ud,k ∈ [0, a];

• A sensor receives the ID’s from all other sensors closer then a deterministic dis-tance ǫ, so if kxi − xjk ≤ ǫ, sensors Si and

Sj are directly connected;

• We use the maximum norm, i.e.,

kxi − xjk = max

k (ui,k − uj,k)

.

Simplicial Homology

Simplicial complexes are structures com-posed by elements named simplices, which can be seen by d-dimensional filled spaces. Exemples of simplexes are given below:

0−simplex 1−simplex 2−simplex 3−simplex

βd denotes the number of d-dimensional holes.

Particularly, β0, β1 and β2 measure,

respec-tively, the connectiveness, the number of holes and the number of voids of a complex.

= 1 :

= 2

β0

Two connex components:

The 3−dimensional being "Blue Point" is trapped, we have one hole: = 1 β₂ The 2−dimensional being "Red Point" is trapped, we have one void β₁

Results: Mean of

k

-simplexes,

s

_k

, and Euler’s Characteristic,

χ

It is possible to calculate the mean of k-simplices given the size of individual coverage ǫ, the density of sensors λ, the dimension d and the sizes of the d-torus, a:

s_k−1 = λ

k_ad

k! k

d _ǫ_k−1d

, ǫ < a/3

Below, we present the variation of sk in function of ǫ for a = 100 and λ = 0.10 in two dimensions.

4 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 s1 ǫ

Comparison between the calculations and the simulations

s1 simulated s1 calculated 6 8 10 12 0 0.5 1 1.5 2 2.5 3 s3 ǫ

s3 simulated s3 calculated 0.2 0.25 0 0.5 1 1.5 2 2.5 3 s6 ǫ

s6 simulated s6 calculated 0 0.05 0.1 0.15 0 2 0

Let Bd be the Bell’s polynomial. Using the mean of k-simplexes, we can calculate the mean of the

Euler’s Characteristic χ: χ = a d_λe_−λǫd −λǫd Bd(−λǫ d )

The variation of χ in function of λ is presented following, for d = 1, d = 2 and d = 3:

10 10 20 30 40 50 60 70 0 0.5 1 1.5 2 2.5 3 3.5 4 M ea n λ Sim χ Calc χ Sim β1 Sim β0 Sim β2

For d = 2, a = 10 and ǫ = 0.5: χ = a2λe−λǫ

2

1 − λǫ2

15 20

λ

For d = 1, a = 40 and ǫ = 0.5: χ = aλe−λǫ

χ Sim β2 Sim β1 Sim χ Calc χ Sim β0 Sim χ Calc χ 0 5 10 15 -800 -600 -400 -200 0 200 400 600 800 1000 0 0.5 1 1.5 2 2.5 3 3.5 4 M ea n λ

For d = 3, a = 10 and ǫ = 0.5: χ = a3λe−λǫ

3 1 − 3λǫ3+ (λǫ3)2 20 25 30 0 5 0

Conjecture:

β

_i

dominance region

Based on simulations and analytical expres-sions, we can conjecture that, given a density of points, there are at most two dominating types of holes. No dominance -10 -8 -6 -4 -2 0 2 4 6 8 10 0 2 4 6 8 10 12 14 16 χ λ

Domination regions of βk when d = 5 in function of λ

Dominating βk β0 β0 and β1 β1 β2 β3 β4 β1 and β2 β2 and β3 β3 and β4 -12

Results: Concentration Inequality

Since a compensated Poisson point process can be seen as a martingale, we can use a concen-tration inequality to find a superior limit for P (β₀ ≥ c) in two dimensions

Distribution β0: λ=2, R=0.5 and a=10

0.05 0.1 0.15 0.2 0.25 0.3 170 175 180 185 190 195 200 205 Psim Psup c P (β0 > c) 0

Interpreting a Sensor Network

We can represent the topology of a sensor net-work by its Rips complex, which is obtained when we consider that whenever k + 1 points are 2 by 2 closer than ǫ between them, they cre-ate a k-simplex. Sensor Coverage 0-simplex 1-simplex 2-simplex 3-simplex

References

[1] R. Ghrist, A. Muhammad. Coverage and

Hole-Detection in Sensor Networks Via Homology In Fourth

International Conference on Information Processing in Sen-sor Networks (IPSN’05), UCLA, 2005.

[2] C. Houdré, N. Privault. Concentration and deviation inaqualities in infinite dimensions via covariance rep-resentations Bernoulli, 2002.

Random Features of the System

• The number of sensors lying over on Tda,

Φ(t), is distributed as poisson with mean λad, where λ is a constant in the model. Indeed, λ represent the density of sen-sors;

• The distribution of the position of each sensor is independent of the other sensors and given by

px(X) =

1[0,t](X)