Poisson Point Processes Applied to Sensor Networks ... The 2âdimensional being "Red Point" is trapped, we have one void β1. Results: Mean ... s6 calculated. 0.
Author manuscript, published in "Spatial Networks Models for Wirelles Communications, Cambridge : United Kingdom (2010)"
Poisson Point Processes Applied to Sensor Networks {eduardo.ferraz, laurent.decreusefond, randriam}@telecom-paristech.fr
Motivation
Simplicial Homology
Interpreting a Sensor Network
The development of wireless ad hoc networking capability together with the decreasing costs and sizes of the electronical circuits allow an increasing using of sensor networks in building, utilities, industrial, home, agriculture, defense and many other contexts. The topology of such networks, particularly the connectiveness and coverage area, are important and, sometimes, critical factors. Recently, many works dealing with topology give interpretations 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.
Simplicial complexes are structures composed by elements named simplices, which can be seen by d-dimensional filled spaces. Exemples of simplexes are given below:
We can represent the topology of a sensor network 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 create a k-simplex.
0−simplex
1−simplex
2−simplex
3−simplex
Sensor
βd denotes the number of d-dimensional holes. Particularly, β0 , β1 and β2 measure, respectively, the connectiveness, the number of holes and the number of voids of a complex.
Coverage
0-simplex 1-simplex 2-simplex 3-simplex
The principal idea of the problem is that sensors {S1 , S2 , ..., Sn } have a power suply allowing them to transmit theirs ID’s and maybe some environmental information (such as temperature, pression, presence/absence of an element 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.
Two connex components: β0 = 2 The 2−dimensional being "Red Point" is trapped, we have one void : β1 = 1 The 3−dimensional being "Blue Point" is trapped, we have one hole: β2 = 1
Results: Mean of k-simplexes, sk , 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: k d
sk−1
Below, we present the variation of sk in function of ǫ for a = 100 and λ = 0.10 in two dimensions.
Tda
25
Comparison between the calculations and the simulations 12
s1 simulated s1 calculated
s6 simulated s6 calculated
0.2
15 s6
0.15 6
10
0.1 4
0
0.05
2
0
0.5
1
1.5
2
2.5
0
3
0
0.5
1
1.5
ǫ
2
2.5
0
3
0
0.5
1
1.5
ǫ
2
2.5
3
ǫ
Let Bd be the Bell’s polynomial. Using the mean of k-simplexes, we can calculate the mean of the Euler’s Characteristic χ:
• We use the maximum norm, i.e.,
−λǫd
d
a λe χ= −λǫd
kxi − xj k = max(ui,k − uj,k ) k
0.25
s3 simulated s3 calculated
8
5
• A sensor receives the ID’s from all other sensors closer then a deterministic distance ǫ, so if kxi − xj k ≤ ǫ, sensors Si and Sj are directly connected;
Comparison between the calculations and the simulations
10
20
s1
• The sensors lie over a d-torus 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 ∈ Tda = (ui,1 , ..., ud,i ), ud,k ∈ [0, a];
Comparison between the calculations and the simulations
s3
Physical Features of the System
λ a d k−1 d = , ǫ < a/3 k ǫ k!
Bd (−λǫd )
for d = 1, d = 2 and d = 3: The variation of χ in function of λ is presented following,
.
For d = 2, a = 10 and ǫ = 0.5: χ = a2 λe−λǫ
For d = 1, a = 40 and ǫ = 0.5: χ = aλe−λǫ
70
30 Sim χ Calc χ
25
50 Mean
15
For d = 3, a = 10 and ǫ = 0.5: χ = a3 λe−λǫ
3
1 − 3λǫ3 + (λǫ3 )2
1000
Sim χ Calc χ Sim β0 Sim β1 Sim β2
60
20
1 − λǫ2
800 600 400
40
Mean
Random Features of the System
2
200
• 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 sensors;
χ
• The distribution of the position of each sensor is independent of the other sensors and given by
Conjecture: βi dominance region
Results: Concentration Inequality
Based on simulations and analytical expressions, we can conjecture that, given a density of points, there are at most two dominating types of holes.
Since a compensated Poisson point process can be seen as a martingale, we can use a concentration inequality to find a superior limit for P (β0 ≥ c) in two dimensions
px (X) =
1[0,t] (X) t
10
0 -200
10 0
15
10 λ
5
0
20
-600 0
0.5
1
1.5
2 λ
2.5
3
3.5
-800
4
1
1.5
2 λ
β0 β0 and β1 β1 β1 and β2 β2 β2 and β3 β3 β3 and β4 β4 No dominance
4 2 0 -2 -4 -6 -8 -10 2
4
6
8 λ
10
12
14
3
3.5
16
0.25 0.2 P (β0 > c)
6
0
2.5
Psim Psup
Dominating βk
8
-12
0.5
0.3
10
References
0
Distribution β0 : λ=2, R=0.5 and a=10
Domination regions of βk when d = 5 in function of λ
[1] R. Ghrist, A. Muhammad. Coverage and HoleDetection in Sensor Networks Via Homology In Fourth International Conference on Information Processing in Sensor Networks (IPSN’05), UCLA, 2005. [2] C. Houdré, N. Privault. Concentration and deviation inaqualities in infinite dimensions via covariance representations Bernoulli, 2002.
Sim χ Calc χ Sim β0 Sim β1 Sim β2
-400
5 0
30 20
χ
hal-00472487, version 1 - 12 Apr 2010
Problem Formulation
0.15 0.1 0.05 0 170
175
180
185
190 c
195
200
205
4
