clt for degrees of random directed geometric networks yilun shang department of mathematics,...

CLT for Degrees of Random Directed Geometric Networks

Yilun Shang

Department of Mathematics, Shanghai Jiao Tong University May 18, 2008

Context

• Background and Motivation

• Model

• Central limit theorems

• Degree distributions

• Miscellaneous

(Static) sensor network

• Large-scale networks of simple sensors

Static sensor network

• Large-scale networks of simple sensors• Usually deployed randomlyUse broadcast paradigms to communicate

with other sensors


• Large-scale networks of simple sensors• Usually deployed randomlyUse broadcast paradigms to communicate

with other sensors• Each sensor is autonomous

and adaptive to environment


• Sensor nodes are densely deployed



• Cheap



• Cheap

• Small size

Communication

• Radio Frequency

omnidirectional antenna

directional antenna

Communication

• Radio Frequency omnidirectional antenna directional antenna• Optical laser beam need line of sight for communication

An illustration

Graph Models

Random (directed) geometric network

• Scatter n points on R2 (n large), X1,X2, …,Xn , i.i.d. with density function f

and distribution F

• Given a communication radius rn, two points are connected if they are at distance ≤rn.

Random geometric network


r

Random directed geometric network

• Fix angle ∈(0,2]. Xn={X1,..,Xn} i.i.d. points in R2, with density f ,distribution F. Let Yn={Y1,..,Yn} be a sequence of i.u.d. angles, let {rn} be a sequence tends to 0. G(Xn ,Yn ,rn) is a kind of random directed geometric network, where (Xi, Xj ) is an arc iff Xj in S i=S(Xi ,Yi ,rn ).

D.,Petit,Serna, IEEE Trans. Mobi. Comp. 2003


Yi

S i

Xi

rn

Each sensor Xi covers a sector S i, defined by rn and with inclination Yi.


• G( Xn ,Yn ,rn ) is a digraph

• If x5 is not in S1 , to communicate from x1 to x5:

Notations and basic facts• For any fixed k N, define ∈ rn=rn(t) by nrn(t)2=t,

for t>0. Here, t is introduced to accommodate the areas of sectors.

• For A in R2, X is a finite point set in R2 and x R∈ 2, let X(A) be the number of points in X located in A,

and Xx=X {x}.∪ • For >0 , let H be the homogeneous Poisson

point process on R2 with intensity .• For k N and ∈ A is a subset of N, set

(k)=P[Poi()=k] and (A)=P[Poi() A].∈

Notations and basic facts

• Let Zn(t) be the number of vertices of out degrees at least k of G( Xn ,Yn ,rn ) , then

Zn(t)=∑ni=1 I{Xn(S(Xi,Yi,rn(t)))≥ k+1}

• Let Wn(t) be the number of vertices of in degrees at least k of G( Xn ,Yn ,rn ) , then

Wn(t)=∑ni=1 I{ ＃ {Xj ∈ Xn|Xi∈ S(Xj,Yj,rn(t))}≥ k+1}

Central limit theorems

• Theorem


• Theorem

Suppose k is fixed. The finite dimensional distributions of the process

n－ 1/2[Zn(t) － EZn(t)], t>0

converge to those of a centered Gaussian process (Z∞(t),t>0) with

E[Z∞(t)Z∞(u)]=∫R2 tf(x)/2([k, ∞))f(x)dx +


(1/4 2) ∫02 ∫0

2∫R2∫R2 g( z, f(x1), y1, y2 )

f 2(x1 )dz dx1 dy1 dy2 － h(t) h(u),

where g( z, , y1, y2 )=

P[{Hz(S(0,y1,t1/2)) ≥k}∩{H

0(S(z,y2 ,u1/2))≥k}] － P[H(S(0,y1,t1/2))≥ k] P[H(S(z,y2 ,u1/2)) ≥k ],

and h(t)= ∫R2{tf(x)/2(k － 1) tf(x)/2

+tf(x)/2([k, ∞))} f(x)dx.


Sketch of the proof

• Compute expectation

• Compute covariance

• Poisson CLT through a dependency graph argument

• Depoissionization


• Wn(t)

• k(n) tends to infinity

• Xn−→Pn , where Pn ={X1,..,XNn } is a Poisson process with intensity function n f(x).

Here, Nn is a Poisson variable with mean n.

Corresponding central limit theorems are obtained

Degree distributions

• For k N 0∈ ∪ , let p(k) be the probability of a typical vertex in G(Xn ,Yn ,rn) having out degree k

• Theorem


• For k N 0∈ ∪ , let p(k) be the probability of a typical vertex in G(Xn ,Yn ,rn) having out degree k

• Theorem

p(k)=∫R2 tf(x)/2(k) f(x)dx ( ＊ )


• Example 1

f=I[0,1]2 uniform


• Example 1

f=I[0,1]2 uniform

p(k)=exp( － t tk/k!

The out degree distribution is Poi(t)


• Example 2

f(x1,x2)=(1/2exp( － (x12+x2

2)/2) normal


• Example 2

f(x1,x2)=(1/2exp( － (x12+x2

2)/2) normal

p(k)=4t － exp( － t/4) ∑ki=0 (t/4i －

1/i!

a skew distribution


• If f is bounded, the degree distribution will never be power law because of fast

decay


• If f is bounded, the degree distribution will never be power law because of fast

decay

• Given p(k)≥0, ∑∞k=0 p(k)=1, it’s very

hard to solve equation ( ＊ ) for getting a f(x)

Miscellaneous

• High dimension

• Angles not uniformly at random

• Dynamic model

(Brownian, Random direction, Random waypoint, Voronoi, etc.)

clt for degrees of random directed geometric networks yilun shang department of mathematics,...

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