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 randomly
Use broadcast paradigms to communicate
with other sensors
Static sensor network
• Large-scale networks of simple sensors
• Usually deployed randomly
Use broadcast paradigms to communicate
with other sensors
• Each sensor is autonomous
and adaptive to environment
Static sensor network
• Sensor nodes are densely deployed
Static sensor network
• Sensor nodes are densely deployed
• Cheap
Static sensor network
• Sensor nodes are densely deployed
• 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
Random geometric network
r
Random geometric network
Random directed geometric network
• Fix angle a ∈(0,2p]. 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. Ga (Xn ,Yn ,rn) is a
kind of random directed geometric network, where
(Xi, Xj ) is an arc iff Xj in Si=S(Xi ,Yi ,rn ).
D.,Petit,Serna, IEEE Trans. Mobi. Comp. 2003
Random directed geometric network
Each sensor Xi covers a sector Si, defined
by rn and a with inclination Yi.
Si
rn
a
Xi
Yi
Random directed geometric network
• Ga ( Xn ,Yn ,rn ) is a digraph
• If x5 is not in S1 , to communicate from x1 to
x5:
Random directed geometric network
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∈R2,
let X(A) be the number of points in X located in A,
and Xx=X∪{x}.
• For l >0 , let Hl be the homogeneous Poisson
point process on R2 with intensity l.
• For k ∈N and A is a subset of N, set
rl(k)=P[Poi(l)=k] and rl(A)=P[Poi(l)∈A].
Notations and basic facts
• Let Zn(t) be the number of vertices of out
degrees at least k of Ga ( 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 Ga ( Xn ,Yn ,rn ) , then
Wn(t)=∑ni=1 I{＃{Xj∈ Xn|Xi∈ S(Xj,Yj,rn(t))}≥ k+1}
Central limit theorems
• Theorem
Central limit theorems
• 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 ratf(x)/2([k, ∞))f(x)dx +
Central limit theorems
(1/4p 2)ּ∫02p ∫02p∫R2∫R2 g( z, f(x1), y1, y2 )
ּf 2(x1 )dz dx1 dy1 dy2－ h(t) h(u),
where g( z, l , y1, y2 )=
z
P[{Hl (S(0,y1,t1/2)) ≥k}∩{Hl0(S(z,y2 ,u1/2))≥k}]－
P[Hl(S(0,y1,t1/2))≥ k]ּP[Hl(S(z,y2 ,u1/2)) ≥k ],
and h(t)= ∫R2{ratf(x)/2(k－1) ּa tf(x)/2
+ratf(x)/2([k, ∞))} f(x)dx.
Central limit theorems
•
•
•
•
Sketch of the proof
Compute expectation
Compute covariance
Poisson CLT through a dependency graph
argument
Depoissionization
Central limit theorems
• 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 Ga (Xn ,Yn ,rn) having out
degree k
• Theorem
Degree distributions
• For k∈N∪ 0, let p(k) be the probability of a
typical vertex in Ga (Xn ,Yn ,rn) having out
degree k
• Theorem
p(k)=∫R2 ratf(x)/2(k) f(x)dx
( ＊)
Degree distributions
• Example 1
f=I[0,1]2
uniform
Degree distributions
• Example 1
f=I[0,1]2
uniform
p(k)=exp(－at/2 ) ּ(at/2) k/k!
The out degree distribution is Poi(at/2)
Degree distributions
• Example 2
f(x1,x2)=(1/2p) exp(－(x12+x22)/2) normal
Degree distributions
• Example 2
f(x1,x2)=(1/2p) exp(－(x12+x22)/2) normal
p(k)=4p/at －exp(－at/4p) ∑ki=0 (at/4p) i－1/i!
a skew distribution
Degree distributions
Degree distributions
• If f is bounded, the degree distribution
will never be power law because of fast
decay
Degree distributions
• 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.)