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 R 2 (n large), X 1,X 2, …,X n, i.i.d. with density function f and distribution F Given a communication radius r n, two points are connected if they are at distance ≤r n.

Random geometric network

r

Random directed geometric network Fix angle  ∈ (0,2  ]. X n ={X 1,..,X n } i.i.d. points in R 2, with density f,distribution F. Let Y n ={Y 1,..,Y n } be a sequence of i.u.d. angles, let {r n } be a sequence tends to 0. G  (X n,Y n,r n ) is a kind of random directed geometric network, where (X i, X j ) is an arc iff X j in S i =S(X i,Y i,r n ). D.,Petit,Serna, IEEE Trans. Mobi. Comp. 2003

Random directed geometric network YiYi  SiSi XiXi rnrn Each sensor X i covers a sector S i, defined by r n and  with inclination Y i.

Random directed geometric network G  ( X n,Y n,r n ) is a digraph If x 5 is not in S 1, to communicate from x 1 to x 5 :

Random directed geometric network

Notations and basic facts For any fixed k ∈ N, define r n =r n (t) by nr n (t) 2 =t, for t>0. Here, t is introduced to accommodate the areas of sectors. For A in R 2, X is a finite point set in R 2 and x ∈ R 2, let X(A) be the number of points in X located in A, and X x =X ∪ {x}. For >0, let H be the homogeneous Poisson point process on R 2 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 Z n (t) be the number of vertices of out degrees at least k of G  ( X n,Y n,r n ), then Z n (t)=∑ n i=1 I {X n (S(X i,Y i,r n (t)))≥ k+1} Let W n (t) be the number of vertices of in degrees at least k of G  ( X n,Y n,r n ), then W n (t)=∑ n i=1 I { ＃ {X j ∈ X n |X i ∈ S(X j,Y j,r n (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 [Z n (t) － EZ n (t)], t>0 converge to those of a centered Gaussian process (Z ∞ (t),t>0) with E[Z ∞ (t)Z ∞ (u)]=∫ R 2   tf(x)/2 ([k, ∞))f(x)dx +

Central limit theorems (1/4  2 ) ּ∫ 0 2  ∫ 0 2  ∫ R 2 ∫ R 2 g( z, f(x 1 ), y 1, y 2 ) ּ f 2 (x 1 )dz dx 1 dy 1 dy 2 － h(t) h(u), where g( z,, y 1, y 2 )= P[{H z (S(0,y 1,t 1/2 )) ≥k}∩{H 0 (S(z,y 2,u 1/2 ))≥k}] － P[H (S(0,y 1,t 1/2 ))≥ k] ּ P[H (S(z,y 2,u 1/2 )) ≥k ], and h(t)= ∫ R 2 {   tf(x)/2 (k － 1) ּ  tf(x)/2 +   tf(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 W n (t) k(n) tends to infinity X n −→P n, where P n ={X 1,..,X N n } is a Poisson process with intensity function n f(x). Here, N n 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  (X n,Y n,r n ) having out degree k Theorem

Degree distributions For k ∈ N ∪ 0, let p(k) be the probability of a typical vertex in G  (X n,Y n,r n ) having out degree k Theorem p(k)=∫ R 2   tf(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( －  t  ּ  t  k /k! The out degree distribution is Poi(  t  )

Degree distributions Example 2 f(x 1,x 2 )=(1/2  exp( － (x 1 2 +x 2 2 )/2) normal

Degree distributions Example 2 f(x 1,x 2 )=(1/2  exp( － (x 1 2 +x 2 2 )/2) normal p(k)=4  t － exp( －  t/4  ) ∑ k i=0 (  t/4  i － 1 /i! a skew distribution

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.)