CLT for Degrees of Random Directed Geometric Networks

1. CLT for Degrees of Random Directed Geometric Networks YilunShang Department of Mathematics, Shanghai Jiao Tong University May 18, 2008

2. Context • Background and Motivation • Model • Central limit theorems • Degree distributions • Miscellaneous

3. (Static) sensor network • Large-scale networks of simple sensors

4. Static sensor network • Large-scale networks of simple sensors • Usually deployed randomly Use broadcast paradigms to communicate with other sensors

5. Static sensor network • Large-scale networks of simple sensors • Usually deployed randomly Use broadcast paradigms to communicate with other sensors • Each sensor is autonomous andadaptive to environment

6. Static sensor network • Sensor nodes are densely deployed

7. Static sensor network • Sensor nodes are densely deployed • Cheap

8. Static sensor network • Sensor nodes are densely deployed • Cheap • Small size

9. Communication • Radio Frequency omnidirectional antenna directional antenna

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

11. An illustration

12. Graph Models Random (directed) geometric network • Scattern points onR2 (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.

13. Random geometric network

14. Random geometric network r

15. Random geometric network

16. 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 geometricnetwork, where (Xi, Xj) is an arc iff Xj in Si=S(Xi ,Yi ,rn ). D.,Petit,Serna, IEEE Trans. Mobi. Comp. 2003

17. Random directed geometric network Each sensor Xi covers a sector Si, defined by rn and a withinclination Yi. Si rn a Yi Xi

18. Random directed geometric network • Ga(Xn,Yn,rn)is a digraph • If x5 is not in S1, to communicate from x1 to x5:

19. Random directed geometric network

20. Notations and basic facts • For anyfixed 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 Hlbe 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].

21. 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}

22. Central limit theorems • Theorem

23. 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 +

24. Central limit theorems (1/4p2)ּ∫02p ∫02p∫R2∫R2g( z, f(x1), y1, y2 ) ּf 2(x1 )dz dx1 dy1 dy2－ h(t) h(u), where g( z, l , y1, y2 )= P[{Hlz(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.

25. Central limit theorems Sketch of the proof • Compute expectation • Compute covariance • Poisson CLT through a dependency graph argument • Depoissionization

26. 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

27. 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

28. 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)=∫R2ratf(x)/2(k) f(x)dx (＊)

29. Degree distributions • Example 1 f=I[0,1]2 uniform

30. Degree distributions • Example 1 f=I[0,1]2 uniform p(k)=exp(－at/2 )ּ(at/2) k/k! The out degree distribution isPoi(at/2)

31. Degree distributions • Example 2 f(x1,x2)=(1/2p) exp(－(x12+x22)/2) normal

32. 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

33. Degree distributions

34. Degree distributions • If f is bounded, the degree distribution will never be power law because of fast decay

35. 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 veryhard to solve equation (＊) for getting a f(x)

36. Miscellaneous • High dimension • Angles not uniformly at random • Dynamic model (Brownian, Random direction, Random waypoint, Voronoi, etc.)

37. Thank you !