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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.
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CLT for Degrees of Random Directed Geometric Networks YilunShang 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 andadaptive 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
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.
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
Random directed geometric network Each sensor Xi covers a sector Si, defined by rn and a withinclination Yi. Si rn a Yi Xi
Random directed geometric network • Ga(Xn,Yn,rn)is a digraph • If x5 is not in S1, to communicate from x1 to x5:
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].
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 (1/4p2)ּ∫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.
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)=∫R2ratf(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 isPoi(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 • 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 veryhard 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.)