Laws of Large Numbers of Subgraphs in Directed Random Geometric Networks

الأصل · EN

Given independent random points Xₙ={X₁,...,Xₙ} in R², drawn according to some probability density function f on R², and a cutoff rₙ>0 we construct a random geometric digraph G(Xₙ,Yₙ,rₙ) with vertex set Xₙ. Each vertex Xᵢ is assigned uniformly at random a sector Sᵢ, of central angle α with inclination Yᵢ, in a circle of radius rₙ (with vertex Xᵢ as the origin). An arc is present from Xᵢ to Xⱼ, if Xⱼ falls in Sᵢ. We also introduce another random geometric digraph G(Xₙ,Rₙ) with vertex set Xₙ={X₁,...,Xₙ} in Rᵈ, d≥1 and an arc present from Xᵢ to Xⱼ if ||Xᵢ-Xⱼ||<Rₙ,ᵢ. Here {Rₙ,ᵢ}ᵢ≥₁ are i.i.d. random variables and we may take an arbitrary norm ||·||. In this paper we investigate two kinds of small subgraphs--induced and isolated--in the above two directed networks, which contribute to understanding the local topology of many spatial networks, such as wireless communication networks. We give some strong laws of large numbers of subgraph counts thus extending those results of Penrose [Random Geometric Graphs, Oxford University Press, 2003].