Laws of Large Numbers of Subgraphs in Directed Random Geometric Networks

Given independent random points ${\mathcal{X}}_n=\{X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n\}$ in ${\mathbb{R}}^2$, drawn according to some probability density function $f$ on ${\mathbb{R}}^2$, and a cutoff $r_n>0$ we construct a random geometric digraph $G({\mathcal{X}}_n,{\mathcal{Y}}_n,r_n)$ with vertex set ${\mathcal{X}}_n$. Each vertex $X_i$ is assigned uniformly at random a sector $S_i$, of central angle $\alpha$ with inclination $Y_i$, in a circle of radius $r_n$ (with vertex $X_i$ as the origin). An arc is present from $X_i$ to $X_j$, if $X_j$ falls in $S_i$. We also introduce another random geometric digraph $G({\mathcal{X}}_n,{\mathcal{R}}_n)$ with vertex set ${\mathcal{X}}_n=\{X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n\}$ in ${\mathbb{R}}^d$, $d\ge 1$ and an arc present from $X_i$ to $X_j$ if $\mathrm{\mid }\mathrm{\mid }{X}_i-X_j\mathrm{\mid }\mathrm{\mid }<R_{n,i}$. Here $\{R_{n,i}{\}}_{i\ge 1}$ are i.i.d. random variables and we may take an arbitrary norm $\mathrm{\mid }\mathrm{\mid }\cdot \mathrm{\mid }\mathrm{\mid }$. 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].