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Given independent random points $\mathcal{X}_n=\{X_1,...,X_n\}$ 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 $\mathcal{X}_n$. Eachvertex $X_i$ 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$\mathcal{X}_n=\{X_1,...,X_n\}$ in $\mathbb{R}^d$, $d\ge1$ and an arc presentfrom $X_i$ to $X_j$ if $||X_i-X_j||<R_{n,i}$. Here $\{R_{n,i}\}_{i\ge1}$ arei.i.d. random variables and we may take an arbitrary norm $||\cdot||$. In thispaper we investigate two kinds of small subgraphs--induced and isolated--in theabove two directed networks, which contribute to understanding the localtopology of many spatial networks, such as wireless communication networks. Wegive some strong laws of large numbers of subgraph counts thus extending thoseresults of Penrose [Random Geometric Graphs, Oxford University Press, 2003].

Laws of Large Numbers of Subgraphs in Directed Random Geometric Networks