Mathematics – Combinatorics
Scientific paper
2009-09-16
IEJPAM vol.2 no. 2, 2010 69-79
Mathematics
Combinatorics
9 pages
Scientific paper
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 $\mathcal{X}_n$. 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 $\mathcal{X}_n=\{X_1,...,X_n\}$ in $\mathbb{R}^d$, $d\ge1$ and an arc present from $X_i$ to $X_j$ if $||X_i-X_j||
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