On the Degree Sequence of Random Geometric Digraphs

Details On the Degree Sequence of Random Geometric Digraphs On the Degree Sequence of Random Geometric Digraphs

2009-09-18

arxiv.org/abs/0909.3344v1

Mathematics

Combinatorics

16 pages

Scientific paper

A random geometric digraph $G_n$ is constructed by taking $\{X_1,X_2,... X_n\}$ in $\mathbb{R}^2$ independently at random with a common bounded density function. Each vertex $X_i$ is assigned at random a sector $S_i$ of central angle $\alpha$ with inclination $Y_i$, in a circle of radius $r$ (with vertex $X_i$ as the origin). An arc is present from vertex $X_i$ to $X_j$, if $X_j$ falls in $S_i$. Suppose $k$ is fixed and $\{k_n\}$ is a sequence with $1\ll k_n\ll n^{1/2}$, as $n\to\infty$. We prove central limit theorems for $k-$ and $k_n-$nearest neighbor distance of out- and in-degrees in $G_n$. We also show that the degree distribution of this model, which varies with the probability distribution of the underlying point processes, can be either homogeneous or inhomogeneous. Our work should provide valuable insights for alternative mechanisms wrapped in real-world complex networks.

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