On the Degree Sequence of Random Geometric Digraphs

Abstract

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.

16 pages

References in corpus (2)

• Evolution of networks
• Focusing of Maximum Vertex Degrees in Random Faulty Scaled Sector Graphs

Cited by in corpus (3)

• Embedding Graphs in Lorentzian Spacetime
• Spatial networks with wireless applications
• Focusing of Maximum Vertex Degrees in Random Faulty Scaled Sector Graphs