Stationary random graphs with prescribed iid degrees on a spatial Poisson process

TL;DR

This paper studies the construction of translation-invariant random graphs on a Poisson point process in Euclidean space, focusing on conditions for finite expected total edge length per vertex based on degree distribution moments.

Contribution

It establishes a precise criterion linking the finiteness of the mean total edge length to the finite moment of the degree distribution of order (d+1)/d.

Findings

Finite mean total edge length per vertex iff degree distribution has finite (d+1)/d moment.

Provides models for translation-invariant graphs with iid degrees on Poisson points.

Analyzes edge length properties in spatial random graphs.

Abstract

Let $[\mathcal{P}]$ be the points of a Poisson process on $\mathbb{R}^d$ and $F$ a probability distribution with support on the non-negative integers. Models are formulated for generating translation invariant random graphs with vertex set $[\mathcal{P}]$ and iid vertex degrees with distribution $F$, and the length of the edges is analyzed. The main result is that finite mean for the total edge length per vertex is possible if and only if $F$ has finite moment of order $(d+1)/d$.