Stationary random graphs on $\mathbb{Z}$ with prescribed iid degrees and finite mean connections

TL;DR

This paper introduces a model for stationary simple graphs on the integer line with a prescribed degree distribution, showing finite second moments are necessary and sufficient for finite expected total edge length per vertex.

Contribution

It establishes a precise condition linking the degree distribution's second moment to the finiteness of the expected total edge length in stationary graph models on or the first time.

Findings

Finite second moment of degree distribution implies finite mean total edge length.

Any model with infinite second moment results in infinite mean total edge length.

Finite second moment is both necessary and sufficient for finite mean total edge length.

Abstract

Let $F$ be a probability distribution with support on the non-negative integers. A model is proposed for generating stationary simple graphs on $\mathbb{Z}$ with degree distribution $F$ and it is shown for this model that the expected total length of all edges at a given vertex is finite if $F$ has finite second moment. It is not hard to see that any stationary model for generating simple graphs on $\mathbb{Z}$ will give infinite mean for the total edge length per vertex if $F$ does not have finite second moment. Hence, finite second moment of $F$ is a necessary and sufficient condition for the existence of a model with finite mean total edge length.