Generating stationary random graphs on $\mathbb{Z}$ with prescribed i.i.d.\ degrees

TL;DR

This paper presents algorithms for generating stationary random graphs on the integer lattice with i.i.d. degrees, analyzing their properties and showing that certain expected edge lengths are infinite.

Contribution

It introduces new algorithms for constructing stationary random graphs with prescribed degree distributions on z, and analyzes their edge length properties and limitations.

Findings

Expected shortest edge length is infinite under the algorithm.

Any stationary pairing algorithm with independent directions results in infinite total edge length.

Connections to finitary isomorphisms between Bernoulli shifts are discussed.

Abstract

Let $F$ be a probability distribution with support on the non-negative integers. Two algorithms are described for generating a stationary random graph, with vertex set $\mathbb{Z}$, so that the degrees of the vertices are i.i.d.\ random variables with distribution $F$. Focus is on an algorithm where, initially, a random number of "stubs" with distribution $F$ is attached to each vertex. Each stub is then randomly assigned a direction, left or right, and the edge configuration is obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second nearest neighbors, and so on. Under the assumption that $F$ has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge of a vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions gives infinite mean for the total length of the edges of a given vertex. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.