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Let $F$ be a probability distribution with support on the non-negativeintegers. Two algorithms are described for generating a stationary randomgraph, with vertex set $\mathbb{Z}$, so that the degrees of the vertices arei.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 eachvertex. Each stub is then randomly assigned a direction, left or right, and theedge configuration is obtained by pairing stubs pointing to each other, firstexhausting all possible connections between nearest neighbors, then linkingsecond nearest neighbors, and so on. Under the assumption that $F$ has finitemean, 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. Itis also shown that any stationary algorithm for pairing stubs with random,independent directions gives infinite mean for the total length of the edges ofa given vertex. Connections to the problem of constructing finitaryisomorphisms between Bernoulli shifts are discussed.

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