Generating stationary random graphs on Z with prescribed i.i.d. degrees [article]

Maria Deijfen, Ronald Meester
2015 arXiv   pre-print
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 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
more » ... 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.
arXiv:1509.06989v1 fatcat:mx7xdmvkrjgardezlf7pexpite