Generating stationary random graphs on Z with prescribed independent, indentically distributed degrees
Artikel i vetenskaplig tidskrift, 2006

Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, 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 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 attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.


Maria Deijfen

Göteborgs universitet

Chalmers, Matematiska vetenskaper, matematisk statistik

R. Meester

VU University Amsterdam

Advances in Applied Probability

0001-8678 (ISSN) 1475-6064 (eISSN)

Vol. 38 287-298