Generating Infinite Random Graphs

Infinite Random Geometric Graphs

We introduce a new class of countably infinite random geometric graphs, whose vertices V are points in a metric space, and vertices are adjacent independently with probability p ∈ (0, 1) if the metric distance between the vertices is below a given threshold. If V is a countable dense set in R equipped with the metric derived from the L∞-norm, then it is shown that with probability 1 such infini...

Infinite Locally Random Graphs

Motivated by copying models of the web graph, Bonato and Janssen [3] introduced the following simple construction: given a graph G, for each vertex x and each subset X of its closed neighbourhood, add a new vertex y whose neighbours are exactly X. Iterating this construction yields a limit graph ↑G. Bonato and Janssen claimed that the limit graph is independent of G, and it is known as the infi...

Generating Random Outerplanar Graphs

Generating Random Regular Graphs Quickly

There are various algorithms known for generating graphs with n vertices of given degrees uniformly at random. Unfortunately, none of them is of practical use for all degree sequences, even for those with all degrees equal. In this paper we examine an algorithm which, although it does not generate uniformly at random, is provably close to a uniform generator when the degrees are relatively smal...

Percolation-type Problems on Infinite Random Graphs

We study some percolation problems on the complete graph over N. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability, such as independency, is made. The main tools are a topological version of Ramsey theory, exchangeability theory and elementary ergodic theory.