On subgraphs with degrees of prescribed residues in the random graph

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

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 random...

The Interpolation Method for Random Graphs with Prescribed Degrees

Subgraphs of random graphs with speci- fied degrees

If a graph is chosen uniformly at random from all the graphs with a given degree sequence, what can be said about its subgraphs? The same can be asked of bipartite graphs, equivalently 0-1 matrices. These questions have been studied by many people. In this paper we provide a partial survey of the field, with emphasis on two general techniques: the method of switchings and the multidimensional s...

Subgraphs of Dense Random Graphs with Specified Degrees

Let d = (d1, d2, . . . , dn) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph. Although there are many results of this kind, they are restricted to the sparse case with only a few exceptions. Our focus is instead on the case where the avera...

Fast generation of random connected graphs with prescribed degrees

We address here the problem of generating random graphs uniformly from the set of simple connected graphs having a prescribed degree sequence. Our goal is to provide an algorithm designed for practical use both because of its ability to generate very large graphs (efficiency) and because it is easy to implement (simplicity). We focus on a family of heuristics for which we prove optimality condi...