The Poisson Cloning Model for Random Graphs, Random Directed Graphs and Random k-SAT Problems

Abstract

In the (classical) random graph model G(n,p) with p=O(1/n), the degrees of the vertices are almost i.i.d Poisson random variables with mean d=pn +O(1/n). Though this fact is useful to understand the nature of the model, it has not been possible to fully utilize properties of i.i.d Poisson random variables. For example, the distribution of the number of isolated vertices is very close to the binomial distribution B(n, de ^{− − d}). In a rigorous proof, however, one has to keep saying how close the distribution is and tracking the effect of the small difference, which is not necessary if the degrees are exactly i.i.d Poisson. Since these kinds of small differences occur almost everywhere in the analysis of the random graph, they make rigorous analysis significantly difficult, if not impossible.

As an approach to minimize such non-essential parts of the analysis, we introduce a random graph model, called the Poisson cloning model, in which all degrees are i.i.d Poisson random variables. Similar models may be introduced for random directed graphs and random k-SAT problems. We will first establish theorems saying that the new models are essentially equivalent to the classical models. To demonstrate how useful the new models are, we will completely analyze some of well-known problems such as the k-core problems of the random graph, the strong component problem of the random directed graph and/or the pure literal algorithm of the random k-SAT problem.

This lecture will be self-contained, especially no prior knowledge about the above mentioned problems are required.