DMCA
Sparse random graphs with clustering (2008)
Cached
Download Links
| Citations: | 9 - 6 self |
BibTeX
@MISC{Bollobás08sparserandom,
author = {Béla Bollobás and Svante Janson and Oliver Riordan},
title = {Sparse random graphs with clustering},
year = {2008}
}
Share
 |
 |
 |
 |
||
OpenURL
Abstract
In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with independent hyperedges, and then replace each hyperedge by a (perhaps complete) graph. Although flexible enough to produce graphs with significant dependence between edges, this model is nonetheless mathematically tractable. Indeed, we find the critical point where a giant component emerges in full generality, in terms of the norm of a certain integral operator, and relate the size of the giant component to the survival probability of a certain (non-Poisson) multi-type branching process. While our main focus is the phase transition, we also study the degree distribution and the numbers of small subgraphs. We illustrate the model with a simple special case that produces graphs with powerlaw degree sequences with a wide range of degree exponents and clustering coefficients.
Keyphrases
sparse random graph degree exponent critical point wide range small graph small subgraphs inhomogeneous random hypergraph giant component phase transition simple special case powerlaw degree sequence multi-type branching process giant component emerges degree distribution basic idea several result general model random graph survival probability independent hyperedges significant dependence main focus certain integral operator full generality
