Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-17T11:42:25.982Z Has data issue: false hasContentIssue false

The Size of the Giant Component of a Random Graph with a Given Degree Sequence

Published online by Cambridge University Press:  01 September 1998

MICHAEL MOLLOY
Affiliation:
Department of Computer Science, University of Toronto, Toronto, Canada (e-mail: molloy@cs.toronto.edu)
BRUCE REED
Affiliation:
Equipe Combinatoire, CNRS, Université Pierre et Marie Curie, Paris, France (e-mail: reed@ecp6.jussieu.fr)

Abstract

Given a sequence of nonnegative real numbers λ0, λ1, … that sum to 1, we consider a random graph having approximately λin vertices of degree i. In [12] the authors essentially show that if [sum ]i(i−2)λi>0 then the graph a.s. has a giant component, while if [sum ]i(i−2)λi<0 then a.s. all components in the graph are small. In this paper we analyse the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine ε, λ′0, λ′1 … such that a.s. the giant component, C, has εn+o(n) vertices, and the structure of the graph remaining after deleting C is basically that of a random graph with n′=n−[mid ]C[mid ] vertices, and with λ′in′ of them of degree i.

Type
Research Article
Copyright
1998 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)