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On the Spread of Random Graphs

Published online by Cambridge University Press:  13 June 2014

LOUIGI ADDARIO-BERRY ,
SVANTE JANSON  and
COLIN McDIARMID
LOUIGI ADDARIO-BERRY
Affiliation:
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montréal, Québec, H3A 2K6, Canada (e-mail: louigi@gmail.comhttp://www.math.mcgill.ca/mytildelouigi/
SVANTE JANSON
Affiliation:
Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden (e-mail: svante.janson@math.uu.sehttp://www.math.uu.se/mytildesvante/
COLIN McDIARMID
Affiliation:
Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK (e-mail: cmcd@stats.ox.ac.ukhttp://www.stats.ox.ac.uk/mytildecmcd/
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Abstract

The spread of a connected graph G was introduced by Alon, Boppana and Spencer [1], and measures how tightly connected the graph is. It is defined as the maximum over all Lipschitz functions f on V(G) of the variance of f(X) when X is uniformly distributed on V(G). We investigate the spread for certain models of sparse random graph, in particular for random regular graphs G(n,d), for Erdős–Rényi random graphs Gn,p in the supercritical range p>1/n, and for a ‘small world’ model. For supercritical Gn,p, we show that if p=c/n with c>1 fixed, then with high probability the spread of the giant component is bounded, and we prove corresponding statements for other models of random graphs, including a model with random edge lengths. We also give lower bounds on the spread for the barely supercritical case when p=(1+o(1))/n. Further, we show that for d large, with high probability the spread of G(n,d) becomes arbitrarily close to that of the complete graph $\mathsf{K}_n$.

Keywords

60C05

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 4 , July 2014 , pp. 477 - 504
Copyright
Copyright © Cambridge University Press 2014 

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