On the spread of random graphs

<p style="text-align:justify;"> 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 distrib...

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Những tác giả chính: McDiarmid, C, Addario-Berry, L, Janson, S
Định dạng: Journal article
Được phát hành: Cambridge University Press 2014
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author McDiarmid, C
Addario-Berry, L
Janson, S
author_facet McDiarmid, C
Addario-Berry, L
Janson, S
author_sort McDiarmid, C
collection OXFORD
description <p style="text-align:justify;"> 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 G<sub>n,p</sub> in the supercritical range p&gt;1/n, and for a ‘small world’ model. For supercritical G<sub>n,p</sub>, we show that if p=c/n with c&gt;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 K<sub>n</sub></p>
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spelling oxford-uuid:d54f9c26-a514-4e11-b086-76e7d5ecfa2f2022-03-27T08:24:57ZOn the spread of random graphsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:d54f9c26-a514-4e11-b086-76e7d5ecfa2fSymplectic Elements at OxfordCambridge University Press2014McDiarmid, CAddario-Berry, LJanson, S <p style="text-align:justify;"> 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 G<sub>n,p</sub> in the supercritical range p&gt;1/n, and for a ‘small world’ model. For supercritical G<sub>n,p</sub>, we show that if p=c/n with c&gt;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 K<sub>n</sub></p>
spellingShingle McDiarmid, C
Addario-Berry, L
Janson, S
On the spread of random graphs
title On the spread of random graphs
title_full On the spread of random graphs
title_fullStr On the spread of random graphs
title_full_unstemmed On the spread of random graphs
title_short On the spread of random graphs
title_sort on the spread of random graphs
work_keys_str_mv AT mcdiarmidc onthespreadofrandomgraphs
AT addarioberryl onthespreadofrandomgraphs
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