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On local weak limit and subgraph counts for sparse random graphs

Part of: Combinatorial probability Graph theory

Published online by Cambridge University Press:  21 July 2022

Valentas Kurauskas
Valentas Kurauskas*
Affiliation:
Faculty of Mathematics and Informatics, Vilnius University
*
*Postal address: Akademijos 4, LT-08412 Vilnius, Lithuania. Email address: valentas@gmail.com
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Abstract

We use an inequality of Sidorenko to show a general relation between local and global subgraph counts and degree moments for locally weakly convergent sequences of sparse random graphs. This yields an optimal criterion to check when the asymptotic behaviour of graph statistics, such as the clustering coefficient and assortativity, is determined by the local weak limit.

As an application we obtain new facts for several common models of sparse random intersection graphs where the local weak limit, as we see here, is a simple random clique tree corresponding to a certain two-type Galton–Watson branching process.

Keywords

Local weak limitsubgraph countclique treerandom intersection graph

MSC classification

Primary: 60C05: Combinatorial probability 05C80: Random graphs
Secondary: 05C82: Small world graphs, complex networks
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 755 - 776
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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