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On local weak limit and subgraph counts for sparse random graphs

Published online by Cambridge University Press:  21 July 2022

Valentas Kurauskas*
Affiliation:
Faculty of Mathematics and Informatics, Vilnius University
*
*Postal address: Akademijos 4, LT-08412 Vilnius, Lithuania. Email address: [email protected]

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.

Type
Original Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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