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Compound Poisson approximation of subgraph counts in stochastic block models with multiple edges

Part of: Distribution theory Operations research and management science Graph theory Limit theorems

Published online by Cambridge University Press:  16 November 2018

Matthew Coulson ,
Robert E. Gaunt  and
Gesine Reinert
Matthew Coulson*
Affiliation:
University of Birmingham
Robert E. Gaunt*
Affiliation:
The University of Manchester
Gesine Reinert*
Affiliation:
University of Oxford
*
* Postal address: School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
** Postal address: School of Mathematics, The University of Manchester, Manchester M13 9PL, UK. Email address: robert.gaunt@manchester.ac.uk
*** Postal address: Department of Statistics, University of Oxford, 24‒29 St Giles', Oxford OX1 3LB, UK
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Abstract

We use the Stein‒Chen method to obtain compound Poisson approximations for the distribution of the number of subgraphs in a generalised stochastic block model which are isomorphic to some fixed graph. This model generalises the classical stochastic block model to allow for the possibility of multiple edges between vertices. We treat the case that the fixed graph is a simple graph and that it has multiple edges. The former results apply when the fixed graph is a member of the class of strictly balanced graphs and the latter results apply to a suitable generalisation of this class to graphs with multiple edges. We also consider a further generalisation of the model to pseudo-graphs, which may include self-loops as well as multiple edges, and establish a parameter regime in the multiple edge stochastic block model in which Poisson approximations are valid. The results are applied to obtain Poisson and compound Poisson approximations (in different regimes) for subgraph counts in the Poisson stochastic block model and degree corrected stochastic block model of Karrer and Newman (2011).

Keywords

Stochastic block modelmultiple edgessubgraph count; compound Poisson approximationStein‒Chen methodpseudo-graph

MSC classification

Primary: 90B15: Network models, stochastic 62E17: Approximations to distributions (nonasymptotic) 60F05: Central limit and other weak theorems 05C80: Random graphs
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 3 , September 2018 , pp. 759 - 782
Copyright
Copyright © Applied Probability Trust 2018 

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