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

Part of: Graph theory Distribution theory Operations research and management science 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: [email protected]
*** 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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References

[1]Airoldi, E. M., Costa, T. B. and Chan, S. H. (2013). Stochastic blockmodel approximation of a graphon: theory and consistent estimation. In Advances in Neural Information Processing Systems 26.Google Scholar
[2]Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: the Chen–Stein method. Ann. Prob. 17, 9–25.Google Scholar
[3]Barbour, A. D. (1982). Poisson convergence and random graphs. Math. Proc. Camb. Philos. Soc. 92, 349–359.Google Scholar
[4]Barbour, A. D. and Chryssaphinou, O. (2001). Compound Poisson approximation: a user's guide. Ann. Appl. Prob. 11, 964–1002.Google Scholar
[5]Barbour, A. D. and Eagleson, G. K. (1983). Poisson approximation for some statistics based on exchangeable trials. Adv. Appl. Prob. 15, 585–600.Google Scholar
[6]Barbour, A. D. and Röllin, A. (2017). Central limit theorems in the configuration model. Preprint. Available at https://arxiv.org/abs/1710.02644.Google Scholar
[7]Barbour, A. D. and Utev, S. (1998). Solving the Stein equation in compound Poisson approximation. Adv. Appl. Prob. 30, 449–475.Google Scholar
[8]Barbour, A. D. and Xia, A. (2000). Estimating Stein's constants for compound Poisson approximation. Bernoulli 6, 581–590.Google Scholar
[9]Barbour, A. D., Chen, L. H. Y. and Loh, W.-L. (1992). Compound Poisson approximation for nonnegative random variables via Stein's method. Ann. Prob. 20, 1843–1866.Google Scholar
[10]Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford University Press.Google Scholar
[11]Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Ann. Prob. 3, 534–545.Google Scholar
[12]Condon, A. and Karp, R. M. (1999). Algorithms for graph partitioning on the planted partition model. In Randomization, Approximation, and Combinatorial Optimization, Springer, Berlin, pp. 221–232.Google Scholar
[13]Coulson, M., Gaunt, R. E. and Reinert, G. (2016). Poisson approximation of subgraph counts in stochastic block models and a graphon model. ESAIM Prob. Statist. 20, 131–142.Google Scholar
[14]Daly, F. (2017). On magic factors in Stein's method for compound Poisson approximation. Electron. Commun. Prob. 22, 67.Google Scholar
[15]Daudin, J.-J., Picard, F. and Robin, S. (2008). A mixture model for random graphs. Statist. Comput. 18, 173–183.Google Scholar
[16]Frank, O. and Strauss, D. (1986). Markov graphs. J. Amer. Statist. Assoc. 81, 832–842.Google Scholar
[17]Girvan, M. and Newman, M. E. (2002). Community structure in social and biological networks. Proc. Nat. Acad. Sci. USA 99, 7821–7826.Google Scholar
[18]Holland, P. W., Laskey, K. B. and Leinhardt, S. (1983). Stochastic blockmodels: first steps. Social Networks 5, 109–137.Google Scholar
[19]Karrer, B. and Newman, M. E. J. (2011). Stochastic blockmodels and community structure in networks. Phys. Rev. E(3) 83, 016107.Google Scholar
[20]Latouche, P. and Robin, S. (2016). Variational Bayes model averaging for graphon functions and motif frequencies inference in W-graph models. Statist. Comput. 26, 1173–1185.Google Scholar
[21]Matias, C. and Robin, S. (2014). Modeling heterogeneity in random graphs through latent space models: a selective review. In MMCS, Mathematical Modelling of Complex Systems, EDP Sciences, Les Ulis, pp. 55–74.Google Scholar
[22]Milo, R. et al. (2002). Network motifs: simple building blocks of complex networks. Science 298, 824–827.Google Scholar
[23]Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6, 161–179.Google Scholar
[24]Nowicki, K. and Snijders, T. A. B. (2001). Estimation and prediction for stochastic blockstructures. J. Amer. Statist. Assoc. 96, 1077–1087.Google Scholar
[25]Olhede, S. C. and Wolfe, P. J. (2014). Network histograms and universality of blockmodel approximation. Proc. Nat. Acad. Sci. USA 111, 14722–14727.Google Scholar
[26]Picard, F. et al. (2008). Assessing the exceptionality of network motifs. J. Comput. Biol. 15, 1–20.Google Scholar
[27]Roos, M. (1993). Stein–Chen method for compound Poisson approximation. Doctoral thesis. University of Zürich.Google Scholar
[28]Roos, M. (1994). Stein's method for compound Poisson approximation: the local approach. Ann. Appl. Prob. 4, 1177–1187.Google Scholar
[29]Sarajlić, A. et al. (2013). Network topology reveals key cardiovascular disease genes. PLoS ONE 8, e71537.Google Scholar
[30]Stark, D. (2001). Compound Poisson approximation of subgraph counts in random graphs. Random Structures Algorithms 18, 39–60.Google Scholar
[31]Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Prob., Vol. 2, University of California Press, Berkeley, pp. 583–602.Google Scholar
[32]Wegner, A. E. et al. (2018). Identifying networks with common organizational principles. To appear in J. Complex Networks. Available at https://doi.org/10.1093/comnet/cny003.Google Scholar