Abstract
We investigate structural properties of large, sparse random graphs through the lens of sampling convergence (Borgs et al. (Ann. Probab. 47 (2019) 2754–2800). Sampling convergence generalizes left convergence to sparse graphs, and describes the limit in terms of a graphex. We introduce a notion of sampling convergence for sequences of multigraphs, and establish the graphex limit for the configuration model, a preferential attachment model, the generalized random graph and a bipartite variant of the configuration model. The results for the configuration model, preferential attachment model and bipartite configuration model provide necessary and sufficient conditions for these random graph models to converge. The limit for the configuration model and the preferential attachment model is an augmented version of an exchangeable random graph model introduced by Caron and Fox (J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 (2017) 1295–1366).
Funding Statement
S. Dhara was supported by an internship at Microsoft Research Lab—New England, and by the Netherlands Organisation for Scientific Research (NWO) through Gravitation Networks, Grant 024.002.003.
Acknowledgments
We thank Samantha Petti for several suggestions on improving an earlier version of the draft, and in particular suggesting a simplification in the coupling in Section 2.1.
Citation
Christian Borgs. Jennifer T. Chayes. Souvik Dhara. Subhabrata Sen. "Limits of sparse configuration models and beyond: Graphexes and MultiGraphexes." Ann. Probab. 49 (6) 2830 - 2873, November 2021. https://doi.org/10.1214/21-AOP1508
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