Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-28T10:42:00.223Z Has data issue: false hasContentIssue false
  • English
  • Français

Strong couplings for static locally tree-like random graphs

Part of: Graph theory Markov processes

Published online by Cambridge University Press:  11 November 2022

Mariana Olvera-Cravioto Open the ORCID record for Mariana Olvera-Cravioto [Opens in a new window]
Mariana Olvera-Cravioto*
Affiliation:
University of North Carolina at Chapel Hill
*
*Postal address: Department of Statistics and Operations Research, University of North Carolina, Chapel HIll. Email: molvera@email.unc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We provide a general purpose result for the coupling of exploration processes of random graphs, both undirected and directed, with their local weak limits when this limit is a marked Galton–Watson process. This class includes in particular the configuration model and the family of inhomogeneous random graphs with rank-1 kernel. Vertices in the graph are allowed to have attributes on a general separable metric space and can potentially influence the construction of the graph itself. The coupling holds for any fixed depth of a breadth-first exploration process.

Keywords

Random graphscomplex networksGalton–Watson processesconfiguration modelinhomogeneous random graphlocal-weak limits

MSC classification

Primary: 05C80: Random graphs
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05C82: Small world graphs, complex networks
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 1261 - 1285
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Prob. 12, 1454–1508.CrossRefGoogle Scholar
Aldous, D. and Steele, J. M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures, ed. H. Kesten. Springer, New York, pp. 1–72.Google Scholar
Benjamini, I. and Schramm, O. (2011). Recurrence of distributional limits of finite planar graphs. In Selected Works of Oded Schramm, eds I. Benjamini and O.Häggström. Springer, New York, pp. 533–545.CrossRefGoogle Scholar
Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Europ. J. Combinatorics 1, 311–316.CrossRefGoogle Scholar
Bollobás, B. (2001). Random Graphs. Cambridge University Press.CrossRefGoogle Scholar
Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31, 3–122.CrossRefGoogle Scholar
Britton, T., Deijfen, M. and Martin-Läf, A. (2006). Generating simple random graphs with prescribed degree distribution. J. Statist. Phys. 124, 1377–1397.CrossRefGoogle Scholar
Chaintron, L.-P. and Diez, A. (2021). Propagation of chaos: A review of models, methods and applications. Preprint, arXiv:2106.14812.Google Scholar
Chen, N., Livtak, N. and Olvera-Cravioto, M. (2017). Generalized PageRank on directed configuration networks. Random Structures Algorithms 51, 237–274.CrossRefGoogle Scholar
Chen, N. and Olvera-Cravioto, M. (2013). Directed random graphs with given degree distributions. Stoch. Syst. 3, 147–186.CrossRefGoogle Scholar
Chung, F. and Lu, L. (2002). Connected components in random graphs with given expected degree sequences. Ann. Combinatorics 6, 125–145.CrossRefGoogle Scholar
Durrett, R. (2007). Random Graph Dynamics. Cambridge University Press.Google Scholar
Erdős, P. E. and Rényi, A. (1959). On random graphs. Publicationes Mathematicae (Debrecen) 6, 290–297.Google Scholar
Fraiman, N., Lin, T. and Olvera-Cravioto, M. (2021). Stochastic recursions on directed random graphs. Preprint, arXiv:2010.09596.Google Scholar
Garavaglia, A., van der Hofstad, R. and Litvak, N. (2020). Local weak convergence for PageRank. Ann. Appl. Prob. 30, 40–79.CrossRefGoogle Scholar
Lacker, D., Ramanan, K. and Wu, R. (2020). Local weak convergence and propagation of ergodicity for sparse networks of interacting processes. Preprint, arXiv:1904.02585.Google Scholar
Lacker, D., Ramanan, K. and Wu, R. (2020). Marginal dynamics of interacting diffusions on unimodular galton-watson trees. Preprint, arXiv:2009.11667.Google Scholar
Lee, J. and Olvera-Cravioto, M. (2020). PageRank on inhomogeneous random digraphs. Stoch. Process. Appl. 130, 1–57.CrossRefGoogle Scholar
Norros, I. and Reittu, H. (2006). On a conditionally Poissonian graph process. Adv. Appl. Prob. 38, 59–75.CrossRefGoogle Scholar
Olvera-Cravioto, M. (2021). PageRank’s behavior under degree correlations. Ann. Appl. Prob. 1, 1403–1442.Google Scholar
van der Hofstad, R. (2017). Random Graphs and Complex Networks, Vol. 1. Cambridge University Press.Google Scholar
van der Hofstad, R. (2021). Random Graphs and Complex Networks, Vol. 2. Cambridge University Press.Google Scholar
van der Hofstad, R., Hooghiemstra, G. and Van Mieghem, P. (2005). Distances in random graphs with finite variance degrees. Random Structures Algorithms 27, 76–123.CrossRefGoogle Scholar
Villani, C. (2009). Optimal Transport, Old and New. Springer, New York.CrossRefGoogle Scholar