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Local weak limit of preferential attachment random trees with additive fitness

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

Published online by Cambridge University Press:  19 January 2024

Tiffany Y. Y. Lo Open the ORCID record for Tiffany Y. Y. Lo [Opens in a new window]
Tiffany Y. Y. Lo*
Affiliation:
Uppsala University
*
*Postal address: Department of Mathematics, Uppsala University, Lägerhyddsvägen 1, 752 37 Uppsala, Sweden. Email address: yin_yuan.lo@math.uu.se
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Abstract

We consider linear preferential attachment trees with additive fitness, where fitness is the random initial vertex attractiveness. We show that when the fitnesses are independent and identically distributed and have positive bounded support, the local weak limit can be constructed using a sequence of mixed Poisson point processes. We also provide a rate of convergence for the total variation distance between the r-neighbourhoods of a uniformly chosen vertex in the preferential attachment tree and the root vertex of the local weak limit. The proof uses a Pólya urn representation of the model, for which we give new estimates for the beta and product beta variables in its construction. As applications, we obtain limiting results and convergence rates for the degrees of the uniformly chosen vertex and its ancestors, where the latter are the vertices that are on the path between the uniformly chosen vertex and the initial vertex.

Keywords

Graph limitcomplex networksdistributional approximation

MSC classification

Primary: 90B15: Network models, stochastic
Secondary: 05C80: Random graphs 60F05: Central limit and other weak theorems
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 40
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Aldous, D. and Steele, J. M. (2004). The objective method: probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures, Springer, Berlin, pp. 1–72.Google Scholar
Barabási, A. L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509–512.CrossRefGoogle ScholarPubMed
Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Prob. 6, article no. 23, 13 pp.Google Scholar
Berger, N., Borgs, C., Chayes, J. T. and Saberi, A. (2014). Asymptotic behavior and distributional limits of preferential attachment graphs. Ann. Prob. 42, 1–40.CrossRefGoogle Scholar
Bhamidi, S. (2007). Universal techniques to analyze preferential attachment trees: global and local analysis. In preparation.Google Scholar
Bloem-Reddy, B. and Orbanz, P. (2017). Preferential attachment and vertex arrival times. Preprint. Available at https://arxiv.org/abs/1710.02159.Google Scholar
Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18, 279–290.CrossRefGoogle Scholar
Borgs, C., Chayes, J., Daskalakis, C. and Roch, S. (2007). First to market is not everything: an analysis of preferential attachment with fitness. In STOC ’07: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, New York, pp. 135–144.CrossRefGoogle Scholar
Borovkov, K. A. and Vatutin, V. A. (2006). On the asymptotic behaviour of random recursive trees in random environments. Adv. Appl. Prob. 38, 1047–1070.CrossRefGoogle Scholar
Bubeck, S., Mossel, E. and Rácz, M. (2015). On the influence of the seed graph in the preferential attachment model. IEEE Trans. Network Sci. Eng. 2, 30–39.CrossRefGoogle Scholar
Dereich, S. (2016). Preferential attachment with fitness: unfolding the condensate. Electron. J. Prob. 21, article no. 3, 38 pp.Google Scholar
Dereich, S. and Ortgiese, M. (2014). Robust analysis of preferential attachment models with fitness. Combinatorics Prob. Comput. 23, 386–411.CrossRefGoogle Scholar
Dharmadhikari, S. W. and Jogdeo, K. (1969). Bounds on moments of certain random variables. Ann. Math. Statist. 40, 1506–1509.CrossRefGoogle Scholar
Durrett, R. (2019). Probability: Theory and Examples, 5th edn. Cambridge University Press.CrossRefGoogle Scholar
Ergün, G. and Rodgers, G. J. (2002). Growing random networks with fitness. Physica A 303, 261–272.CrossRefGoogle Scholar
Garavaglia, A. (2019). Preferential attachment models for dynamic networks. Doctoral Thesis, Technische Universiteit Eindhoven.Google Scholar
Garavaglia, A., Hazra, R. S., van der Hofstad, R. and Ray, R. (2022). Universality of the local limit of preferential attachment models. Preprint. Available at https://arxiv.org/abs/2212.05551 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
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 (draft). Available at https://www.win.tue.nl/rhofstad/NotesRGCNII.pdf.Google Scholar
Holmgren, C. and Janson, S. (2017). Fringe trees, Crump–Mode–Jagers branching processes and m-ary search trees. Prob. Surveys 14, 53–154.CrossRefGoogle Scholar
Iyer, T. (2023). Degree distributions in recursive trees with fitnesses. Adv. Appl. Prob. 55, 407–443.CrossRefGoogle Scholar
Karp, R. M. (1990). The transitive closure of a random digraph. Random Structures Algorithms 1, 73–93.CrossRefGoogle Scholar
Lo, T. Y. Y. (2023). Supplementary material for ‘Local weak limit of preferential attachment random trees with additive fitness’. Available at https://doi.org/10.1017/apr.2023.54.CrossRefGoogle Scholar
Lodewijks, B. and Ortgiese, M. (2020). A phase transition for preferential attachment models with additive fitness. Electron. J. Prob. 25, article no. 146, 54 pp.Google Scholar
Lukacs, E. (1955). A characterization of the gamma distribution. Ann. Math. Statist. 26, 319–324.CrossRefGoogle Scholar
Mahmoud, H. (2008). Pólya Urn Models. CRC Press, Boca Raton.CrossRefGoogle Scholar
Móri, T. F. (2005). The maximum degree of the Barabási–Albert random tree. Combinatorics Prob. Comput. 14, 339–348.CrossRefGoogle Scholar
Pain, M. and Sénizergues, D. (2020). Correction terms for the height of weighted recursive trees. Ann. Appl. Prob. 32, 3027–3059.Google Scholar
Peköz, E. A., Röllin, A. and Ross, N. (2013). Total variation error bounds for geometric approximation. Bernoulli 19, 610–632.CrossRefGoogle Scholar
Peköz, E. A., Röllin, A. and Ross, N. (2016). Generalized gamma approximation with rates for urns, walks and trees. Ann. Prob. 44, 1776–1816.CrossRefGoogle Scholar
Peköz, E. A., Röllin, A. and Ross, N. (2017). Joint degree distributions of preferential attachment random graphs. Adv. Appl. Prob. 49, 368–387.CrossRefGoogle Scholar
Peköz, E. A., Röllin, A. and Ross, N. (2019). Pólya urns with immigration at random times. Bernoulli 25, 189–220.CrossRefGoogle Scholar
Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, Berlin, Heidelberg.Google Scholar
Rudas, A., Tóth, B. and Valkó, B. (2007). Random trees and general branching processes. Random Structures Algorithms 31, 186–202.CrossRefGoogle Scholar
Sénizergues, D. (2021). Geometry of weighted recursive and affine preferential attachment trees. Electron. J. Prob. 26, article no. 80, 56 pp.Google Scholar
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