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On the dense preferential attachment graph models and their graphon induced counterpart

Part of: Graph theory

Published online by Cambridge University Press:  30 July 2019

Ágnes Backhausz  and
Dávid Kunszenti-Kovács
Ágnes Backhausz*
Affiliation:
Eötvös Loránd University and MTA Alfréd Rényi Institute of Mathematics
Dávid Kunszenti-Kovács*
Affiliation:
MTA Alfréd Rényi Institute of Mathematics
*
*Postal address: Eötvös Loránd University, Pázmány Péter Sétány 1/c, 1117 Budapest, Hungary.
**Postal address: MTA Alfréd Rényi Institute of Mathematics, PO Box 127, 1364 Budapest, Hungary.
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Abstract

Letting ℳ denote the space of finite measures on ℕ, and μλ ∊ ℳ denote the Poisson distribution with parameter λ, the function W : [0, 1]2 → ℳ given by W(x, y) = μc log x log y is called the PAG graphon with density c. It is known that this is the limit, in the multigraph homomorphism sense, of the dense preferential attachment graph (PAG) model with edge density c. This graphon can then in turn be used to generate the so-called W-random graphs in a natural way, and similar constructions also work in the slightly more general context of the so-called PAGκ models. The aim of this paper is to compare these dense PAGκ models with the W-random graph models obtained from the corresponding graphons. Motivated by the multigraph limit theory, we investigate the expected jumble-norm distance of the two models in terms of the number of vertices n. We present a coupling for which the expectation can be bounded from above by O(log3/2n · n−1/2), and provide a universal lower bound that is coupling-independent, but without the logarithmic term.

Keywords

Dense graph limitPólya urn processcut normjumble norm

MSC classification

Primary: 05C80: Random graphs
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 2 , June 2019 , pp. 590 - 601
Copyright
© Applied Probability Trust 2019 

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References

Von Bahr, B. (1965). On the convergence of moments in the central limit theorem. Ann. Math. Statist. 36, 808–818.CrossRefGoogle Scholar
Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286 (5439), 509–512.CrossRefGoogle ScholarPubMed
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
Blackwell, D. and Macqueen, J. B. (1973). Ferguson distributions via Polya urn schemes. Ann. Statist. 2, 353–355. Available at https://projecteuclid.org/euclid.aos/1176342372.Google Scholar
Borgs, C., Chayes, J., Cohn, H. and Lovász, L. M. (2018). Identifiability for graphexes and the weak kernel metric. Available at arXiv:1804.03277.Google Scholar
Borgs, C., Chayes, J., Lovász, L., Sós, V. and Vesztergombi, K. (2011). Limits of randomly grown graph sequences. Eur. J. Combin. 32 (7), 985–999.CrossRefGoogle Scholar
Degroot, M.H. (1970). Optimal Statistical Decisions. McGraw-Hill, New York.Google Scholar
Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer, New York.CrossRefGoogle Scholar
Durrett, R. (2007). Random Graph Dynamics (Cambridge Series in Statistical and Probabilistic Mathematics). Cambridge University Press, Cambridge.Google Scholar
Elwes, R. (2015). Preferential attachment processes approaching the Rado multigraph. Available at arXiv:1502.05618.Google Scholar
Elwes, R. (2016). A linear preferential attachment process approaching the Rado graph. Available at arXiv:1603.08806.Google Scholar
Frieze, A. and Karoński, M. (2015). Introduction to Random Graphs. Cambridge University Press, Cambridge.Google Scholar
Hall, P. (1983). On the rate of convergence of moments in the central limit theorem for lattice distributions. Trans. Amer. Math. Soc. 278 (1), 169–181.CrossRefGoogle Scholar
Kunszenti-Kovács, D., Lovász, L. and Szegedy, B. (2014). Multigraph limits, unbounded kernels, and Banach space decorated graphs. Available at arXiv:1406.7846.Google Scholar
Lovász, L. (2012). Large Networks and Graph Limits. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Pemantle, R. (1988). Random processes with reinforcement. Doctoral thesis, Massachusetts Institute of Technology.Google Scholar
Ráth, B. and Szakács, L. (2012). Multigraph limit of the dense configuration model and the preferential attachment graph. Acta Math. Hung. 136, 196–221.CrossRefGoogle Scholar
Wong, T.-T. (1998). Generalized Dirichlet distribution in Bayesian analysis. Appl. Math. Comput. 97, 165–181.Google Scholar