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Consistency of Hill estimators in a linear preferential attachment model

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Abstract

Preferential attachment is widely used to model power-law behavior of degree distributions in both directed and undirected networks. Statistical estimates of the tail exponent of the power-law degree distribution often use the Hill estimator as one of the key summary statistics. The consistency of the Hill estimator for network data has not been explored and the major goal in this paper is to prove consistency in certain models. To do this, we first derive the asymptotic behavior of the degree sequence via embedding the degree growth of a fixed node into a birth immigration process and then show the convergence of the tail empirical measure. From these steps, the consistency of the Hill estimator is obtained. Simulations are provided as an illustration for the asymptotic distribution of the Hill estimator.

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Acknowledgments

This work was supported by Army MURI grant W911NF-12-1-0385 to Cornell University. The authors would like to thank the editor and referees for their helpful comments.

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Authors and Affiliations

  1. School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, 14853, USA

    Tiandong Wang & Sidney I. Resnick

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  1. Tiandong Wang
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  2. Sidney I. Resnick
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Correspondence to Tiandong Wang.

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Wang, T., Resnick, S.I. Consistency of Hill estimators in a linear preferential attachment model. Extremes 22, 1–28 (2019). https://doi.org/10.1007/s10687-018-0335-7

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