Consistency of Hill Estimators in a Linear Preferential Attachment Model

TL;DR

This paper investigates whether the Hill estimator remains consistent when used on network data generated by a linear preferential attachment model, extending its theoretical justification beyond iid data.

Contribution

It provides a proof of the Hill estimator's consistency in a specific non-iid network setting and analyzes the degree sequence's asymptotic behavior.

Findings

Hill estimator is consistent in the linear preferential attachment model

Degree sequence converges to a birth immigration process

Simulation supports theoretical results

Abstract

Preferential attachment is widely used to model power-law behavior of degree distributions in both directed and undirected networks. Practical analyses on the tail exponent of the power-law degree distribution use the Hill estimator as one of the key summary statistics, whose consistency is justified mostly for iid data. The major goal in this paper is to answer the question whether the Hill estimator is still consistent when applied to non-iid network data. 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. We also need to show the convergence of the tail empirical measure, from which the consistency of Hill estimators is obtained. This step requires checking the concentration of degree counts. We give a proof for a particular linear preferential attachment model and use simulation results as an illustration in other choices of models.