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.

中文翻译：

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线性优先依恋模型中Hill估计的一致性

优先附件被广泛用于对有向和无向网络中度数分布的幂律行为进行建模。幂律度分布的尾部指数的统计估计通常使用Hill估计器作为关键摘要统计之一。尚未探索Hill估计器用于网络数据的一致性，并且本文的主要目标是证明某些模型的一致性。为此，我们首先通过将固定节点的度增长嵌入到出生移民过程中来推导度序列的渐近行为，然后显示尾部经验测度的收敛性。通过这些步骤，获得了Hill估计量的一致性。提供模拟作为希尔估计量渐近分布的说明。