We extend the well-known $\beta$-model for directed graphs to dynamic network setting, where we observe snapshots of adjacency matrices at different time points. We propose a kernel-smoothed likelihood approach for estimating $2n$ time-varying parameters in a network with $n$ nodes, from $N$ snapshots. We establish consistency and asymptotic normality properties of our kernel-smoothed estimators as either $n$ or $N$ diverges. Our results contrast their counterparts in single-network analyses, where $n\to\infty$ is invariantly required in asymptotic studies. We conduct comprehensive simulation studies that confirm our theory's prediction and illustrate the performance of our method from various angles. We apply our method to an email data set and obtain meaningful results.

翻译：我们将有向图的经典 $\beta$ 模型推广到动态网络场景，在此场景中我们观测不同时间点的邻接矩阵快照。我们提出一种核平滑似然方法，从 $N$ 个快照中估计具有 $n$ 个节点的网络中的 $2n$ 个时变参数。我们建立了当 $n$ 或 $N$ 发散时，核平滑估计量的一致性和渐近正态性。我们的结果与单网络分析中的对应结果形成对比，后者在渐近研究中恒要求 $n\to\infty$。我们开展了全面的模拟研究，验证了理论预测，并从多个角度展示了方法的性能。我们将该方法应用于电子邮件数据集，获得了有意义的结果。