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 dataset and obtain meaningful results.

中文翻译：

---

动态有向网络的时变 β 模型

我们扩展了众所周知的$\beta$-动态网络设置的有向图模型，我们在其中观察不同时间点的邻接矩阵的快照。我们提出了一种核平滑似然方法来估计$2n$网络中的时变参数$n$节点，从$氮$快照。我们将核平滑估计量的一致性和渐近正态性属性建立为$n$或者$氮$发散。我们的结果与单网络分析中的对应结果进行了对比，其中$n\to \mathrm{无穷大}$在渐近研究中始终需要。我们进行了全面的模拟研究，证实了我们的理论预测，并从各个角度说明了我们方法的性能。我们将我们的方法应用于电子邮件数据集并获得有意义的结果。