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:我们将著名的有向图$\beta$模型扩展到动态网络设置中,其中我们观察到在不同时间点的邻接矩阵快照。我们提出了一种核平滑似然方法,用于从$N$个快照中估计$2n$个时变参数,其中$n$是节点数。我们证明了我们的核平滑估计器具有一致性和渐进正态性质,当$n$或$N$发散时。这与单网络分析中它们的类似物形成对比,其在渐近研究中不变地要求$n\to\infty$。我们进行了全面的模拟研究,证实了我们理论的预测,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,并获得了有意义的结果。