Longitudinal network consists of a sequence of temporal edges among multiple nodes, where the temporal edges are observed in real time. It has become ubiquitous with the rise of online social platform and e-commerce, but largely under-investigated in literature. In this paper, we propose an efficient estimation framework for longitudinal network, leveraging strengths of adaptive network merging, tensor decomposition and point process. It merges neighboring sparse networks so as to enlarge the number of observed edges and reduce estimation variance, whereas the estimation bias introduced by network merging is controlled by exploiting local temporal structures for adaptive network neighborhood. A projected gradient descent algorithm is proposed to facilitate estimation, where the upper bound of the estimation error in each iteration is established. A thorough analysis is conducted to quantify the asymptotic behavior of the proposed method, which shows that it can significantly reduce the estimation error and also provides guideline for network merging under various scenarios. We further demonstrate the advantage of the proposed method through extensive numerical experiments on synthetic datasets and a militarized interstate dispute dataset.

翻译：纵向网络由多个节点间按时间顺序观测到的时序边构成，这些时序边随时间实时生成。随着在线社交平台与电子商务的兴起，纵向网络已变得无处不在，但相关研究在文献中仍较为匮乏。本文提出一种面向纵向网络的高效估计框架，融合了自适应网络合并、张量分解与点过程的优势。通过合并相邻稀疏网络，该方法可增加观测边数量并降低估计方差，同时利用局部时序结构实现自适应网络邻域选择，从而有效控制网络合并引入的估计偏差。我们设计了一种投影梯度下降算法以促进估计，并建立了每次迭代中估计误差的上界。通过严谨的理论分析量化了所提方法的渐近行为，证明其能显著降低估计误差，同时为不同场景下的网络合并提供指导准则。基于合成数据集与军事化国家间争端数据集的数值实验进一步验证了该方法的优势。