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的时序边构成，其边以实时方式呈现。随着在线社交平台与电子商务的兴起，此类网络已变得无处不在，但相关文献对此研究尚不充分。本文提出一种高效估计框架，融合自适应网络合并、张量分解与点过程的优势。该框架通过合并邻近稀疏网络以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，以控制网络合并引入的估计偏差。我们提出投影梯度下降算法辅助估计，并建立了各迭代步骤中估计误差的上界。通过详尽分析刻画了所提方法的渐近性质，表明该方法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。进一步基于合成数据集与军事化国家间争端数据集的大量数值实验，验证了该方法的优越性。