  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.
