Multi-relational networks among entities are frequently observed in the era of big data. Quantifying the effects of multiple networks have attracted significant research interest recently. In this work, we model multiple network effects through an autoregressive framework for tensor-valued time series. To characterize the potential heterogeneity of the networks and handle the high dimensionality of the time series data simultaneously, we assume a separate group structure for entities in each network and estimate all group memberships in a data-driven fashion. Specifically, we propose a group tensor network autoregression (GTNAR) model, which assumes that within each network, entities in the same group share the same set of model parameters, and the parameters differ across networks. An iterative algorithm is developed to estimate the model parameters and the latent group memberships simultaneously. Theoretically, we show that the group-wise parameters and group memberships can be consistently estimated when the group numbers are correctly- or possibly over-specified. An information criterion for group number estimation of each network is also provided to consistently select the group numbers. Lastly, we implement the method on a Yelp dataset to illustrate the usefulness of the method.

翻译：大数据时代，实体间的多重关系网络频繁出现。量化多个网络的影响近年来引起了广泛研究兴趣。本文通过张量值时间序列的自回归框架对多重网络效应进行建模。为同时刻画网络潜在异质性并处理时间序列数据的高维性，我们假设每个网络中的实体具有独立的群结构，并以数据驱动方式估计所有群的隶属关系。具体而言，我们提出群张量网络自回归（GTNAR）模型，该模型假设在每个网络内，同一群中的实体共享相同的模型参数集，且不同网络的参数存在差异。我们开发了一种迭代算法以同时估计模型参数和潜在群隶属关系。理论上，我们证明当群数量被正确指定或可能过度指定时，群参数和群隶属关系可被一致估计。此外，还提供了用于估计每个网络群数量的信息准则，以实现群数的一致选择。最后，我们将该方法应用于Yelp数据集，以说明其实用性。