  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.
