  We propose a first-order autoregressive (i.e. AR(1)) model for dynamic
network processes in which edges change over time while nodes remain unchanged.
The model depicts the dynamic changes explicitly. It also facilitates simple
and efficient statistical inference methods including a permutation test for
diagnostic checking for the fitted network models. The proposed model can be
applied to the network processes with various underlying structures but with
independent edges. As an illustration, an AR(1) stochastic block model has been
investigated in depth, which characterizes the latent communities by the
transition probabilities over time. This leads to a new and more effective
spectral clustering algorithm for identifying the latent communities. We have
derived a finite sample condition under which the perfect recovery of the
community structure can be achieved by the newly defined spectral clustering
algorithm. Furthermore the inference for a change point is incorporated into
the AR(1) stochastic block model to cater for possible structure changes. We
have derived the explicit error rates for the maximum likelihood estimator of
the change-point. Application with three real data sets illustrates both
relevance and usefulness of the proposed AR(1) models and the associate
inference methods.
