  The original generalized network autoregressive models are poor for modelling
count data as they are based on the additive and constant noise assumptions,
which is usually inappropriate for count data. We introduce two new models
(GNARI and NGNAR) for count network time series by adapting and extending
existing count-valued time series models. We present results on the statistical
and asymptotic properties of our new models and their estimates obtained by
conditional least squares and maximum likelihood. We conduct two simulation
studies that verify successful parameter estimation for both models and conduct
a further study that shows, for negative network parameters, that our NGNAR
model outperforms existing models and our other GNARI model in terms of
predictive performance. We model a network time series constructed from
COVID-positive counts for counties in New York State during 2020-22 and show
that our new models perform considerably better than existing methods for this
problem.
