  We study general nonlinear models for time series networks of integer and
continuous valued data. The vector of high dimensional responses, measured on
the nodes of a known network, is regressed non-linearly on its lagged value and
on lagged values of the neighboring nodes by employing a smooth link function.
We study stability conditions for such multivariate process and develop quasi
maximum likelihood inference when the network dimension is increasing. In
addition, we study linearity score tests by treating separately the cases of
identifiable and non-identifiable parameters. In the case of identifiability,
the test statistic converges to a chi-square distribution. When the parameters
are not-identifiable, we develop a supremum-type test whose p-values are
approximated adequately by employing a feasible bound and bootstrap
methodology. Simulations and data examples support further our findings.
