  The edges in networks are not only binary, either present or absent, but also
take weighted values in many scenarios (e.g., the number of emails between two
users). The covariate-$p_0$ model has been proposed to model binary directed
networks with the degree heterogeneity and covariates. However, it may cause
information loss when it is applied in weighted networks. In this paper, we
propose to use the Poisson distribution to model weighted directed networks,
which admits the sparsity of networks, the degree heterogeneity and the
homophily caused by covariates of nodes. We call it the \emph{network Poisson
model}. The model contains a density parameter $\mu$, a $2n$-dimensional node
parameter ${\theta}$ and a fixed dimensional regression coefficient ${\gamma}$
of covariates. Since the number of parameters increases with $n$, asymptotic
theory is nonstandard. When the number $n$ of nodes goes to infinity, we
establish the $\ell_\infty$-errors for the maximum likelihood estimators
(MLEs), $\hat{\theta}$ and $\hat{{\gamma}}$, which are $O_p( (\log n/n)^{1/2}
)$ for $\hat{\theta}$ and $O_p( \log n/n)$ for $\hat{{\gamma}}$, up to an
additional factor. We also obtain the asymptotic normality of the MLE.
Numerical studies and a data analysis demonstrate our theoretical findings. )
for b{\theta} and Op(log n/n) for b{\gamma}, up to an additional factor. We
also obtain the asymptotic normality of the MLE. Numerical studies and a data
analysis demonstrate our theoretical findings.
