  The $p_0$ model is an exponential random graph model for directed networks
with the bi-degree sequence as the exclusively sufficient statistic. It
captures the network feature of degree heterogeneity. The consistency and
asymptotic normality of a differentially private estimator of the parameter in
the private $p_0$ model has been established. However, the $p_0$ model only
focuses on binary edges. In many realistic networks, edges could be weighted,
taking a set of finite discrete values. In this paper, we further show that the
moment estimators of the parameters based on the differentially private
bi-degree sequence in the weighted $p_0$ model are consistent and
asymptotically normal. Numerical studies demonstrate our theoretical findings.
