  Modelling edge weights play a crucial role in the analysis of network data,
which reveals the extent of relationships among individuals. Due to the
diversity of weight information, sharing these data has become a complicated
challenge in a privacy-preserving way. In this paper, we consider the case of
the non-denoising process to achieve the trade-off between privacy and weight
information in the generalized $\beta$-model. Under the edge differential
privacy with a discrete Laplace mechanism, the Z-estimators from estimating
equations for the model parameters are shown to be consistent and
asymptotically normally distributed. The simulations and a real data example
are given to further support the theoretical results.
