When the network is reconstructed, two types of errors can occur: false positive and false negative errors about the presence or absence of links. In this paper, the influence of these two errors on the vertex degree distribution is analytically analyzed. Moreover, an analytic formula of the density of the biased vertex degree distribution is found. In the inverse problem, we find a reliable procedure to reconstruct analytically the density of the vertex degree distribution of any network based on the inferred network and estimates for the false positive and false negative errors based on, e.g., simulation studies.