In biology, networks serve as powerful representations of systems across various scales, from genetic interactions (GI) in organisms to food webs in ecological communities. However, biological networks are often plagued by multiple sources of error, including measurement inaccuracies, sampling biases, and incomplete data. While sophisticated algorithms have been developed to address these challenges, yielding notable successes [1], existing methods fail to fully exploit the rich second-moment statistical information (variances and covariances) present in biological data. Filling this gap requires methodological advancements. We develop a network version of the generalized Wiener filter [2], specifically tailored for filtering edge noise in biological networks. The core technical obstacle arises from the absence of a natural distance metric in network

settings, distinguishing this task from traditional signal and image processing applications. Depending on the scenario, we resolve this issue by either uncovering the complete covariance structure of the network data or employing a network theoretic ansatz.