This paper introduces a methodology to conduct robust inference in bipartite networks under local misspecification. We focus on a class of dyadic network models with misspecified conditional moment restrictions. The framework of misspecification is local, as the effect of misspecification varies with the sample size. We utilize this local asymptotic approach to construct a robust estimator that is minimax optimal for the mean square error within a neighborhood of misspecification. Additionally, we introduce bias-aware confidence intervals that account for the effect of the local misspecification. These confidence intervals have the correct asymptotic coverage for the true parameter of interest under sparse network asymptotics. Monte Carlo experiments demonstrate that the robust estimator performs well in finite samples and sparse networks. As an empirical illustration, we study the formation of a scientific collaboration network among economists.