With the recent progress of data acquisition technology, classification of data exhibiting relational dependence, from online social interactions to multi-omics studies to linkage of electronic health records, continues to gain an ever increasing attention. By introducing a robust and inherently geometric concept of data depth we propose a new type of geometrically-enhanced classification method for relational data that are in a form of a complex network with multiple node attributes. Starting from a logistic regression to describe the relationship between the class labels and node attributes, the key approach is based on modeling the link probability between any two nodes as a function of their class labels and their data depths within the respective classes. The approximate prediction rule is then obtained according to the posterior probability of the class labels. Integrating the depth concept into the classification process allows us to better capture the underlying geometry of the relational data and, as a result, to enhance its finite sample performance. We derive asymptotic properties of the new classification approach and validate its finite sample properties via extensive simulations. The proposed geometrically-enhanced classification method is illustrated in application to user analysis of the one of the largest Chinese social media platforms, Sina Weibo.