While node semantics have been extensively explored in social networks, little research attention has been paid to profile edge semantics, i.e., social relations. Ideal edge semantics should not only show that two users are connected, but also why they know each other and what they share in common. However, relations in social networks are often hard to profile, due to noisy multi-modal signals and limited user-generated ground-truth labels. In this work, we aim to develop a unified and principled framework that can profile user relations as edge semantics in social networks by integrating multi-modal signals in the presence of noisy and incomplete data. Our framework is also flexible towards limited or missing supervision. Specifically, we assume a latent distribution of multiple relations underlying each user link, and learn them with multi-modal graph edge variational autoencoders. We encode the network data with a graph convolutional network, and decode arbitrary signals with multiple reconstruction networks. Extensive experiments and case studies on two public DBLP author networks and two internal LinkedIn member networks demonstrate the superior effectiveness and efficiency of our proposed model.