In this paper we develop a Gaussian process (GP) framework to model a collection of reciprocal random variables defined on the \emph{edges} of a network. We show how to construct GP priors, i.e.,~covariance functions, on the edges of directed, undirected, and bipartite graphs. The model suggests an intimate connection between \emph{link prediction} and \emph{transfer learning}, which were traditionally considered two separate research topics. Though a straightforward GP inference has a very high complexity, we develop an efficient learning algorithm that can handle a large number of observations. The experimental results on several real-world data sets verify superior learning capacity.