Social platforms have paved the way in creating new, modern ways for users to communicate with each other. In recent years, multiple platforms have introduced ‘‘Stories’’ features, which enable broadcasting of ephemeral multimedia content. Specifically, ‘‘Friend Stories,’’ or Stories meant to be consumed by one’s close friends, are a popular feature, promoting significant user-user interactions by allowing people to see (visually) what their friends and family are up to. A key challenge in surfacing Friend Stories for a given user, is in ranking over each viewing user’s friends to efficiently prioritize and route limited user attention. In this work, we explore the novel problem of Friend Story Ranking from a graph representation learning perspective. More generally, our problem is a link ranking task, where inferences are made over existing links (relations), unlike common node or graph-based tasks, or link prediction tasks, where the goal is to make inferences about non-existing links. We propose ELR, an edge-contextual approach which carefully considers local graph structure, differences between local edge types and directionality, and rich edge attributes, building on the backbone of graph convolutions. ELR handles social sparsity challenges by considering and attending over neighboring nodes, and incorporating multiple edge types in local surrounding egonet structures. We validate ELR on two large country-level datasets with millions of users and tens of millions of links from Snapchat. ELR shows superior performance over alternatives by 8% and 5% error reduction measured by MSE and MAE correspondingly. Further generality, data efficiency and ablation experiments confirm the advantages of ELR.