Caching popular files in the storage of edge networks, namely edge caching, is a promising approach for service providers (SPs) to reduce redundant backhaul transmission to edge nodes (ENs). It is still an open problem to design an efficient incentive mechanism for edge caching in 5G networks with a large number of ENs and mobile users. In this paper, an edge network with one SP, a large number of ENs and mobile users with time-dependent requests is investigated. A convergent and scalable Stackelberg game for edge caching is designed. Specifically, the game is decomposed into two types of sub-games, a storage allocation game (SAG) and a number of user allocation games (UAGs). A Stackelberg game-based alternating direction method of multipliers (Stackelberg game-based ADMM) is proposed to solve either the SAG or each UAG in a distributed manner. Based on both analytical and simulation results, the convergence speed, the optimum of the entire game, and the amount of information exchange are linearly (or sublinearly) related to the network size, which indicates that this framework can potentially cope with large-scale caching problems. The proposed approach also requires less backhaul resource than the existed approaches.