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This paper investigates a cellular edge caching problem under a very large number of small base stations (SBSs) and users. In this ultra-dense edge caching network (UDCN), conventional caching algorithms are inapplicable as their computational complexity increases with the number of small base stations (SBSs). Furthermore, the performance of UDCN is highly sensitive to the dynamics of user demand. To overcome such difficulties, we propose a distributed caching algorithm under a stochastic geometric network model, as well as a spatio-temporal user demand model that characterizes the content popularity dynamics. By leveraging mean-field game (MFG) theory, the complexity of the proposed UDCN caching algorithm becomes independent of the number of SBSs. Numerical evaluations validate this consistent complexity of the proposed algorithm with respect to the number of SBSs. Also, it shows that the …