Over the past few years Content-Centric Networking, a networking architecture in which host-to-content communication protocols are introduced, has been gaining much attention. A central component of such an architecture is a large-scale interconnected caching system. To date, the way these Cache Networks operate and perform is still poorly understood. Following the work of Cruz on queueing networks, in this paper we develop a network calculus for bounding flows in LRU cache networks of arbitrary topology. We analyze the tightness of these bounds as a function of several system parameters. Also, we derive from it several analytical results regarding these systems: the uniformizing impact of LRU on the request stream, and the significance of cache and routing diversity on performance.