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This paper proposes a new methodology to model the distribution of finite-size content to a group of users connected through an overlay network. Our methodology describes the distribution process as a constrained stochastic graph process (CSGP), where the constraints dictated by the content distribution protocol and the characteristics of the overlay network define the interaction among nodes. A CSGP is a semi-Markov process whose state is described by the graph itself. CSGPs offer a powerful description technique that can be exploited by Monte Carlo integration methods to compute in a very efficient way not only the mean but also the full distribution of metrics such as the file download times or the number of hops from the source to the receiving nodes. We model several distribution architectures based on trees and meshes as CSGPs and solve them numerically. We are able to study scenarios with a very large number of nodes, and we can precisely quantify the performance differences between the tree-based and mesh-based distribution architectures.