This paper presents an analytical framework to model fault-tolerance in unstructured peer-to-peer overlays, represented as complex networks. We define a distributed protocol peers execute for managing the overlay and reacting to node faults. Based on the protocol, evolution equations are defined and manipulated by resorting to generating functions. Obtained outcomes provide insights on the nodes' degree probability distribution. From the study of the degree distribution, it is possible to estimate other important metrics of the peer-to-peer overlay, such as the diameter of the network. We study different networks, characterized by three specific desired degree distributions, i.e. nets with nodes having a fixed desired degree, random graphs and scale-free networks. All these networks are assessed via the analytical tool and simulation as well. Results show that the approach can be factually employed to dynamically tune the average attachment rate at peers so that they maintain their own desired degree and, in general, the desired network topology.