Exact solution of Random Graph fragmentation and physical, chemical and biological applications

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Random graphs (RG), also called mean field percolation in the frame of mesoscopic physics, are a basic model composed solely of connected entities called nodes. The connections, called bonds, can be active of broken. When the number of broken bonds is too large, the system of entities separates into a set of fragments called a partition. The exact solution for the micro-canonical partition probabilities of finite size systems was yet unresolved, thus a series of fundamental questions about the model could not be answered. We have established the exact equations of RG partition probabilities as a function of the number of nodes and of the number of broken bonds. From these probabilities, it is also possible to deduce intrinsic properties of RG. Many actual networks, while composed of complex interactions, behave like RG. We show examples where information was deduced, using RG, from systems consisting of sets of nucleons, atoms or termite nest chambers.