Abstract
Random Graphs and Mean Field Percolation are two names given to the most general mathematical model of systems composed of a set of connected entities. It has many applications in the study of real life networks as well as physical systems. The model shows a precisely described phase transition, but its solution for finite systems was yet unresolved. However, atomic nuclei, as well as other mesoscopic objects (e.g. molecules, nano-structures), cannot be considered as infinite and their fragmentation does not necessarily occur close to the transition point. Here, we derive for the first time the exact solution of Mean Field Percolation for systems of any size, as well as provide important information on the internal structure of Random Graphs. We show how these equations can be used as a basis to select non-trivial correlations in systems and thus to provide evidence for physical phenomena.
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