We first deduce the exact differential equation for the SIS prevalence (i.e. the average number of infected nodes) on any network, which illustrates both the importance of the cut-set (i.e. set of infective links with one infected node) and the "local-rule, global emergent property" of the SIS/SIR class, leading to phase-transitions. A spectral decomposition of the exact time-varying prevalence leads to the "tanh-formula" for any network. I will briefly illustrate the potential of the so-called "tanh-formula".

After this introduction, we propose a new approximation framework, the Universal Mean-Field Framework (UMFF), that unifies and generalizes a number of existing mean-field approximation methods for the SIS epidemic model on complex networks. The main novelty of UMFF lies in the topological approximation of the SIS epidemic process by graph partitioning and by the famous isoperimetric inequality. These deep network concepts, related to Szemeredi’s regularity lemma, allow us to bound the approximation errors of UMFF and thus of the existing mean-field methods, like our N-Intertwined Mean-Field Approximation (NIMFA) and the Heterogeneous Mean-Field (HMF, Pastor-Satorras & Vespignani), that are particular cases of UMFF.