Complex networks offer a powerful paradigm to study the structure of complex systems on a common basis, using the same concepts and nomenclature to represent systems as various as human interactions, ecosystems, the brain or the Internet. Although this impressive feature suggests some sort of universality, the available toolbox to characterize these structures is incomplete since it does not provide a thorough understanding of their organization. Indeed, many key properties have been identified as fundamental structural features, but they only offer a partial glimpse at the global picture and we are still unable to accurately predict the evolution of many dynamical processes taking place on real networks. In other words, we still lack a comprehensive and intensive way to capture and synthesize the full essence of these structures.
In this presentation, I will discuss two promising approaches synthesizing the macroscopic organization of real complex networks into a set of local properties, which in turn naturally define random graph ensembles reproducing the said macroscopic features based on local connection rules only. I will then discuss how the various tools developed to unveil this effective structure of networks can be used to shed light on new phenomena in Epidemiology and Neuroscience. This will be illustrated via ongoing projects dealing with the current threat of a Zika epidemic and the organization of the connectome across species.