Random networks with concave preferential attachment rule

Many of the phenomena in the complex world in which we live have a rough description as a large network of interacting components. Random network theory tries to describe the global structure of such networks from basic local prin- ciples. One such principle is the preferential attachment paradigm which suggests that networks are built by adding nodes and links successively, in such a way that new nodes prefer to be connected to existing nodes if they have a high degree. Our research gives the first comprehensive and mathematically rigorous treatment of the case when this preference follows a nonlinear, or more precisely concave, rule. We survey results obtained so far and some ongoing developments.