Random Networks with Concave Preferential Attachment Rule

Abstract

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 principles. 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.