Mathematically well-understood random graphs serve as reference models which can guide the analysis and design of (models for) real world networks such as communication networks, social networks, or food chains. This project aims at developing and advancing mathematical tools and techniques for analyzing various random graphs, including "dynamic" models (i.e., graphs that grow step-by-step in a random way). The dynamic aspect is conceptually important since (a) many real-world networks also grow over time, whereas (b) the majority of the existing mathematical random graphs literature focuses on "static" models. This project thus contributes to the interdisciplinary effort of understanding networks, which are nowadays omnipresent in real life as well as in data science, complex networks, and many other disciplines.
This project explores a topic which is central to the theory of random graphs: the phase transition phenomenon in the component structure (from only "small" components to a "giant" component dominating most of the graph). The PI will investigate this fundamental and striking phenomenon in several difficult-to-analyze random graph models, with particular emphasis on (the finite-size scaling behavior of) the size of the largest component in models with non-trivial dependencies between the edges. Two principle aims are (i) to advance our general understanding of the phase transition, and (ii) to improve the mathematical proof techniques in the area, in particular to make them more robust (so that they apply to a wider range of models).
