Phase Transition Phenomena in Random Graphs

数学上理解得很好的随机图可以作为参考模型，指导现实世界网络（如通信网络、社交网络或食物链）的分析和设计。该项目旨在开发和推进用于分析各种随机图的数学工具和技术，包括“动态”模型（即以随机方式逐步增长的图）。动态方面在概念上很重要，因为(a)许多现实世界的网络也随着时间的推移而增长，而(b)大多数现有的数学随机图文献关注的是“静态”模型。因此，该项目有助于理解网络的跨学科努力，网络如今在现实生活中无处不在，也在数据科学、复杂网络和许多其他学科中无处不在。本项目探讨了随机图理论的一个核心主题：组件结构中的相变现象（从只有“小”组件到主导大部分图的“巨型”组件）。PI将在几个难以分析的随机图模型中研究这一基本而引人注目的现象，特别强调在边缘之间具有非平凡依赖关系的模型中最大组件的大小（有限大小的缩放行为）。两个主要目标是：(i)促进我们对相变的一般理解，（ii）改进该领域的数学证明技术，特别是使它们更健壮（以便它们适用于更广泛的模型）。

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).