Phase Transition Phenomena in Random Graphs

数学上很好理解的随机图作为参考模型，可以指导分析和设计真实世界的网络(模型)，例如通信网络、社交网络或食物链。该项目旨在开发和推进用于分析各种随机图形的数学工具和技术，包括“动态”模型(即以随机方式逐步增长的图形)。动态方面在概念上很重要，因为(A)许多真实世界的网络也会随着时间的推移而增长，而(B)现有的大多数数学随机图文献都集中在“静态”模型上。因此，该项目有助于理解网络的跨学科努力，目前网络在现实生活中无所不在，在数据科学、复杂网络和许多其他学科中也是如此。这个项目探索了一个对随机图理论至关重要的主题：成分结构中的相变现象(从只有“小”成分到主宰图形大部分的“大”成分)。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，特别强调边之间具有非平凡依赖关系的模型中最大组件的大小(有限大小缩放行为)。两个主要目标是(I)促进我们对相变的一般理解，和(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).