Random Graphs: A Mathematical Physics Perspective

这个项目的目的是研究大指数随机图的结构和行为，这是最近在理论和实验上都进行了深入研究的主题。它们的受欢迎程度在于，它们通过一组易于处理的局部特征来表示复杂的全球结构，从而捕捉了各种常见的网络趋势，例如连通性和互惠性。PI主要从数学物理的角度来看待这些模型，并引入了各种先进的统计物理工具，如集团膨胀方法和重整化群技术，来研究这些模型。特别强调的是“相变”的定量理论的发展，因为在附近的相变，即使是一个微小的变化，在一些局部功能可能会导致整个系统的急剧变化。在这个项目中提出了多个问题和研究途径。这类问题的具体例子是描述吸引和排斥指数随机图的相结构，并推导出收敛的幂级数展开的限制freeenergy.The拟议的工作是由数学，物理学和计算机科学之间的思想交流的动机。PI将建立指数随机图中相变的精确定义，并探索它们与其他数学物理模型的联系。所使用的主要技术将是平衡统计物理学的变体。许多正在考虑的问题有广泛的应用到不同的数学领域，包括组合数学，概率论和图论。拟议的研究的一个深远的潜在好处将是更好地了解不同的本地功能对现实世界的网络，如社会和生物网络，其研究仍处于起步阶段的全球结构的影响。该项目的更广泛的影响将通过将研究纳入课堂教学和让学生参与学习和发现来实现。还将通过PI继续参加国家和国际的跨学科会议和讲习班来广泛传播拟议的研究。

This project aims to investigate the structure and behavior of large exponential random graphs, which have recently been the subject of intense research both theoretically and experimentally. Their popularity lies in the fact that they capture a wide variety of common network tendencies, for example connectivity and reciprocity, by representing a complex global structure through a set of tractable local features. The PI views these models mainly from a mathematical physics perspective, and introduces various advanced statistical physics tools, such as cluster expansion methods and renormalization group techniques, to the study of these models. Special emphasis is given to the development of a quantitative theory of "phase transitions", since in the vicinity of a phase transition, even a tiny change in some local feature may result in a dramatic change of the entire system. Multiple problems and avenues for research are presented in this project. Particular examples of such problems are to characterize the phase structure of attractive and repulsive exponential random graphs, and to derive a convergent power series expansion for the limiting free energy.The proposed work is motivated by the interchange of ideas between mathematics, physics, and computer science. The PI will establish a precise definition of phase transitions in exponential random graphs and explore their connections to other mathematical physics models. The main techniques used will be variants of equilibrium statistical physics. Many of the questions under consideration have broad applications to different areas of mathematics including combinatorics, probability, and graph theory. A far reaching potential benefit of the proposed research will be a better understanding of the influence of different local features on the global structure of real-world networks, such as social and biological networks, whose study is still in its infancy. The broader impacts of the project will be achieved through integrating research into classroom teaching and engaging students in learning and discovery. Broad dissemination of the proposed research will also be realized through the PI's continued participation in interdisciplinary conferences and workshops both nationally and internationally.