Random Graphs: A Mathematical Physics Perspective

本项目旨在研究大型指数随机图的结构和行为，这是近年来理论和实验研究的热点。它们的受欢迎程度在于，它们通过一组可处理的局部特征来表示复杂的全球结构，从而捕获了各种各样的共同网络趋势，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，如聚类展开方法和重整化群技术来研究这些模型。特别强调了“相变”定量理论的发展，因为在相变附近，即使某些局部特征的微小变化也可能导致整个系统的剧烈变化。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，并导出极限自由能的收敛幂级数展开式。这项提议的工作是由数学、物理学和计算机科学之间的思想交流所激发的。PI将在指数随机图中建立相变的精确定义，并探索它们与其他数学物理模型的联系。使用的主要技术将是平衡统计物理的变体。考虑中的许多问题在数学的不同领域有广泛的应用，包括组合学、概率论和图论。拟议研究的一个深远的潜在好处将是更好地理解不同的局部特征对现实世界网络（如社会和生物网络）的全球结构的影响，这方面的研究仍处于起步阶段。该项目的更广泛影响将通过将研究融入课堂教学和吸引学生学习和发现来实现。提议的研究也将通过PI继续参加国内和国际的多学科会议和讲习班来广泛传播。

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.