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.
