This award will support research in the structures, called phases, which emerge in systems with many components as the number of components grows. We will study complex networks, simple models consisting only of abstract nodes and edges between them. These models are simpler than those traditionally used to analyze phases, and the simplicity will allow us to better understand how phases transition between one another as control parameters vary, a study of wide application in the sciences and technology. Possible applications range from management of urban construction projects to astrophysics computations and understanding behavior of granular media, such as sand and grain piles.
The prototype of emergent systems are the phases in molecular materials in thermal equilibrium, and they have been studied intensively, but only special sorts of phases and transitions have been successfully modeled; in particular there has never been a satisfactory model of the emergence of the solid phase, or the fluid-solid transition. We will modify the microcanonical ensemble of statistical mechanics to study complex networks, which exhibit phases analogous to solid and fluid phases, and we will exploit the relative simplicity of these models to better understand such phases and the phase transitions between them.
