Phases and Phase Transitions in Complex Networks

该奖项将支持结构的研究，称为阶段，这出现在系统中的许多组件的数量， 组件增长。我们将研究复杂的网络，简单的模型只包括抽象的节点和它们之间的边缘。这些 模型比传统上用于分析相的模型简单，这种简单性将使我们能够更好地理解当控制参数变化时相之间如何过渡，这是在科学和技术中广泛应用的研究。可能的应用范围从城市建设项目的管理到天体物理学计算和理解颗粒介质的行为，如沙子和谷物堆。涌现系统的原型是分子材料中处于热平衡的相，并且它们已经被研究 密集，但只有特殊种类的阶段和过渡已成功建模;特别是从来没有一个 固相出现或流体-固体转变的满意模型。我们将修改统计力学的微正则系综来研究复杂的网络，它表现出类似于固体和流体相的相，我们将利用 这些模型的相对简单性，以更好地理解这些阶段和它们之间的相变。

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.