Phases and Phase Transitions in Complex Networks

该奖项将支持在具有许多组件的系统中随着组件数量的增长而出现的称为相位的结构的研究。我们将研究复杂的网络，简单的模型只包括抽象节点和它们之间的边。这些模型比传统的相位分析模型更简单，并且这种简单性使我们能够更好地理解随着控制参数的变化，相位如何在彼此之间转变，这是一项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（如沙子和谷物堆）的行为。涌现系统的原型是处于热平衡状态的分子材料中的相，人们对它们进行了深入的研究，但只有一些特殊的相和转变被成功地模拟出来；特别是从来没有一个令人满意的模型来解释固相的出现，或流固转变。我们将修改统计力学的微正则系综来研究复杂网络，这些网络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来更好地理解这些相及其之间的相变。

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.