Broad-Scale Modeling of Complex Networks

技术、科学和医学中的许多感兴趣的对象可以表示为网络，包括互联网、电网、大脑中的神经网络，以及疾病传播所在的个人之间的联系网络。网络理论是本研究项目的主题，它提供了此类系统的数学表示，有助于我们理解和预测它们的行为。这个项目的重点是使用数学模型和计算机模型对网络系统进行建模。近年来，网络模型在计算机、信息网络和流行病学等领域取得了令人印象深刻的成功，但以前工作的一个根本缺陷是无法同时准确地表示小尺度和大尺度的网络结构。该项目开发了实现这一目标的新的模型类别，从而更准确地表示网络在现实生活中出现的情况，并提高我们对其行为的理解和预测能力。该项目的具体目标包括：开发新的联网系统的多尺度数学和计算机模型；测试和验证模型，以显示它们如何很好地捕捉它们所代表的系统的特征；确定如何最好地代表特定系统的模型选择方法；网络中的异常检测；改进的计算方法，使计算能够在目前的计算机硬件上有效地运行；以及一系列具体应用，例如用于模拟网络复原力和疾病的传播。这个项目将开发和应用新的数学模型来表示和分析网络系统。网络模型广泛用于分析网络数据的方法，如社区检测、嵌入和可视化，并作为网络过程的模拟和建模的基础，如系统对其组件故障的恢复能力、疾病在接触网络上的传播，或网络协议和算法的设计和改进。然而，当前网络模型的一个根本缺陷是它们不能准确地捕捉小尺度和大尺度上的网络结构。基于局部结构主题的模型，如构型模型或子图模型，能很好地捕捉小尺度结构，但不能捕捉大尺度结构，如社区、分层或核心-外围结构。捕捉大尺度结构的模型，如块模型，通常是局部树状的，因此很难捕捉到小尺度。这个项目将开发一类新的随机图模型，它自然地整合了大和小，以及分析模型属性和快速蒙特卡罗采样的方法。将开发最大似然拟合方法，使模型与真实世界的网络数据相匹配，从而实现准确的概化、链接预测和网络重建。还将开发模型的统计方法，包括拟合优度测试和模型选择方法，以及具体应用，例如异常检测、网络复原力和流行病建模。

Many objects of interest in technology, science, and medicine can be represented as networks, including the internet, the power grid, neural networks in the brain, and the contact networks between individuals over which diseases spread. Network theory, which is the subject of this research project, provides a mathematical representation of systems like these which helps us to understand and predict their behavior. This project focuses on modeling networked systems, using mathematical models and computer models. Network models have seen impressive successes in recent years, in areas such as computer and information networks and epidemiology, but a fundamental shortcoming of previous work has been an inability to accurately represent network structure at both small and large scales simultaneously. This project develops new classes of models that achieve this goal, thereby more accurately representing networks as they appear in real life and improving our understanding and ability to predict their behavior. Specific goals of the project include: development of new multiscale mathematical and computer models of networked systems; testing and validation of models to demonstrate how well they capture the features of the systems they represent; model selection methods for determining how best to represent a particular system; anomaly detection in networks; improved computational methods to allow calculations to run efficiently on current computer hardware; and a range of specific applications, for instance to modeling of network resilience and the spread of disease. This project will develop and apply new classes of mathematical models for representing and analyzing networked systems. Network models find wide uses in methods for the analysis of network data, such as community detection, embedding, and visualization, and as the foundation for simulation and modeling of network processes, such as resilience of systems to failure of their components, the spread of diseases over contact networks, or the design and refinement of network protocols and algorithms. A fundamental shortcoming of current network models, however, is their inability to capture network structure accurately on both small and large scales. Models based on local structural motifs, such as the configuration model or subgraph models, capture small-scale structure well, but fail with large-scale structure such as communities, stratification, or core-periphery structure. Models that capture large-scale structure, such as block models, are normally locally tree-like and hence fail badly to capture the small scale. This project will develop a new class of random graph models that naturally integrates the large and small, along with methods for analyzing the models' properties and for rapid Monte Carlo sampling. Maximum-likelihood fitting methods will be developed to fit models to real-world network data, enabling accurate generalization, link prediction, and network reconstruction. Statistical methods for the models will also be developed, including goodness-of-fit tests and model selection methods, along with specific applications, for instance to anomaly detection, network resilience, and epidemic modeling.