Partial Differential Equation Models for Large Networks

该项目涉及使用偏微分方程(PDE)的大型网络的连续介质模型。前提是，超大型网络的某些全局特征可以通过考虑组件行为的连续介质模型来捕获，这些模型考虑的是集合而不是个体的规模。该项目涉及一个跨学科的团队，其中包括一名在网络方面具有显著专长的电气工程师，一名在随机过程渐近分析和统计建模方面经验丰富的统计学家，以及一名在涉及非线性偏微分方程的建模和数值模拟方面具有专长的应用数学家。成功完成此项目将使您能够使用传统蒙特卡罗模拟无法完成的计算工作来分析大型网络的性能。智力优势：此项目在工程和数学方面都具有重要的智力优势。从数学的角度来看，创建大型网络连续统行为的数学描述和分析由此产生的微分方程式都是真正的挑战。从工程学的角度来看，连续体建模将需要在最小的几个节点的最小尺度上详细了解网络行为，并确定可以建模的全局特征。广泛的影响：该项目还将对主要研究人员的当前学生和未来学生的培养产生直接的积极影响。致力于解决网络建模中产生的问题的学生在他们的论文中将有非常宝贵的实践部分。这将极大地拓宽他们在行业和跨学科环境中的职业机会。这反过来将为科罗拉多州立大学未来提供一个强大的招生工具。

This project involves continuum models of large networks using partial differential equations (PDEs). The premise is that some global characteristics of very large networks can be captured by continuum models that consider the behavior of the components on the scale of the aggregate rather than of the individual. The project involves an interdisciplinary team of an electrical engineer with notable expertise in networks, a statistician experienced in asymptotic analysis of stochastic processes and statistical modeling, and an applied mathematician with expertise in modeling and numerical simulation involving nonlinear PDEs. Successful completion of this project will result in the ability to analyze the performance of large networks with a computational effort that is not possible using conventional Monte Carlo simulation.Intellectual Merit: This project contains significant intellectual merit for engineering as well as mathematics. From the mathematical point of view, both creating mathematical descriptions of the continuum behavior of large networks and analyzing the resulting differential equations pose real challenges. From the engineering point of view, continuum modeling will require a detailed understanding of network behavior on the smallest scale of a few nodes and identification of global characteristics that can be modeled.Broader Impacts:This project will also have both an immediate positive impact on the current students of the principal investigators and on training future students. The students working on problems originating in the modeling of networks will have an invaluable practical component to their theses. This will significantly widen their career opportunities in industry and interdisciplinary settings. This in turn will provide a powerful recruiting tool for Colorado State University in the future.