Partial Differential Equation Models for Large Networks

本项目涉及使用偏微分方程（PDEs）的大型网络的连续统模型。前提是，超大型网络的一些全局特征可以通过连续体模型来捕捉，连续体模型考虑的是整体而不是个体尺度上的组成部分的行为。该项目涉及一个跨学科团队，其中包括一位在网络方面具有显著专业知识的电气工程师，一位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，以及一位在非线性偏微分方程建模和数值模拟方面具有专业知识的应用数学家。这个项目的成功完成将使我们能够通过计算来分析大型网络的性能，这是使用传统的蒙特卡罗模拟无法实现的。智力优势：该项目在工程和数学方面都具有显著的智力优势。从数学的角度来看，创建大型网络连续行为的数学描述和分析由此产生的微分方程都是真正的挑战。从工程的角度来看，连续体建模将需要在几个节点的最小尺度上详细了解网络行为，并识别可以建模的全局特征。更广泛的影响：这个项目也将对当前的主要研究人员的学生和培养未来的学生产生直接的积极影响。研究源自网络建模的问题的学生将为他们的论文提供宝贵的实践组成部分。这将大大拓宽他们在工业和跨学科环境中的职业机会。这反过来又将为科罗拉多州立大学未来的招生提供一个强大的工具。

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.