Interactions among Analysis, Optimization, and Network Science

由于移动、交通和信息科学的进步，我们的世界变得越来越相互联系，网络的作用在全球范围内变得越来越突出。这个项目将把度量空间和复数分析的研究进展与凸分析、概率和图论的工具结合起来，以解决网络科学中新出现的问题。特别是，该项目将弥合某些源自数学分析学科的强大工具与其在离散和应用数学领域的对应工具之间的差距。该项目还将开发数值和算法方法，将分析中的抽象方法与网络科学、数据科学和应用数学中的其他领域的实际问题联系起来。例如，在该项目中开发的用于研究网络科学的工具可能对流行病传播的建模、预测和缓解有用。该项目将以堪萨斯州立大学网络优化设计和探索(NODE)研究小组的工作为基础并加以扩大，该研究小组是一个由来自数学、电气和计算机工程、计算机科学和统计学的研究人员组成的跨学科团队。该项目将支持教师、博士后、研究生和本科生在这个跨学科研究小组的框架内从事分析、应用数学和网络科学的各种主题的工作。研究网络上的传播现象的一个关键方面是对互联性的理解，即由网络中的节点代表的个人与其他节点交互和接触的多种方式。传统上，在对子种群间传播过程的定量研究中，研究人员使用了数学概念，如最短路径的长度(用来衡量一个人必须走多远)或最小切割的大小(用来捕捉可用的不同路径的数量)。有效电阻的概念起源于电网络理论，它在这两个极端之间提供了一种必要的妥协，平衡了路径的短小性和路径的多样性。从某种意义上说，电力能够找到许多最优路径，从而洞察网络的内在结构。从数学的角度来看，电流与随机游动和经典扩散过程有很深的联系。在以前的工作中，PI和他们的合作者发展了网络上的p-模理论，该理论将最短路径、最小割和有效阻力这三个经典度量统一在一个框架中。P-模的概念起源于复分析，但当应用于网络研究时，它成为一种非常灵活的工具，可以以多种方式衡量许多不同对象家族的丰富度。当前项目的目标是使用离散模数理论在分析中获得新的结果，并证明离散环境中的某些量在特定的近似方案或尺度限制下收敛于它们的连续对应项。该项目由数学科学部门的分析计划和既定的激励竞争研究计划(EPSCoR)共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

As our world becomes more and more interconnected, thanks to advances in mobility, transportation, and information science, the role of networks becomes increasingly salient at a global scale. This project will integrate advances in the study of metric spaces and complex analysis with tools from convex analysis, probability, and graph theory to address emerging questions in network science. In particular, the project will bridge the gap between certain powerful tools originating within the subject of mathematical analysis, and their counterparts in discrete and applied mathematics. The project will also develop numerical and algorithmic approaches that will connect abstract methods in analysis with practical questions of network science, data science, and other areas in applied mathematics. For example, tools developed in this project for the study of network science may be useful for the modeling, prediction, and mitigation of the spread of epidemics. The project will build upon and amplify the work of the Network Optimization Design and Exploration (NODE) research group at Kansas State University, an interdisciplinary team of researchers from mathematics, electrical and computer engineering, computer science, and statistics. The project will support the involvement of faculty, postdocs, graduate students, and undergraduate students, working on a diverse range of topics in analysis, applied mathematics, and network science within the framework of this interdisciplinary research group.A key aspect in the study of spreading phenomena on a network is an understanding of interconnectedness, i.e., the many ways in which individuals, represented by nodes within the network, interact with and come into contact with other nodes. Classically, in the quantitative study of spreading processes among subpopulations, researchers have used mathematical concepts such as the length of a shortest path (to measure how far one must travel) or the size of minimum cuts (to capture the number of different pathways available). The notion of effective resistance, which originated in the theory of electrical networks, provides a needed compromise between these two extremes, balancing shortness of paths with variety of pathways. In a sense, electricity is able to find many optimal routes and thus yields insights into the intrinsic structure of networks. From a mathematical point of view, electrical current has deep connections to random walks and to classical diffusion processes. In previous work, the PIs and their collaborators have developed the theory of p-modulus on networks, which unifies in a single framework three classical measures: shortest paths, mincuts, and effective resistance. The concept of p-modulus originated in complex analysis, but when applied in the study of networks becomes an extremely flexible tool that can measure the richness of many different families of objects in multiple ways. The goal of the current project is to use the discrete theory of modulus to obtain new results in analysis, and also to show that certain quantities in the discrete context converge to their continuous counterparts under specific approximation schemes or scaling limits.This project is jointly funded by the Analysis Program in the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.