Interactions among Analysis, Optimization, and Network Science

Kansas State University, Manhattan KS

Investigators

Abstract

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.