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Metric and Topological Approaches to Network Data Analysis

Chowdhury, Samir
http://rave.ohiolink.edu/etdc/view?acc_num=osu1555420352147114

Abstract Details

2019, Doctor of Philosophy, Ohio State University, Mathematics.
Network data, which shows the relationships between entities in complex systems, is becoming available at an ever-increasing rate. In particular, advances in data acquisition and computational power have shifted the bottleneck in analyzing weighted and directed network datasets towards the domain of available mathematical methods. Thus there is a pressing need to develop mathematical foundations for analyzing such datasets. In this thesis, we present methods for applying one of the flagship tools of topological data analysis---persistent homology---to weighted, directed network datasets. We ground these methods in a network distance that had appeared in a restricted context in earlier literature, and is now fully developed in this thesis. This development independently provides metric methods for network data analysis, including and invoking methods from optimal transport. In our framework, a network dataset is represented as a set of points (equipped with the minimalistic structure of a first countable topological space) and a (continuous) real-valued edge weight function. With this terminology, a finite network dataset is viewed as a finite sample from some infinite underlying process---a compact network. This perspective is especially appropriate for data streams that are so large that they are "essentially infinite", or are perhaps being generated continuously in time. We show that the space of all compact networks is the completion of the space of all finite networks. We develop the notion of isomorphism in this space, and explore a range of different geodesics that exist in this space. We develop sampling theorems and explain their use in obtaining probabilistic convergence guarantees. Several persistent homology methods---notably including persistent path homology---are also developed. By virtue of the sampling theorems, we are able to define these methods even for infinite networks. Our theoretical contributions are complemented by software packages that we developed in the course of producing this thesis. We illustrate the theory and implementations via experiments on simulated and real-world data.
Facundo Mémoli (Advisor)
241 p.
Mathematics
network distance; persistent homology; metric spaces

Recommended Citations

Citations

  • Chowdhury, S. (2019). Metric and Topological Approaches to Network Data Analysis [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1555420352147114

    APA Style (7th edition)

  • Chowdhury, Samir. Metric and Topological Approaches to Network Data Analysis. 2019. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1555420352147114.

    MLA Style (8th edition)

  • Chowdhury, Samir. "Metric and Topological Approaches to Network Data Analysis." Doctoral dissertation, Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1555420352147114

    Chicago Manual of Style (17th edition)

osu1555420352147114
736
© 2019, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.