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The Ohio State University

1. Chowdhury, Samir. Metric and Topological Approaches to Network Data Analysis.

Degree: PhD, Mathematics, 2019, The Ohio State University

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. Advisors/Committee Members: Mémoli, Facundo (Advisor).

Subjects/Keywords: Mathematics; network distance; persistent homology; metric spaces

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APA (6th Edition):

Chowdhury, S. (2019). Metric and Topological Approaches to Network Data Analysis. (Doctoral Dissertation). The Ohio State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=osu1555420352147114

Chicago Manual of Style (16th Edition):

Chowdhury, Samir. “Metric and Topological Approaches to Network Data Analysis.” 2019. Doctoral Dissertation, The Ohio State University. Accessed September 21, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1555420352147114.

MLA Handbook (7th Edition):

Chowdhury, Samir. “Metric and Topological Approaches to Network Data Analysis.” 2019. Web. 21 Sep 2019.

Vancouver:

Chowdhury S. Metric and Topological Approaches to Network Data Analysis. [Internet] [Doctoral dissertation]. The Ohio State University; 2019. [cited 2019 Sep 21]. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1555420352147114.

Council of Science Editors:

Chowdhury S. Metric and Topological Approaches to Network Data Analysis. [Doctoral Dissertation]. The Ohio State University; 2019. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1555420352147114

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