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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

URL: http://rave.ohiolink.edu/etdc/view?acc_num=osu1555420352147114

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 (6^{th} 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 (16^{th} 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 (7^{th} 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