  This study details an approach for the analysis of social media collected
political data through the lens of Topological Data Analysis, with a specific
focus on Persistent Homology and the political processes they represent by
proposing a set of mathematical generalizations using Gaussian functions to
define and analyze these Persistent Homology categories. Three distinct types
of Persistent Homologies were recurrent across datasets that had been plotted
through retweeting patterns and analyzed through the k-Nearest-Neighbor
filtrations. As these Persistent Homologies continued to appear, they were then
categorized and dubbed Nuclear, Bipolar, and Multipolar Constellations. Upon
investigating the content of these plotted tweets, specific patterns of
interaction and political information dissemination were identified, namely
Political Personalism and Political Polarization. Through clustering and
application of Gaussian density functions, I have mathematically characterized
each category, encapsulating their distinctive topological features. The
mathematical generalizations of Bipolar, Nuclear, and Multipolar Constellations
developed in this study are designed to inspire other political science digital
media researchers to utilize these categories as to identify Persistent
Homology in datasets derived from various social media platforms, suggesting
the broader hypothesis that such structures are bound to be present on
political scraped data regardless of the social media it's derived from. This
method aims to offer a new perspective in Network Analysis as it allows for an
exploration of the underlying shape of the networks formed by retweeting
patterns, enhancing the understanding of digital interactions within the sphere
of Computational Social Sciences.
