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\begin{document}

\title{Using Mapper Graphs to Reveal Morphological Relationships in Passiflora
Leaves}



\author[1,1]{Sarah Percival}%
\author[1,1]{Beronda Montgomery L.}%
\author[1,1]{Daniel H. Chitwood}%
\author[1,1]{Arjun Krishnan}%
\author[1,1]{Erik J. Amézquita}%
\author[1,1]{Sourabh Palande}%
\author[1,1]{Elizabeth Munch}%
\author[2,2]{Aman Husbands}%
\affil[1]{Michigan State University}%
\affil[2]{The Ohio State University}%


\vspace{-1em}



  
  \date{May 16, 2024}


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\begin{abstract}
As collections of data grow in size, it is increasingly important to
have an efficient means of analyzing large data sets. Topological data
analysis (TDA) applies concepts from the mathematical field of topology
to not only efficiently examine large data sets, but to make inferences
related to the overall ``shape'' of data. In this project, we use
Mapper, a tool from TDA that summarizes data into a graph, to discover
an underlying structure relating the shapes of more than 3,300
Passiflora leaves from 40 different species. We choose to study leaves
of the Passiflora species in particular due to their extraordinary
diversity of shape. As the Mapper graph has a structure, or ``shape'' of
its own, we think of it as a ``shape of shapes'' that provides
information on the interplay between the developmental processes
determining leaf shape within a single plant and the evolutionary
processes between species. In particular, we examine the interactions
between leaf species and both heteroblasty and leaf area by constructing
a Mapper graph for each measure. For each node in the resulting graphs,
we then compute the average leaf shape to obtain a graph structure that
reveals how morphometric differences between species relate to the
developmental changes that must occur for those shapes to be realized.%
\end{abstract}%



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