DSI | Hypergraphs and Topology for Data Science | By Emilie Purvine
Data scientists and applied mathematicians must grapple with complex data when analyzing complex systems. Analytical methods almost always represent phenomena as a much simpler level than the complex structure or dynamics inherent in systems, through either simpler measured or sampled data, or simpler models, or both. As just one example, collaboration data from publications databases are often modeled as graphs of authors, in which pairs of authors (vertices) are connected if they published a paper together, perhaps weighted by the number of such papers. This graph view is also commonly found when analyzing many other kinds of data including biological, cyber, and social. But to better represent inherent complexity, researchers are striving to adopt hypergraphs, representing connections not only as pairwise, but as mulit-way or higher order.
In bibliometrics, where papers have multiple authors, and authors write multiple papers, hypergraphs can natively capture the complex ways that groups of authors form into collaborations as sets of authors on papers, where traditional collaboration networks can only do so via complex coding schemes. Our recent work has focused on first developing and implementing methods that extend common graph methods to hypergraphs – e.g., distance, diameter, centrality – and then using such methods to study real data sets from biology to cyber security. Moreover, the complexity of hypergraphs imbues them with significant topological properties, and we have been active in developing a theory and interpretation of hypergraphs homology, through abstract simplicial complexes and other topological representations. Additionally, graphs and hypergraphs both arise in data systems with more than two dimensions, for example adding keywords or institutions to papers and authors. These four dimensions – authors, papers, keywords, and institutions –now can form a combinatorial number of hypergraphs (e.g., author vs. papers, papers vs. keywords, institutions vs. authors, etc.). But what mathematical structure can be formed when we consider all these dimensions simultaneously? Tensors may be one such structure, but even they may be too restrictive since tensors represent a multi-relation among all dimensions, and data may only be available on certain projections. In this talk I will provide an overview of our work on hypergraphs and topology for data science, including both theory and practice of the methods we have been developing, and provide some thoughts on going beyond hypergraphs.
Dr. Emilie Purvine is a Senior Data Scientist at Pacific Northwest National Laboratory. Although her academic background is in pure mathematics, with a BS from University of Wisconsin - Madison and a PhD from Rutgers University, her research since joining PNNL in 2011 has focused on applications of combinatorics and computational topology together with theoretical advances needed to support the applications. Over her time at PNNL Emilie has been both PI and technical staff on a number of projects in applications ranging from computational chemistry and biology to cyber security and power grid modeling. She has authored over 40 technical publications and is currently an associate editor for the Notices of the American Mathematical Society. Emilie also coordinates PNNL’s Postgraduate Organization which plans career development seminars, an annual research symposium, and promotes networking and mentorship for PNNL’s post bachelors, post masters, and post doctorate research associates.
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LLNL-VIDEO-828614
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