Networks are mathematical formalisms that capture relations within and between data sets. Over the past several decades, much progress has been made in explaining real-world phenomena in the biological, physical, and social sciences using network theory, and in particular, graph theory. The complexity of data has increased rapidly in recent years, partly because of the following reasons: (1) It is now easier to produce large volumes of data, e.g. in social networks, and (2) Technological progress has enabled better data capture and extraction, e.g. in biological networks. The objective of the growing field of Network Data Analysis is to devise methods for analyzing such complex network data. Some aspects of these methods are listed below.

Our research is supported by NSF-IIS grant 1422400 and NSF-DMS grant 1723003.

Network metrics
A natural question to ask when comparing two networks is: are these networks the same, or are they different? The difference between two networks can be formally quantified by developing metrics on the collections of all networks. Understanding the behavior of such metrics is one of our ongoing projects.

Clustering networks
Clustering methods, in particular hierarchical clustering methods, can be very useful in exploratory data analysis. Classical hierarchical clustering reveals the presence of clusters across a range of scales for metric datasets (i.e. "well-behaved" datasets), from which researchers are often able to find hidden groupings and patterns. However, general network data tends to be more "wild." We are interested in being able to provide hierachical clustering schemes that simplify the visualization of even very general network data, while providing theoretical guarantees on the worst-case distortion that can occur from applying such clustering methods.

Persistent Homology of Networks
An ongoing project in our group is to better understand the persistent homology of a general network. In addition to devising robust theoretical methods for producing persistence diagrams from networks, we are interested in using our methods to glean new insights into the structures of a wide range of network datasets.

Zigzag Persistent Homology of Dynamic Networks
When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of animals in different groups. Motivated by this, we study the problem of obtaining persistent homology based summaries of time-dependent metric data. Read more in