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 a Network

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.