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. See the related papers
[1],
[2], and
[3].

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. See
[1] and
[2].

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. See the related works
[1],
[2], and
[3].

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. Our research has produced the notions of formigrams, and distances between dynamic metric spaces, and dynamic graphs. Fot the interphase with the data anlysis part, we use concepts from zigzag persistent homology.
Read more in
https://research.math.osu.edu/networks/formigrams/.

Zigzag Persistent Homology for Neural Data

We apply Zigzag Persistent Homology towards understanding
the amount of information contained in the spike trains
of hippocampal place cells. Previous work has established
that simply knowing which groups of place cells fire together
in an animal's hippocampus is sufficient to extract the
global topology of the animal's physical environment. We
model a system where collections of place cells group and
ungroup according to short-term plasticity rules. We
obtain the surprising result that in experiments with
spurious firing, the accuracy of the extracted topological
information decreases with the persistence (beyond a
certain regime) of the cell groups. This suggests that
synaptic transience, or forgetting, is a mechanism by which
the brain counteracts the effects of spurious place cell
activity.
PLoS ONE link.

Metric graph approximation of geodesic spaces

A standard result in metric geometry is that every compact geodesic metric space can be approximated arbitrarily well by finite metric graphs in the Gromov-Hausdorff sense. It is well known that the first Betti number of the approximating graphs may blow up as the approximation gets finer.
In our work, given a compact geodesic metric \(X\), we define a sequence \((\delta^X_n)_{n \geq 0}\) of non-negative real numbers by $$\delta^X_n:=\inf \{d_{GH}(X,G): G \text{ a finite metric graph, } \beta_1(G)\leq n \} .$$
By construction, and the above result, this is a non-increasing sequence with limit \(0\). We study this sequence and its rates of decay with \(n\). We also identify a precise relationship between the sequence and the first Vietoris-Rips persistence barcode of \(X\). Furthermore, we specifically analyze \(\delta_0^X\) and find upper and lower bounds based on hyperbolicity and other metric invariants. As a consequence of the tools we develop, our work also provides a Gromov-Hausdorff stability result for the Reeb construction on geodesic metric spaces with respect to the function given by distance to a reference point.
Arxiv link.