Current Research Projects

To view my full CV and published papers please click the Research link at the top of the page. If you are an undergraduate or graduate student at WSU interested in working on one of these projects, please send me an email.

Main Research Projects

The focus of my research is on applications of algebraic and combinatorial methods to problems that arise in data analysis. Some of the specific topics that I am currently studying are sketched below. For software and data related to these projects, see this page.

Political Redistricting

Political redistricting can be abstracted as a graph partitioning problem subject to a variety of legal constraints. My main research in this area focuses on developing methods for efficient sampling of graph partitions usiing Markov chain techniques. I am also interested in shape based analysis of districting plans and metrics defined on the space of permissible graph partitions. For more details, see the description and notes from my 2019 MIT IAP course here.

There are many interesting, open problems in this domain that are well suited for student research projects. I also maintain an active list of open mathematical problems related to redistricting here, please feel free to email me for more details if any of these catch your interest.

Multiplex Networks

While standard network models consider a set of nodes an a binary relation between them, multiplex networks allow for many different types of connections between the nodes. For example, we might consider a social multiplex whose nodes are people and where different types of edges represent familial ties, coworkers, and friendship relations.

I am particularly interested in dynamical models on these networks, such as information flow through our social network example, and the properties of their associated operators. Current research includes developing clustering algorithms that respect multiplex structure and better understanding the effects of modeling choices in application domains, as well as machine learning approaches for inferring inter-layer edge weights.

Random Dot Product Graphs

The Random Dot Product Graph (RDPG) model is a latent space model for complex networks, where each node is associated to a vector in \(\mathbb{R}^d\) and nodes are connected with probability equal to the dot product of their respective vectors. The geometry of this relationship provides natural connections to notions of comunity assignment and centrality as well as allowing for the construction of multiresolution models with community structure. This model has proven useful theoretically, for proving consistency results about spectral embeddings, as well as practically, with recent applications to many machine learning problems on graphs. I am particularly interested in methods for learning RDPG representations from empirical networks as well as proving results relating the geometric properties of the embeddings to the combinatorial structure of the associated graphs.

Representations and Eigenvalues

 The connection between the Fourier transform on finite groups and eigenvalues of Cayley graphs leads to many natural questions, both in graph theory and representation theory. I am particularly interested in questions relating to the convergence of the spectral measures for sequences of these graphs. One way to formalize this quesion is to ask for a combinatorial characterization of increasing graph sequences that satisy for all \(\varepsilon > 0\) there exists a finite set \(\Lambda \subseteq \mathbb{R}\) and an integer \(N\) such that for all \(n > N\) we have \(\dfrac{|\{ \lambda \in \operatorname{spec}(G_n): \lambda \notin \Lambda\}|}{|\operatorname{spec}(G_n)|} < \varepsilon \). The future work sections of the papers linked above provide sketches of additional current research questions. Addtionally, various bases for the dihedral groups lead to families of interger spectra graphs, while others define a periodic spectral structure that appears to obey a well-defined limit law. Explaining these phenomena is an ongoing research project, see here and here for related software and figures.

Entropy in Time Series

 Time series data consists of real valued samples, indexed by a time parameter, such as daily stock returns or temperature values. I am particularly interested in probabilistic generative models for this data and their relation to entropy measures. Current work with Kate Moore focuses on clustering and detecting distributional changes using information-theoretic metrics and Markov based null models. These null models represent a given time series by a Markov chain defined on permutations, whose transition probabilities are inferred from the data. This model allows for both theoretical results for known distributions as well as an empirical test for divergence between the steady state of the Markov chain and the observed distribution.

Other Projects:

This section contains links to brief descriptions of various research projects, outside of my main areas, that have also captured my interest. Each .pdf contains some background material, thoughts on possible approaches, and a bibliography.