Directed Graph Symmetrization
From social networks to electrical grids to the human brain, networked systems are extremely prevalent in the world around us. Because of this ubiquity, it is natural to seek a mathematical understanding of networked systems and ask questions such as: Is a city’s power grid robust to faults or failure? How central is a user or set of users in a social network ? Will a sickness spread through a population? One way in which researchers have begun to develop answers to these and other big-picture questions is by using methods from the broad mathematical field of graph theory.
In general, a large number of these graph-theoretic problems are well studied and understood in the context of undirected graphs, but less so in the context of directed graphs. These problems rely on algebraic and spectral graph theory that has been derived for undirected graphs, but cannot be immediately applied to directed graphs due to the loss of symmetry in the Laplacian matrix. This research project focuses on developing a way to symmetrize a directed Laplacian matrix, while preserving a notion of distance between nodes in the graph. The preservation of this graph property makes it possible to apply algebraic and spectral graph tools to the symmetrized graph and meaningfully interpret the results in the context of the original directed graph. Applications include graph clustering & partitioning, vertex sparsification, and signal processing on graphs.
A paper on this work is under review, a draft can be viewed on arXiv.
Humanitarian Operations Research
Mathematical optimization has a large and relatively untapped potential to help develop better responses to humanitarian problems. For example, the distribution of food after a natural disaster or the optimal routing for emergency response vehicles. These optimization problems typically have a different objective function from those commonly found in operations research and are further distinguished by important ethical considerations.
I am currently advising Master's students and leading two projects in this field. The first project uses data from homelessness shelters in Western New York and investigates the optimal placement of homeless clients in shelters such that overflow & time waiting for a shelter bed are minimized. The second project is on techniques for determining matches & cycles in a kidney exchange network, and a study on the benefits of establishing a kidney exchange network in Germany
Networked Multi-Agent Systems
The field of networked multi-agent systems represents a fusion of topics from dynamics and control theory as well as algebraic and spectral graph theory. The solutions to many problems rely on an understanding of the dynamics of a network of agents (or nodes) as well as the role of the underlying network graph. My research in this field aims to characterize how a node, or set of nodes, influences a network with stochastic dynamics as a function of the network topology. More specifically, in this context influencing a network is studied as A. Robustly distributing information across a noisy network and B. Optimizing metrics for network controllability.