Research

Current Projects

Coevolving Voter Models
Voter Models were originally conceived as spatial conflict models by Clifford and Sudburry in 1973. In a static voter model, one has a graph of voters and relationships, a set of opinions such that each voter is assigned an opinion, and an update rule to re-assign opinions. These models have been studied extensively and many of their features are well understood. For a coevolving voter model, in addition to having a dynamical process on the network, the network itself is dynamic such that the update rule may also create or break relationships.

I study both the transient and terminal-state properties of coevolving voter models. I have compared the properties of different versions of models, governed by various rules that have been proposed across multiple sources in the literature. One version, for example, involves a nonlinear rewiring mechanism, where the probabilities for which action a node will take depend on a local vote of the node’s neighbors. This nonlinear version has a particularly interesting behavior in which a minority opinion can effectively spread to the entire network under certain conditions.

To study coevolving voter models, I use techniques from probability, dynamical systems, and statistical mechanics. Given the large size and complexity of the systems, I have also made significant use of Monte Carlo simulations for gaining insights and evaluating hypotheses. In addition to their intrinsic mathematical value, I think these models can help develop a better understanding of the interplay between social influence and network topology in the context of competing beliefs and decision-making.

Mason Porter and I uploaded our recent paper ``Fitting In and Breaking Up: A Nonlinear Version of Coevolving Voter Models'' to arXiv.



Topological Data Analysis and IC Manufacturability (w)

Problems in simulation, in physical defects, or in electrical failures of the IC devices generally occur at the boundaries of dimensional tolerances, such as the minimum width and space. However, for layout configurations with four or more critical dimensions, simple minimums are insufficient to characterize dimensional coverage. Persistent homology is a multi-resolution analysis technique which robustly summarizes dimensional coverage. We apply this technique to compare dimensional coverage of IC design configurations, on the same layer, on different layers, and on different designs, yielding results both expected and unexpected based on manufacturing process and design rule knowledge.

This work started during my summer internships at Motivo, Inc. We presented some of results at SPIE Advanced Lithography, the world's premier semiconductor lithography conference and exhibition in February 2019. Our paper (with Dr. Vito Dai and Dr. Luigi Capodieci) is available on the SPIE Digital Library.

Past Projects

Research in Industrial Projects for Students

From left: Mohit Agrawal, Leah Rosenbaum, Leah Birch, and Yacoub Kureh In the summer of 2011, I participated in Research in Industrial Projects for Students (RIPS), a summer research program for undergraduate students in mathematics. Students work in teams of four on projects proposed by industry sponsors, such as IBM, the Los Angeles Police Department, and S&P. Each team is advised by an academic mentor and works closely with one or more mentors from the industry sponsor. RIPS is organized by the Institute for Pure and Applied Mathematics at UCLA.

I worked with a team of students on a project for The Aerospace Corporation, a federal R&D center that manages the design, launch, and maintenance of satellites. Below is the abstract of that work, along with our paper, our final presentation and PowerPoint, and a draft of a paper that we submitted to an IEEE conference. Leah Rosenbaum and I participated in the student poster session at the Joint Mathematics Meetings in Boston where we won Outstanding Presentation awards.

Abstract: Many satellite communication (SATCOM) systems currently use a static resource allocation scheme, which is known to provide excellent service to high-priority jobs. However, this static allocation method is inefficient, as it makes valuable network resources inaccessible to low-priority jobs even when resources are available. We developed stochastic models that approximate the SATCOM systems and studied their statistical properties. We began by reviewing the standard M/M/m model from classic queueing theory and used Markov chains to extend the classic models to include multi-class jobs, more suitably approximating SATCOM systems. Then, by way of simulation, we isolated and characterized the system’s key performance trends in terms of its statistical characteristics. Properties under consideration include the probability of call blocking, preemption, satisfaction, and the expected server utilization, all as a function of priority and bandwidth.

Files: Presentation Slides, Final Report, IEEE Paper, and Video of our final presentation (Thanks to Mohit Agrawal for posting).