Numerical Linear Algebraic Analyses of Opinion Dynamics on Networks

Online social media networks have become extremely influential sources of news and information. Given the large audience and the ease of sharing content online, the content that spreads on online social networks can have important consequences on public opinion, policy, and voting. To better understand the online content spread, mathematical modeling of opinion dynamics is becoming an increasingly popular field of study. 

This project focuses on a special class of mathematical models of opinion dynamics on networks called bounded-confidence models.  One fundamental question for these models is whether the collection of opinions on the network will converge to a consensus (or other stationary profile) and how quickly that convergence will occur. Many of the mathematical analyses of the dynamics of these models utilize well-known and classical tools from linear algebra and algebraic graph theory.  Recently, however, a framework for analysis has been developed that relates the dynamics of these models to iterative methods in numerical linear algebra (Kaczmarz method, Jacobi method, Gauss-Seidel method) and allows for finer-grained analyses of the dynamics via modern numerical algebraic techniques. 

In this project, students will relate bounded-confidence models (and variants) to the classical Jacobi and Gauss-Seidel iterative methods (and variants), and will identify directions for novel theoretical convergence analysis of the dynamics of these models.  Our group will explore these models both empirically and theoretically, utilizing numerical simulations to guide, develop, and investigate hypotheses for later formal mathematical investigation.

Student researchers in this project will develop mathematical skills in areas like numerical linear algebra, mathematical modeling, and optimization.  They will build and strengthen skills in literature review, scientific reading, technical writing and presentation, and programming and package development.

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Interested applicants should submit a CV, the name of a reference, and answers to the following prompts:

  • Why are you interested in this project?  What makes you a good fit?
  • What skills do you bring to this project?  What skills do you hope to develop?
  • What are your goal outcomes for your summer research project?
Name of research group, project, or lab
matH of Algorithms, Data & Decisions (HADD) and Nonlinear & Complex Systems Research Groups
Why join this research group or lab?

This project is a collaboration between the matH of Algorithms, Data & Decisions (HADD) Research Group (https://jamiehaddock.com/people/) and the Nonlinear & Complex Systems Research Group (https://math.hmc.edu/hzinnbrooks/research-group/).

Nonlinear & Complex Systems Research Group: 

You'll be working alongside a fun and supportive group of mathematicians who study a wide variety of social and biological applications. Projects in my group are often driven by student interest, so you'll have autonomy in your research experience.

You will have the opportunity to meet our collaborators to learn how your research fits into the larger program. Students may continue their work in a senior thesis, present their findings at conferences, and/or coauthor resulting publications.

matH of Algorithms, Data & Decisions (HADD) Research Group: 

In the matH of Algorithms, Data & Decisions (HADD) research group, we consider problems motivated by the study of real-world data. We consider the mathematics of data, models for making decisions with data, and methods for training such models.  We consist of fun and passionate people who encourage one another and help each other develop as mathematicians, data scientists, and researchers.  Our group has students working on various projects, and often interacts with collaborators from other institutions (graduate students, postdoctoral researchers, and faculty).  Students may continue their work in a senior thesis, present their findings at conferences, and/or coauthor resulting publications.

We work in areas like mathematical data science, optimization, and applied convex geometry, leveraging mathematical tools like probability, combinatorics, and convex geometry, on problems in data science and optimization. Our group has been active recently in randomized numerical linear algebra, combinatorial methods for convex optimization, tensor decomposition for topic modeling, network consensus and ranking problems, and community detection on graphs and hypergraphs.

Representative publication
Logistics Information:
Project categories
Mathematics
Algorithms
Numerical Modeling
Optimization
Student ranks applicable
First-year
Sophomore
Junior
Senior
Student qualifications

Required: completion of core courses in calculus and linear algebra, programming experience (Python and/or Matlab preferred)

Highly desireable (but not required): numerical analysis/scientific computing, differential equations

Helpful (but not required): dynamical systems, graph theory, operations research, probability, analysis 

Time commitment
Summer - Full Time
Compensation
Paid Research
Number of openings
2
Techniques learned

Students will build and strengthen skills in:

  • numerical linear algebra
  • mathematical modeling
  • dynamical systems
  • optimization
  • literature review
  • scientific reading
  • technical writing and presentation
  • programming and package development
Contact Information:
Mentor name
Jamie Haddock
Mentor email
jhaddock@hmc.edu
Mentor position
Principal Investigator
Name of project director or principal investigator
Jamie Haddock and Heather Zinn-Brooks
Email address of project director or principal investigator
jhaddock@g.hmc.edu
2 sp. | 17 appl.
Hours per week
Summer - Full Time
Project categories
Optimization (+3)
MathematicsAlgorithmsNumerical ModelingOptimization