Mathematical models 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. 

These summer projects focus on a special class of mathematical models of opinion dynamics on networks called bounded-confidence models. An area of interest in my research group is characterizing the dynamics of bounded-confidence models whose outcomes are driven primarily by key agents in the system (which have distinct properties as compared to the remaining agents). This is particularly well-suited to study by mechanistic models, where we can explicitly control inputs, apply techniques from dynamical systems and network theory, and carefully highlight underlying assumptions.

Possible project directions:

(1) Tuning media impact: Via Monte Carlo simulations of an agent-based bounded-confidence network model, we could investigate strategies for depolarization of synthetic networks and quantify conditions under which polarization and depolarization occur in these networks. We can also explore the effects of more realistic media forcing drawn from Twitter data on the activity of media outlets.

(2) Competition, cooperation, and influence maximization of information cascades: Messages in an online social network are not spreading in a vacuum; they are competing and interacting with other active messages on the network. Under what conditions do we expect the messages to “compete”? When will one message be limited by another? When will they spread independently? Modeling the context under which messages are affected by multiple cooperating or competing diffusion processes is an important extension of online content spreading models. Another interesting potential direction is to explore is how the choice of seed node affects the properties of the resulting dissemination tree (e.g., number of messages spread, width, and diameter).

(3) Parameter fitting and sensitivity analysis for information cascades: While members of my group generally study these questions from a theoretical perspective, we are also motivated by modeling the spread of information in real-world networks. In this project, the student would use Twitter retweet data to fit parameters our information cascade models. Given these data, we would extraction information about the resulting dissemination tree and use parameter fitting techniques and perform a sensitivity analysis as a means to validate or invalidate existing models.

(4) Other projects in mathematical modeling of complex systems: I am open to and excited about other projects in math modeling and dynamical systems with biological and social applications. Active research directions include mathematical modeling of gender representation in mathematics, polarization in two-party political systems, pattern formation and agent-based models in biological systems, and more (see website). If you are interested in a project in along one of these lines, feel free to "pitch" your project idea.


Interested candidates should submit a CV, the name of a reference, and a response to the following questions:

  • Which of the projects listed above are you interested in working on and why?
  • What skills and expertise would you bring to this project?
  • What do you hope to get out of the research experience?
Name of research group, project, or lab
Nonlinear & Complex Systems Research Group
Why join this research group or lab?

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.

Representative publication
Logistics Information:
Project categories
Mathematical Biology
Numerical Modeling
Student ranks applicable
Student qualifications
  • Required: completion of core classes in Calculus and Linear Algebra 
  • Highly desirable: Differential Equations, programming experience (particularly MATLAB) 
  • Helpful (but not required): Dynamical Systems, Graph Theory, Operations Research, Probability, Analysis 
Time commitment
Summer - Full Time
Paid Research
Number of openings
Techniques learned

A summer student in my research group will learn:

  • Development and analysis of mathematical models of opinion dynamics on networks.
  • Numerical simulation and Monte Carlo methods for studying dynamical systems on networks.
  • Literature reading, technical writing, and presentation skills.
Contact Information:
Mentor name
Heather Zinn Brooks
Mentor email
Mentor position
Professor of Mathematics
Name of project director or principal investigator
Prof. Heather Zinn Brooks
Email address of project director or principal investigator
4 sp. | 28 appl.
Hours per week
Summer - Full Time
Project categories
Mathematics (+2)
MathematicsMathematical BiologyNumerical Modeling