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 include (but are not limited to):

(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) Heterogeneity and homophily: When studying bounded-confidence models on networks, it is common to assume that all nodes have the same properties. Perhaps a more realistic social-network setting would be to relax this assumption and study the case where a node’s local network structure, its opinion state, and/or the states of its neighbors affect that node’s receptiveness to new content or ideologies. Furthermore, there is evidence that online social networks exhibit a high degree of homophily: that is, nodes are more likely to be connected to nodes with similar opinion states. By comparing dissemination on networks with high homophily (where agents preferentially attach to other agents whose states are closer in opinion space) to those that do not, we could determine the situations under which homophily facilitates or hinders the spread of content.

(3) Message mutation: Another benefit of studying content spread with bounded-confidence mechanisms is that it allows for a natural generalization to incorporate mutations or shifts in the ideology of a message’s content. For example, we may suppose that if an agent is receptive to a message, it will pass on a version of that message with a modification in ideology. Given an initial opinion distribution and network structure, we could study the parameter regimes under which mutation enhances the spread of a message (if any).

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 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
Numerical Modeling
Student ranks applicable
Student qualifications
  • Required: completion of core classes in Linear Algebra and Differential Equations
  • Highly desirable: programming experience (particularly MATLAB, but Python is also good) 
  • 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
2 sp. | 22 appl.
Hours per week
Summer - Full Time
Project categories
Numerical Modeling (+1)
MathematicsNumerical Modeling