From the ORC: "Disciplines such as anthropology, economics, sociology, psychology, and linguistics all now make extensive use of mathematical models, using the tools of calculus, probability, game theory, network theory, often mixed with a healthy dose of computing. This course introduces students to a range of techniques using current and relevant examples."

Prerequisites:  MATH 13 and MATH 20.

Lecture Information:  9L (MWF 8:50–9:55), Haldeman 028.

Please hold the 9LX hour (Thursdays 9:05–9:55) open for possible use including one-on-one and small group meetings. Any full class meetings at the 9LX time will be indicated on the Calendar.

Office Hours:  TBD in Kemeny 240, or by appointment.

Course Objectives: 

The course aims to consider a variety of mathematical modeling types frequently used in the social sciences, including cellular automata, dynamics on and of networks, and agent based models. We will use simple computational simulations as well as analytical techniques in developing a better understanding of these complex systems. Topics of particularly timely interest that may be considered in varying depth by different students include models for the spread of disease and/or information, and models of opinion dynamics and voting. The goal is for you to develop a broad overview understanding of a variety of these models and methods and, through the course project, a detailed expertise on a selected topic.


We will organize our explorations of different topics and methods using parts of different textbooks that are available to you online, either from the author or through the Dartmouth library, including

We will also consider some articles that will be listed in References (which you can also find under Pages in the left sidebar). Note that you do not need to purchase anything.

Assignments, Projects, and Grading: 

Homework assignments will include short "daily assignments" and longer "weekly assignments". The "daily assignments" are due by the next class meeting (unless indicated within 2 class meetings) and are worth 1–2 points each. The "weekly assignments" will have longer times to their due dates, and will be worth more. Unless otherwise specified, all daily assignments are due by the start of the corresponding class period (8:50am) and all weekly assignments are due by the end of the day (11:59pm).

Each student will engage in a course project on a topic selected by the student in consultation with the instructor. Possible topics include deep dives into another chapter in one of the textbooks or an appropriate journal article. The project will include multiple milestones assigned as weekly assignments, as well as a poster (15 points) and final report (25 points, in lieu of an in-person final exam).

The instructor reserves the right to modestly re-weight the separate components of the course (daily problems, weekly problem sets, project milestones, poster, final report) to ensure each contributes fairly to the overall course grade. The course grade will then be determined by a traditional 90/80/etc. scale from the final total.

Attendance Policy: 

Class participation is essential to our exploration of the course material. Please reach out to me in a timely, responsible way to pre-approve any excused absences as needed. After 3 unexcused absences, each additional unexcused absence will be penalized by a 1 point deduction in determining the course grade.

You are expected to attend class in person unless you have made alternative arrangements due to, e.g., illness, medical reasons, or the need to isolate due to COVID-19. For the health and safety of our class community please do not attend class when you are sick, nor when you have been instructed by Student Health Services to stay home. If possible, please let me know in advance in a timely way if you are sick and believe you might miss an upcoming class.

There are many potential reasons to request an excused absence. Please just reach out to me ahead of time and follow up after to discuss how to best catch up on any course material that you miss. In short, be responsible.

Religious Observances: 

Dartmouth has a deep commitment to support students’ religious observances and diverse faith practices. Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me as soon as possible — before the end of the second week of the term at the latest — to discuss appropriate course adjustments.

Academic Honor Principle: 

The Academic Honor Principle is an essential tenet of the Dartmouth community. Collaboration is strongly encouraged in this course. At the same time, ultimately, all assignments submitted must represent your own understanding of the material. Be generous and honest in your citing assistance and input from your fellow students. You should also be sure to always fairly and completely cite your sources.

Use of Generative Artificial Intelligence: 

As machine learning continues to advance, large language models (LLMs), such as ChatGPT, and other Generative AI (GAI) technologies are becoming more widespread. These models can at times be useful tools to accelerate productivity and understanding. The use of such technologies is permitted for the assignments in our course, so long as the following guidelines are adhered to:

  • When using an LLM or other GAI to aid in completion of an assignment, all prompts and output should be saved and submitted as part of the assignment. This may be in the form of a screenshot, copy and paste, PDF, etc.
  • The work that you submit should reflect your own understanding of the assignment.
  • Copying the output from an LLM or other GAI and handing it in as your own work is not permitted, similarly to how copying a peer's work and submitting it as your own is not allowed.

Examples of situations where you might find it useful to use GAI in your work include when you know what kind of calculation you want to do but you don't know all of the details or syntax for how to code it, or you forgot an idea or concept from a previous class that is needed for the current item you are working on. Many other reasonable examples are surely possible; you are strongly encouraged to share your experiences using such tools. Please be aware that in many cases these technologies can give answers to a prompt that are completely incorrect (and sometimes wildly so).  As such, you should always be skeptical of any GAI output you see and verify the veracity of the information contained within. If you have any questions about the use of GAI in the class, please reach out to the instructor.

Student Accessibility and Accommodations: 

Students requesting disability-related accommodations and services for this course are required to register with Student Accessibility Services (SAS; Apply for Services webpage;; 1-603-646-9900) and to request that an accommodation email be sent to me in advance of the need for an accommodation. Then, students should schedule a follow-up meeting with me to determine relevant details such as what role SAS or its Testing Center may play in accommodation implementation. This process works best for everyone when completed as early in the quarter as possible. If students have questions about whether they are eligible for accommodations or have concerns about the implementation of their accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential. 

Mental Health: 

The academic environment at Dartmouth is challenging, our terms are intensive, and classes are not the only demanding part of your life. There are a number of resources available to you on campus to support your wellness, including the undergraduate deans, Counseling Center, and Student Wellness Center. I encourage you to use these resources to take care of yourself throughout the term, and to speak to me if you experience any difficulties. 

Diversity and Inclusion: 

Dartmouth is committed to maintaining a diverse and inclusive workplace and welcomes all members of Dartmouth's scholar-educator community to join in cultivating a culture that values and rewards teaching and welcomes diversity in its many aspects. I acknowledge that the distribution of authorship of the books, articles, and original materials referenced therein do not reflect that desired diversity, especially insofar as we consider historical references, but not only as such. I encourage you to talk to me if anything in class or out of class makes you uncomfortable or if you have any suggestions to improve our environment or the quality of the course materials.

Title IX: 

At Dartmouth, we value integrity, responsibility, and respect for the rights and interests of others, all central to our Principles of Community. We are dedicated to establishing and maintaining a safe and inclusive campus where all have equal access to the educational and employment opportunities Dartmouth offers. We strive to promote an environment of sexual respect, safety, and well-being. In its policies and standards, Dartmouth demonstrates unequivocally that sexual assault, gender-based harassment, domestic violence, dating violence, and stalking are not tolerated in our community.

The Sexual Respect Website at Dartmouth provides a wealth of information on your rights with regard to sexual respect and resources that are available to all in our community.

Please note that, as a faculty member, I am a mandatory reporter obligated to share disclosures regarding conduct under Title IX with Dartmouth's Title IX Coordinator. Confidential resources are also available, and include licensed medical or counseling professionals (e.g., a licensed psychologist), staff members of organizations recognized as rape crisis centers under state law (such as WISE), and ordained clergy (see Links to an external site.).

Should you have any questions, please feel free to contact Dartmouth's Title IX Coordinator. Their contact information can be found on the sexual respect website at:



Course Schedule: Note future topics listed are tentative and will move around as time permits
Week Topics

A Very Brief History of Population Dynamics
See Bacaër chapters 1–6 and 21.

  • Introductions, syllabus
  • Population dynamics, life tables, annuities, Leslie matrix
  • Obstacles to geometric growth, logistic equation
  • Compartmental model for smallpox inoculation

We will cover later: Bacaër chapters 16, 22; Kermack & McKendrick (1927).


Cellular Automata
See Sayama chapter 11. If you are using PyCX, see also chapter 10.

  • How to read a scientific paper (see Pain, 2016)
  • Cellular automata simulations in PyCX, NetLogo, NetLogo Web
  • The "majority rule" CA (a.k.a. "Voting"), with different variations on the state-transition function
  • The "Fire" model for epidemic spread and "Percolation"

See Voting random.nlogo.


Branching Processes, Renormalization Group, Finite-Size Effects
See Sayama chapter 12 (but not 12.3 yet) and Bacaër chapter 22.

  • Estimate critical points of phase transitions in Fire and Percolation CA: branching process arguments and renormalization group calculations
  • Finite-size effects in percolation
  • Probabilistic extension of Fire model as an example of a stochastic state transitions: simulation and renormalization group

See FireFiniteSize notebook and data in Files.


Agent Based Models
See Sayama chapter 19.

  • Examples drawn from NetLogo Models Library:
    Segregation, Ising, Flocking, epiDEM Basic, Virus, Virus on a Network

See Sayama sections 15.1, 15.2, 16.1, 16.3, 17.1 and 17.4.
If you are generally interested in learning more about networks, the quickest introduction I recommend is the book by Kolaczyk & Csárdi.

  • Examples drawn from NetLogo Models Library:
    Giant Component, Preferential Attachment, Small Worlds

Re-Introduce Mean Field Approximation
See Bacaër chapter 16.


Mean Field Approximation
See Sayama section 12.3 and Kermack & McKendrick (1927).

  • MF for Majority ("Voting") model and for stochastic version
  • SIR model: phase transition and numerical solution (see Colab notebook Integrate SIR.ipynb)
  • SI and SIS
  • What do we expect to happen on a network?

No class on Monday 10/9.
Optional class meeting in Thursday 10/12 X-hour for group discussion about projects and computations.


Mean Field Approximations for Dynamics on Networks
See Sayama sections 18.4–18.6.

  • MF discrete-time SIS
  • Continuous-time SI and SIS
  • Regular graphs
  • Friendship paradox
  • Heterogeneous MF

Additional information about other continuous-state dynamics is available in Sayama 18.1–18.3. If you are interested, I also recommend the Porter & Gleeson reference for a review different methods of studying binary-state dynamics on networks.


Heterogeneous Mean Field for SIS (continued from last week)

Pair Approximation
See Supporting Information of Durrett et al. (2012) for pair approximation for voter models. See the Colab notebook SIR_Pair_Approximation.ipynb to experiment with PA for the SIR model.

  • Pair approximation and moment closures
  • Recall limited information from MF for voter model
  • Voter models with rewiring ("co-evolving"/"adaptive")

Approximate Master Equations
See Gleeson (2013).

Cascades and Complex Contagions
See Easley & Kleinberg chapters 16 and 19.

Continuous-State Dynamics
See Sayama 16.2, 18.2 and 18.3.


Voting and Ranking
See Easley & Kleinberg Chapter 23 on Voting, Sections 14.3 & 14.6 on PageRank, and Section 13.4 on The Bow-Tie Structure of the Web.

See Holme, Porter & Sayama (2019) for more about PageRank.
See De Bacco, Larremore & Moore (2018) about SpringRank

Friday:  Course project presentations

10 Monday (LDOC):  Continue course project presentations