Theory of Complex Systems

Course manual 2024/2025

Course content

The course introduces students to the theory of complex systems with a main focus on critical phenomena. We will discuss the following topics: from microscopic to macroscopic description of complex systems, emergence, phase transitions, critical phenomena, universality, scale invariance, mean-field theory, renormalisation, and self-organized criticality. The course will focus mainly on equilibrium phase transitions, but will also touch upon phase transitions in dynamical systems.

We will work with simple models of complex systems, such as percolation and epidemic models, the Ising model, the linear voter model, Metropolis and Glauber dynamics, the TASEP model (a simple traffic model), the sandpile model, simple models of collective behavior, etc. Numerical simulations of the studied models will be briefly discussed in class, but not implemented. The course will be illustrated with examples of complex systems taken from various fields of science (e.g., physics, neuroscience, social sciences, animal collective behavior, earth science). Contemporary research directions will be explored through the review of recent papers, as well as with a few invited guest lecturers throughout the course.

The course and the assignments will be mainly theoretical (problems with analytical answers, open answers). Assignments can involve implementing simple numerical simulations. For this reason, it is recommended to have basic programming skills prior to taking this class.

Study materials

Literature

• Entropy, Order Parameters, and Complexity, by J. Sethna: PDF of the book available online on the webpage of the author

• Website Complexity Explorables, which as a large collection of interactive simulations of complex systems.

Other

• The course material (slides, tutorials, assignments, and references) will be published every week on Canvas, in the "module" section of the course.

Objectives

• Explain what are complex systems using examples of real complex systems, and be familiar with several simple models of complex systems.
• Recognise if a phase transition is continuous or discontinuous; list and recognize properties of a system at criticality (i.e. of a continuous phase transition)
• Use tools from statistical physics (ensemble averages, partition function, free energy, Shannon entropy) to connect the microscopic to the macroscopic description of a complex system at equilibrium.
• Use their knowledge of Poisson processes (distribution of waiting time, and Poisson distribution) to solve simple problems;
• Write down a master equation to characterize the dynamics of a continuous-time Markovian process, and use the detailed balance approximation;
• Use the mean-field approximation to obtain the equations for average quantities describing the system
• Find the equilibrium solutions of a system and analyse their stability.
• characterize the properties of a system at a critical point (obtain the scaling form and compute critical exponents); Explain the concept of universality.
• Explain the notion of emergence and give examples of emergent phenomena
• Explain the notion of self-organized criticality and give examples
• Students will be able to understand classical results in critical phenomena, and apply key concepts of critical phenomena in complex systems;

Teaching methods

• Lecture
• Computer lab session/practical training
• Self-study
• Seminar

Lectures will introduce you to the general concepts and tools of the course.
During practical sessions, exercises will guide you through practicing the mathematical tools seen in class, and help you get ready for the final written exam.
Several seminars, spread out over the second half of the course and given by guest lecturers, will help you get a more general idea of the field of complex systems and of applications in research of the concepts and tools learned during the course.

Learning activities

Attendance

This programme does not have requirements concerning attendance (Ter part B).

Assessment

Assessment: 1 homework assignment; 3 regular quizzes; 1 final exam.

Quizzes: There will be 3 quizzes spread out over the 7 weeks. Only your two best scores will be taken into account for your final quiz grade. This grade will count for 15% of your final grade.

Homework assignment (and group project): There will be one homework assignment during the first part of the course, and one optional group project during the second part of the course. The result from these two assignments will count for 25% of your final grade.

Final exam: there will be a final written exam, which will count for 60% of your final grade.

Note: The quizzes will regularly assess your general understanding of the concepts seen during the lectures and the tutorials, while the homework and the final exam will assess your understanding of the technical and mathematical tools seen in class. Finally, the group project will allow you to apply these tools and concepts to the understanding of a research paper in complex systems.

Inspection of assessed work

Quizzes: There will be a quiz every two weeks about the content of the course of the two previous weeks (+ two questions taken from the past quizzes). The three quizzes will be taken in class in Canvas. You will have direct access to your grade just after you have taken the quiz. There will be a few minutes in class to reflect back on the quiz just after the quiz has taken place.

Homework assignment: You will be asked to submit your homework assignment on Canvas, and will receive your grade directly on Canvas. After checking your result, please feel free to ask us in class if you have any questions about your grading.

Final Exam: The grade of your final exam will be published in Canvas about one to two week(s) after the exam. There will then be an in-person session organized so that you can consult the copy of your exam and ask your questions (if any) regarding your final grade. Please contact me quickly if the time does not work for you due to overlap with another course.

Assignments

Quizzes: will be taken individually in class directly on canvas. You will have to log in Canvas with your student account. You will need to bring in your own laptop or tablet to take the quiz. Each quiz takes less than 15min, and the result will be directly available afterwards. Quizzes are graded.

Homework assignment: The homework assignment will take place over the first part of the course. The exact deadline of the assignment will be discussed in class (typically around the break of week 5). Your assignment must be submitted on Canvas before the decided deadline. You can work through the solutions of the homework in groups. However, you must hand in your own written solutions to the homework, and we ask you to write these solutions on your own. Writing the answers of the homework on your own will help you prepare for the final exam.  This homework is graded.

Final exam: will take place during week 8 and will be individual. It is a written exam. More details will be provided during the two weeks preceding the exam.

Tutorial exercises: Each module will have its own set of exercises, which will not be graded. Exercises will be discussed in class during tutorials. Students are expected to finish the exercises at home (we advise you to work in groups to help each other). Written corrections for each set of exercises will be provided once all the exercises of the module have been discussed in class. Students are encouraged to check their answers to the exercises, and to discuss them in group or in class if there is anything they haven't understood in the correction. Working through these exercises is a very good preparation for the final exam.

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Contact information

Coordinator

• dr. Clelia de Mulatier

If you have any questions, you can contact me or the TA, preferably through Canvas' messaging system.