Center for Models of Life
Niels Bohr Institute

Topics in Complex Systems

Fall, 2017



Weeks 36-41, 43-44, 2017
Tuesday : 9:15-10:00
Thursday : 10:15-12:00

Weeks 36-41, 43-44, 2017
Tuesday : 10:30-12:00
Thursday : 13:15-15:00


All lectures and tutorials take place in Aud C, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen.


All lectures and problem sessions will be given in English. The final oral examination can be held in either English or Danish (it is up to the student).

Scope and background

Topics in Complex Systems (7.5 ECTS) is one of the courses offered yearly by the Center for Models of Life of the Niels Bohr Institute for both Master and PhD students. The course focuses on a wide variety of complex phenomena ranging from phase transitions and critical phenomena, to complex networks and econophysics. Common themes will be self-organization (complex patterns and properties arising from simple rules) and universality, i.e. that broad classes of phenomena can be captured by the same simple model. Students may benefit from a background knowledge of dynamical systems, e.g., Dynamical Systems and Chaos, but this is not a necessity. Also students from economics or applied mathematics may find the course useful.

This course is ideal for students who plan to write their thesis in complex systems and bio-complexity.

Exam requirements

The final examination will be based on homework assignments as well as the questions/subjects from the list of 9 subjects below; the questions should be prepared using the indicated materials (notice the updated version of the notes). Students will draw one question and are expected to answer it within 10-15 minutes (by giving a small summary of the specific topic), followed by 15 minutes questions in other subjects (along the lines of the homework problems). The grade is set solely by the final examination, i.e., there are no midterm reports of any kind. Attendance to lectures and problem sessions is beneficial but not mandatory; it therefore does not influence the final grade. The students will be graded in the exam.

List of Course Topics (and exam questions, examin in Aud B or Aud C at NBI)

  1. Partition function and free energy, definition of phase transitions, definition of the Ising Model, mean field solution of the Ising model (Jan Haerter's lecture notes: chapters 2, 3, 4, 6.2)
  2. 1D Ising Model (exact solution using transfer matrices), high and low temperature series expansions (Jan's lecture notes: chapters 5 and 9)
  3. The Monte Carlo method: Ergodicity, Detailed balance and equilibrum distribution, Metropolis algorithm and heat bath, show your own Monte Carlo curves (on paper), auxiliary notes on critical slowing down are not mandatory (Jan's lecture notes, chapter 8)
  4. Networks: Scale-free networks, Analysis of network topologies, Percolation on a network, Disease, information or Diffusion Dynamics on a Network (Sneppen notes, Chapter 4 first part)
  5. Networks: Models for Scale free networks (Sneppen notes, Chapter 4 last part), first return of a random walker (chapter 2, appendix, or for a short derrivation assuming a power law, please see footnote in last subchapter in chapter 6)
  6. Gillespie algorithm, how to implement it in a epidemic model, SIR Epidemic model, Discrete model of excitable media (Sneppen notes, Chapter 5 and a little in chapter 6)
  7. Agent based models: Schelling model, voter model, persistantly competing states, Threshold dynamics and Self organized criticality (avalanches and firt return of random walker) (Sneppen notes, Chapter 6)
  8. Game Theory: Information games (when to bluff) , Unfair wealth distribution from a fair game, Bet-hedging (Sneppen notes, Chapter 7)
  9. Econophysics: Hurst exponent H from a time series, future predictions for H not equal a 1/2, Fear-Factor model, Bet-hedging in random walk markets, Collapse and Exponential Growth (Sneppen notes, Chapter 8)
  10. Topics 1-3 will be covered within the first three weeks of the course (instructor: Jan O. Haerter)

    Topics 4-9 will be covered in weeks 4-8 (instructor: Kim Sneppen)


Kim Sneppen (, course responsible
Jan O. Haerter (
Mogens Hogh Jensen (