Skip to main content Skip to navigation

MA4J5 Structures of Complex Systems

Lecturer: Markus Kirkilionis 

Term: Term 1

Status for Mathematics students: List C

Commitment: 30 lectures

Assessment: 80% 3 hour examination 20% Project

Formal registration prerequisites: None

Assumed knowledge:

MA398 Matrix Analysis and Algorithms:

  • Methodological foundations in linear algebra and matrix algorithms as well as hands-on experience in programming

ST112 Probability B:

  • Basic probability theory
  • Random variables

Useful background:

ST202 Stochastic Processes:

  • Markov processes and Markov chains

MA241 Combinatorics:

  • Foundations of graph theory

MA252 Combinatorial Optimisation:

  • Algorithms in graph theory and NP-hard problems

Synergies: The following modules go well together with Structures of Complex Systems:

Leads to: The following modules have this module listed assumed knowledge or useful background:

Part A: Complex Structures

Graphs, the language of relations:

  • Introduction to graph theory
  • Degree distributions, their characteristics, examples from real world complex systems (social science, infrastructure, economy, biology, internet)
  • Introduction to algebraic and computational graph theory

Evolving graph structures:

  • Stochastic processes of changing graph topologies
  • Models and applications in social science, infrastructure, economy and biology
  • Branching structures and evolutionary theory

Graphs with states describing complex systems dynamics:

  • Stochastic processes defined on vertex and edge states
  • Models and applications in social science and game theory, simple opinion dynamics
  • Opinion dynamics continued

Graph applications:

  • Graphs and statistics in social science
  • Graphs describing complex food webs
  • Graphs and traffic theory

Extension of graph structures:

  • The general need to describe more complex structures, examples, introduction to design
  • Hypergraphs and applications
  • Algebraic topology and complex structures

Part B: Complex Dynamics:

Agent-based modelling:

  • Introduction to agent-based modelling
  • Examples from social theory
  • Agent-based modelling in economy

Stochastic processes and agent-based modelling:

  • Markov-chains and the master equation
  • Time-scale separation
  • The continuum limit (and ‘inversely’ references to numerical analysis lectures)

Spatial deterministic models:

  • Reaction-diffusion equations as limit equations of stochastic spatial interaction
  • Basic morphogenesis
  • The growth of cities and landscape patterns

Evolutionary theory I:

  • Models of evolution
  • Examples of complex evolving systems, biology and language
  • Examples of complex evolving systems, game theory

Evolutionary theory II:

  • Basic genetic algorithms
  • Basic adaptive dynamics
  • Discussion and outlook


  • To introduce mathematical structures and methods used to describe, investigate and understand complex systems
  • To give the main examples of complex systems encountered in the real world
  • To characterize complex systems as many component interacting systems able to adapt, and possibly able to evolve
  • To explore and discuss what kind of mathematical techniques should be developed further to understand complex systems better

Objectives: By the end of the module the student should be able to:

  • Know basic examples of and important problems related to complex systems
  • Choose a set of mathematical methods appropriate to tackle and investigate complex systems
  • Develop research interest or practical skills to solve real-world problems related to complex systems
  • Know some ideas how mathematical techniques to investigate complex systems should or could be developed further

Books: There are currently no specialized text books in this area available, but all the standard textbooks related to the prerequisite modules indicated are relevant.

Additional Resources