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MA4J5 Structures of Complex Systems

Lecturer: Markus Kirkilionis 

Term: Term 1

Status for Mathematics students: List C

Commitment: 30 lectures

Assessment: 80% project and 20% assignments

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:

The module is structured into three parts, structural modelling, dynamic modelling and learning/data analysis. All of these parts have proven to be necessary for any complex systems modelling, sich as models in the Life Sciences, in the Social Sciences, in Economy & Finance or Ecology and Infectious Diseases.

In the lectures we will learn how to start the modelling process by thinking about the model's static structure, which then in a dynamic model gives rise to the choice of variables. Finally, with the dive into mathematical learning theories, the students will understand that a mathematical model is never finished, but needs recursive learning steps to improve its parametrisation and even structure.

A very important aspect of the lecture is the smooth transition from static to dynamic stochastic models with the help of rule-based system descriptions which have evolved from the modelling of chemical reactions.


  • 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


Weekly Overview


Week 1: Mathematical Modelling, Past, Present and Future

  • What is Mathematical Modelling?
  • Why Complex Systems?..
  • Philosophy of Science, Empirical Data and Prediction.
  • About this course.

Part I Structural Modelling

Week 2: Relational Structures

  • Relational family: hypergraphs, simplicial complexes and hierachical hypergraphs.
  • Graph characteristics, examples from real world complex systems (social science, infrastructure, economy, biology, internet).
  • Introduction to algebraic and computational graph theory.

Week 3: Transformations of Relational Models

  • Connections between graphs, hypergraphs, simplicial complexes and hierachical hypergraphs.
  • Applications of hierachical hypergraphs.
  • Stochastic processes of changing relational model topologies.

Part II Dynamic Modelling

Week 4: Stochastic Processes

  • Basic concepts, Poisson Process.
  • Opinion formation: relations and correlations.
  • Master eqation type-rule based stochastic collision processes.

Week 5: Applications of type-rule based stochastic collision processes

  • Chemical reactions and Biochemistry.
  • Covid-19 Epidemiology.
  • Economics and Sociology, Agent-based modelling.

Week 6: Dynamical Systems (single compartment)

  • Basic concepts, examples.
  • Relation between type-rule-based stochastic collision processes in single compartments and ODE
  • Applications, connections between dynamical systems and structural modelling (from Part I), the interaction graph, feedback loops.
  • Time scales: evolutionary outlook.

Week 7: Spatial processes and Partial Differential Equations:

  • Type-rule-based multi-compartment models.
  • Reaction-Diffusion Equations.
  • Applications.

Part III Data Analysis and Machine Learning

Week 8: Statistics and Mathematical Modelling

  • Statistical Models and Data.
  • Classification.
  • Parametrisation.

Week 9: Machine Learning and Mathematical Modelling:

  • Mathematical Learning Theory.
  • Bayesian Networks.
  • Bayesian Model Selection.

Week 10: Neural Networks and Deep Learning:

  • Basic concepts.
  • Neural Networks and Machine Learning.
  • Discussion and outlook.

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