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

ST120 Introduction to Probability:

• 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:

• MA4E7 Population Dynamics: Ecology and Epidemiology
• MA4M1 Epidemiology by Example
• MA4M4 Topics in Complexity Science

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.

Aims:

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

Content:

Weekly Overview

Introduction:

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