Warwick Lecture
Academic Year 2022/2023

MA4J5 Mathematical Structures
of Complex Systems


Location: Warwick Mathematics Institute
Zeeman Building
Term 1, Academic Year 2022-2023
Wednesdays 10:00-12:00 Room B1.01
Fridays 10:00 - 11:00  Room A1.01

This should be one of the world's most sophisticated courses on mathematical modelling, combined with applications on complex systems. The current crisis, the global pandemic, economic transitions and climate change have brought complex systems theory once more into the spotlight, and their mathematical modelling is key to their understanding and possible management, especially accessing risks for society.


In this course we discuss the foundations of how to set up modern IT-based expert systems for tackling crisis. We discuss how mathematical modelling and data analysis help to support expert panels in solving global crisis.

We keep also this year the partition of the lecture 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 this lecture 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.


The course is managed via the University of Warwick Moodle pages. Students get notifications via Warwick Moodle.

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, current societal challenges. How mathematical modelling can help?

Video link to Lecture 1
Video link to Lecture 2

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.

Video to Lecture 4

Week 3: Relational Modelling: Generalisations and Extensions
• Connections between graphs, hypergraphs, simplicial complexes and hierachical hypergraphs.
• Applications of hierachical hypergraphs.
• Stochastic processes of changing relational model topologies. 

Video to Lecture 6

Part II Dynamic Modelling

Week 4: Stochastic Processes
• Basic concepts, Poisson Process.
• Collision processes, the Gillespie algorithm.
• Master eqation  for reaction systems and general rule-based stochastic collision processes.

Video link to Lecture 8

Week 5: Applications of type-rule based stochastic collision processes
• Chemical reactions and Biochemistry.
• Covid-19 Epidemiology.
• Economics and Sociology, Agent-based modelling.

Video link to Lecture 9
Video link to Lecture 10
Video link to Lecture 16

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.

Video link to Lecture 11

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.

Video Link to Lecture 13

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.

Lecture Partition - Topics

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Part 1
Structural Modelling

In this part we introduce modelling with structural or relational models, such as graphs.
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Part 2
Dynamic Modelling

Most real-world systems have dynamic aspects, i.e. they change their state with time. Dynamic descriptions can be either stochastic or deterministic.
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Part 3
Data Analysis and Machine Learning

In emprical science we need to test our models with data. The data themselves have usually first to be analysed with statistical techniques. A modern technique for data analysis is machine learning (ML)
Learn More