# 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

- Basic probability theory
- Random variables

**Useful background: **

- Markov processes and Markov chains

- 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:**

- 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

**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.