# 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

- 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

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

**Content:
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

**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

**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.