# MA4J5 Structures of Complex Systems

Lecturer: Markus Kirkilionis

Term(s): Term 1

Status for Mathematics students: List C

Commitment: 30 lectures

Assessment: 3 hour examination (80%), project (20%)

Prerequisites: There are no formal pre-requisites, but the following background will be assumed:
Familiarity with basic programming and programming languages, e.g. MA117 Programming for Scientists; Knowledge of basic stochastic processes, e.g. ST202 Stochastic Processes; Some basic statistics and differential equations e.g. ST111/112 Probability A and B, MA131 Differential Equations.

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.
• Discussion and outlook.

Aims:
1. To introduce mathematical structures and methods used to describe, investigate and understand complex systems.
2. To give the main examples of complex systems encountered in the real world.
3. To characterize complex systems as many component interacting systems able to adapt, and possibly able to evolve.
4. 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.