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.