# 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

ST112 Probability B:

• Basic probability theory
• Random variables

Useful background:

ST202 Stochastic Processes:

• Markov processes and Markov chains

MA241 Combinatorics:

• 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:

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
• 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.