Introductory description

N/A

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Part A Complex Structures

Week 1: 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.

Week 2: Evolving graph structures:

• Stochastic processes of changing graph topologies.
• Models and applications in social science, infrastructure, economy and biology.
• Branching structures and evolutionary theory.

Week 3: 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.

Week 4: Graph applications:

• Graphs and statistics in social science.
• Graphs describing complex food webs.
• Graphs and traffic theory.

Week 5: 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:

Week 6: Agent-based modelling:

• Introduction to agent-based modelling.
• Examples from social theory.
• Agent-based modelling in economy.
  Module Content and Teaching

Week 7: 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)

Week 8: Spatial deterministic models:

• Reaction-diffusion equations as limit equations of stochastic spatial interaction.
• Basic morphogenesis.
• The growth of cities and landscape patterns.

Week 9: Evolutionary theory I:

• Models of evolution.
• Examples of complex evolving systems, biology and language.
• Examples of complex evolving systems, game theory.

Week 10: Evolutionary theory II:

• Basic genetic algorithms.
• Basic adaptive dynamics.
• Discussion and outlook.

Learning outcomes

By the end of the module, students 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.

Interdisciplinary

This module is highly interdisciplinary, as is the nature of mathematical modelling lectures. Mathematical Modelling is applied to complex systems, which can be in nearly any scientific area, and combinations of scientific areas. Examples are Biology, Chemistry, Biochemistry, Medicine, Social Sciences, Economy and Finance.

International

As we like to move more towards online lectures, we like to offer this module to Warwick teaching partners online.

Subject specific skills

See learning outcomes.

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.