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CO905 Stochastic models of complex systems

THIS MODULE DOES NOT RUN IN 2013/14!

Term 2, 10 weeks, oral examination at end of term

Module Leader: Stefan Grosskinsky (Mathematics and Complexity)

link to Online Course Materials

Assessment information 2013:

Week

Assessment

Issued

Deadline

how assessed

%credit

2

homework assignments
(calculations)

TBA
TBA

written script

12.5

7 homework assignments
(calculations + simulations)
TBA TBA written script 20
10 homework assignments
(calculations + simulations)
    written script 17.5

10

Oral Examination

TBA

Oral examination

50

 

 

Taken by students from:

Code Degree Title Year of study core or option credits
P-F3P4(5) Complexity Science MSc (+PhD)

1

option

18

P-G1P8(9) Complexity Science MSc (+PhD)

1

option

18

P-G3G1 Maths and Stats MSc (+PhD) 1 option 18
P-F3P6(7) MSc in Complex Systems Science 1 or 2 option 18

 

Context: This is part of of the Complexity DTC taught programme.

 

Module Aims:

This module covers the mathematical description and analysis of complex systems with stochastic time evolution.

 

Syllabus:

  1. Short introduction to basic theory
    Markov processes, graphical construction, semigroups and generators, stationary distributions and reversibility, conservation laws, symmetries, absorbing states
  2. Population models
    branching processes, Moran model, Wright-Fisher diffusion and duality with Kingman’s coalescent, fixation times, diffusion limits
  3. Epidemic models
    Contact process, survival and extinction, mean field rate equations, critical values, general remarks on the DP universality class
  4. Interacting random walks
    exclusion processes, stationary currents and conservation laws, hydrodynamic limits, dynamic phase transition

Theoretical techniques (introduced along in lectures):
scaling limits and Fokker-Planck approximations, mean-field rate equations, generating functions, duality;

Computational techniques (covered in classes):

  • how to simulate discrete and continuous-time models: random sequential update and other update rules, sampling rates and jump chains, construction with Poisson processes and rejection
  • how to measure: stationary averages, ergodic theorem, equilibration times
  • maybe also: classical Monte Carlo with heat-bath and Metropolis algorithm
  • implementation of 2 simulations with measurements and plots (homework, basic codes will be provided in C to be adapted)

 

 

Illustrative Bibliography:

  • Gardiner: Handbook of Stochastic Methods (Springer).
  • Grimmett, Stirzaker: Probability and Random Processes (Oxford).
  • Grimmett: Probability on Graphs (CUP). (available online here)

 

Teaching:

Lectures per week

3 hours

Classwork sessions per week

1 hour

Module duration

10 weeks

Total contact hours

40

Private study and group working

140