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)
Assessment information 2013:
Week 
Assessment 
Issued 
Deadline 
how assessed 
%credit 
2 
homework assignments 
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 
PF3P4(5)  Complexity Science MSc (+PhD) 
1 
option 
18 
PG1P8(9)  Complexity Science MSc (+PhD) 
1 
option 
18 
PG3G1  Maths and Stats MSc (+PhD)  1  option  18 
PF3P6(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:
 Short introduction to basic theory
Markov processes, graphical construction, semigroups and generators, stationary distributions and reversibility, conservation laws, symmetries, absorbing states  Population models
branching processes, Moran model, WrightFisher diffusion and duality with Kingman’s coalescent, fixation times, diffusion limits  Epidemic models
Contact process, survival and extinction, mean field rate equations, critical values, general remarks on the DP universality class  Interacting random walks
exclusion processes, stationary currents and conservation laws, hydrodynamic limits, dynamic phase transition
Theoretical techniques (introduced along in lectures):
scaling limits and FokkerPlanck approximations, meanfield rate equations, generating functions, duality;
Computational techniques (covered in classes):
 how to simulate discrete and continuoustime 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 heatbath 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