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Networks-and-Random-Processes

Notebooks will be uploaded for each support class.

For Latex, I recommend you to use Overleaf (particularly useful for RSG as can have multiple people working on the same document at the same time - plus you can chat to each other on there): https://www.overleaf.com/

For GitHub help, check out https://www.gitkraken.com/

Steps to download GitKraken:

  • Download the .deb file
  • open terminal and go into your downloads (cd Downloads)
  • sudo dpkg -i name.deb
  • log in with you github

Useful links for the assignments:

Detail of each notebook:

Support Class 1.ipynb

  • basic linear algebra in python
  • simple random walk animation
  • function SRW, a simple random walk with arguements p (probability steping up), tmax (when to terminate the walk) and N (number of replications)
  • empirical distribution calculated at a fixed time (hist plot over possible states at time n)
  • simulation with periodic boundary conditions (modulus L (L =10))
  • simulation with closed boundary conditions (reflects at 10 and 0)

Support Class 2.ipynb

  • Geometric random walk
  • ergodic average
  • empirical tail (1 - CDF)
  • Wright-fisher model ( heatmaps)
  • time to reach steady state
  • Gershgorin disk theorem

Support Class 3.ipynb

Support Class 4.ipynb

  • Kingman's Coalescent
  • CTMC (waiting times, exponentially distributed)

Support Class 5.ipynb

  • Orenstein-Uhlenbeck Process
  • simulated by finite difference approximation (taking the Weiner incremenet by sampling from normal distributioon with zero mean and dt vaiance)
  • simulated using sdeint (python stochastic differential equations, numerical integration)

Support Class 6.ipynb

  • Moran model (similar to wright-fisher but continuous time)
  • CTMC with waiting times
  • introduction to networks and using the networkx package in python
  • degree distribution, clustering, transitivity, distance, largest component

Support Class 7.ipynb

  • Erods-Renyi random graphs
  • compare degree distribution to binomial distribution
  • expected size of largest component for multiple realisations
  • expected local clustering coefficient for multiple realisations
  • Wigner semi-circle law

Support Class 8.ipynb

  • compare degree distribution to poisson distribution
  • Barabsai-albert model
  • empirical tail distribution
  • expected degree of nearest neighbour given node has degree k

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