Notebook for Stochastic Modelling and Random Processes module

Setup

• Create a new empty GitHub Repo for this module
• Clone this repo git clone <url to this github repo>
• Rename the origin of this github to upstream git remote rename origin upstream
• Add your new empty GitHub Repo as the origin git remote add origin <url to new github repo>
• Initialise your work with git add ., git commit -m "first commit"
• Push the work to your own repo git push origin master
• Every time this repo is updated, use git pull upstream master (get locally) and then git push origin master (to GitHub repo)

Notebooks will be uploaded for each support class.

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

• Kingsman's Coalescent
• Ornstein-Uhlenbeck Process
• Fractional Brownian Motion
• 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 4.ipynb

• Kingman's Coalescent
• Geometric Random Walks

Support Class 5.ipynb

• Moran model (similar to wright-fisher but continuous time)
• CTMC with waiting times

Support Class 6.ipynb

• 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

• Barabsai-albert model
• empirical tail distribution
• expected degree of nearest neighbour given node has degree k
• contact process
• mean-field contact process

Support Class 9.ipynb

• Directed Networks
• Trophic Analysis

Support Class 10.ipynb

• Community Detection in Networks
• Girvan-Newman Algorithm
• Hierarchical Clustering
• Markov Clustering

Other

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:

• Log Normal Distribution: https://en.wikipedia.org/wiki/Log-normal_distribution
• Log Normal Scipy Documentstion: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lognorm.html
• KDE plot visualisation: https://mathisonian.github.io/kde/
• Fokker-Planck Equation: book Stochastic Processes in physics and chemistry - N.G. Van Kampen
• Ornstein-Uhlenbeck process: https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process#Fokker%E2%80%93Planck_equation_representation
• Colour options in matplotlib https://xkcd.com/color/rgb/