Tutorial 2

In this tutorial, we will learn to plot figures.

Markov-Cayley tree-shifts

Please go to https://s92077.github.io/blog/julia-learning/ to download the code \"Markov-Cayley tree-shifts\" to proceed with this tutorial. The code basically computes the Entropy of Markov tree-shifts over Markov-Cayley trees ^[1], and

Problem

What can we say about the convergence of topological entropy using the above method?

Spread model simulation

Please go to https://s92077.github.io/blog/julia-learning/ to download the code \"spread model simulation\" and model files \"model 1-3\" to proceed with this tutorial. The code basically simulate the multi-type Galton-Watson processes, in the almost sure sense, involves finite patterns of reproduction.

Model files *.json

In the model files, we need to specify the following fields:

• label: the set of types of individual

• initial: the number of ancestors of the population

• transition: reproduction patterns

It is not hard to see the syntax of the json files from the given examples.

The model struct

The an object of the model struct stores parsed and preprocessed information from the *.json files. To learn more about the struct in Julia, check out the documentation.

simulate, run_single_trial, produce_children

• To run the code, we use simulate to simulate the growth of a population by specifying gen_num, the number of generations, and trial_num, the number of trials.

• In simulate, the function run_single_trial is called for trial_num times and return the empirical averages of the number of indiviuals as well as the theoretical growth on average. The produce_children determines

• In run_single_trial, iteratively and randomly produce childrens up to generation gen_num, and for each generation, produce_children is called to determine how parents reproduce its children.

plot_simulation_proportion

The function plot_simulation_proportion plots the output of the function of simulate. To learn more about the plot function

The function uses the Plots.jl package to create the plots, which provides a unified collection of APIs of several different backends. To learn more about it, one can check out the official tutorial. To create a good-looking plot, one might also look into the attributes of the plots.

Parallelism

The code use the packages Base.Threads and LoopVectorization to speed up the computation. One can look into the documentation to learn more about parallel computing.

Problem

What if one wants to simulate a multi-type Galton-Watson process with infinitely many patterns of reproduction?

Final notes

[1]	One can find more details about Markov-Cayley trees and its entropy in https://arxiv.org/abs/2110.08960.