21.170 • Mutating Exceptional Structures

foundational exercise for Wildberger dynamics on graphs course

June 19, 2021

review of concept for exceptional structures and the mutation game

Mutation Explorer for Simple Graphs

Mutation Explorer is a browser-based application that allows one to run the mutation game on predefined simple graph structures for a specified number of cycles or turns.

Simple Graphs are defined as a set of vertices V and edges E which in this case is a bidirectional relationship between two vertices.

The mutation game starts by setting an initial value of 0 to each of the vertices except for the first vertex, which receives a value of 1.

To mutate a vertex, we negate its value, and add the sum of its neighbors to it.

The set of values at a point in time is called a population
Our starting population is called a singleton population.
Each unique population derived from successive mutations of the graph starting with the singleton population is called a root population.

On each turn, Mutation Explorer

  • selects a vertex,
  • mutates it,
  • records the new population to a history table,
  • and then plots the total of each population.

Both the table and plot are arranged so the most recent data is at the top.

Mutation Explorer currently has two modes for deciding which vertex to mutate on each cycle:

  • Random selects a random vertex,
  • Round selects each vertex round-robin in the order they were defined.

You might ask why someone would endeavor to create such a contraption.

Exceptional Structures from Dynamics on Graphs

Well, it's all because Dr. Norman J. Wildberger.

I have been following Dr. Wildberger on YouTube for years, and recently joined his Wild Egg Maths channel, where he is starting new curriculum on Exploring Research Level Math.

About a month ago, he launched a new course called Exceptional Structures from Dynamics on Graphs.

In the first lecture, Dr. Wildberger discusses the mutation game on simple graphs and presents the following exercise / challenge:

Which simple graphs X have a finite number of root populations?

Tells us this is a computational problem, so we must roll up our sleeves and do the work.

I've have watched countless hours of his lectures, but have never done the work.

Exploring Graphs for Patterns

Building a Framework for Exploring Exceptional Structures

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