Lecture 11

October 8, 2019

For today

1. Read Chapter 7 and do the reading quiz
2. Turn in a link to your project notebook on Binder

Today

1. Chapter 7
2. Draft report
3. Notebook 7

For next time:

1. Read Chapter 8 and do the reading quiz
2. Turn in a draft project report

Physical Models

CAs as physical models

1) Diffusion-reaction

2) Percolation (and fractals)

3) Forest fire model (Chapter 7 notebook)

4) Self-organized criticality (sand pile models)

Heavy-tailed distributions, fractals, and pink noise!

Dimension

https://en.wikipedia.org/wiki/Dimension

"In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it."

Very often mathematical entities have a natural definition in terms of one type (like int), and we create interesting new math by extending it to another type (like real).

For example, the gamma function is the generalization of factorial to real numbers.

Exponentiation extends repeated multiplication to real exponents.

Complex exponentials are the extension of exponentiation to complex exponents.

So how do we extend the concept of dimension to reals?

Fractal Dimension

https://en.wikipedia.org/wiki/Fractal_dimension

"The concept of a fractal dimension rests in unconventional views of scaling and dimension.[22] As Fig. 4 illustrates, traditional notions of geometry dictate that shapes scale predictably according to intuitive and familiar ideas about the space they are contained within, such that, for instance, measuring a line using first one measuring stick then another 1/3 its size, will give for the second stick a total length 3 times as many sticks long as with the first. This holds in 2 dimensions, as well. If one measures the area of a square then measures again with a box of side length 1/3 the size of the original, one will find 9 times as many squares as with the first measure. Such familiar scaling relationships can be defined mathematically by the general scaling rule in Equation 1, where the variable

stands for the number of sticks,

for the scaling factor, and for the fractal dimension:

The symbol

above denotes proportionality."

For objects that are defined mathematically, we can compute the fractal dimension analytically. For example, the Koch snowflake has dimension

which is 1.26 for the standard version.

Box counting

To look at data and estimate the fractal dimension, we need some kind of empirical metric. There are several, of which the box counting dimension is the most common:

In the context of CAs, it is relatively easy to implement box-counting algorithms, by either:

1) Generating CAs in a range of sizes and counting the number of cells.

2) Generating a large CA and counting the number of cells covered by boxes with increasing sizes.

Then plotting the counts versus the sizes on a log-log scale. The slope is the fractal dimension.

The details are sometimes tricky:

1) In the case of percolation, there is some spatial correlation to deal with.

2) In the case of the single source sand pile, we need a radial version.

Also, interpreting the results can be tricky:

1) The lines are not always straight.

2) The estimated slopes are not very precise.

3) When the estimate is near an integer, it's hard to say for sure whether the data "are fractal".

But that might not be a meaningful claim anyway.

Reading quiz question: If I double the size of a 2D CA and the number of live cells triples, what is its fractal dimension?

Project 1 Draft Report

Due before class on Friday. This deadline is strict, because another group has to read your draft. Basically, turn in what you have.

Audience and goal

The audience for the report is other students in the class and external readers who know a little bit about complexity science. The goal of the report is to present enough background and information about your experiments that the reader can understand what your experiments are intended to do and what we can learn from the results.

In the draft final report, and the final report:

1) The presentation should be organized around the experiments, not a narrative description of what you did.

2) Presentation of each experiment should follow the QMRI structure (see below), implicitly or explicitly.

3) You should write in the active voice and primarily in the present tense. Avoid using the future tense for things coming later in the paper. You can use the past tense for things you did, but prefer the present tense when possible.

4) As much as possible, show your results in a way that allows the reader to reach the conclusion you are trying to reach (as opposed to describing results and asserting conclusions).

The logical flow of the report should generally follow QMRI structure:

Q: motivating question; what is the purpose of the experiment?

M: methodology; how did you implement the experiment (at an appropriate level of detail)?

R: results; what happened when you ran the experiment?

I: interpretation; how do you interpret the result as an answer to the question?

Content

The draft final report should contain the following elements:

1) A meaningful title (not "Draft Final Report") and the full names of the authors.

2) An abstract that identifies the topics you investigate and the tools you use.

3) An annotated bibliography of 1-3 papers that relate to your topic and/or tools. Explain what the papers are about, what experiments they report, and what their primary conclusions are.

4) For each experiment you performed, present your question, methodology, results, and interpretation. Your draft final report should include at least one experiment that is substantially complete. If you have additional experiments in progress, you should draft those sections and include place-keepers for the results and interpretation.