Lecture 12

October 11, 2019

For today

• 1. Read Chapter 8 and do the reading quiz

  2. Turn in a draft project report

Today

• 1. Chapter 8

  2. Notebook 7

For next time:

• 1. Turn in Notebook 7

  2. Read another team's draft and write a review

  3. Prepare for a quiz on Chapters 7 and 8

Self-organized criticality

SOC is an answer to the following mystery: Why do so many natural systems exhibit characteristics of criticality?

1) Heavy-tailed distributions,

2) Fractal geometry

3) Pink noise

These features appear naturally in a system at a critical point like a phase change.

But most critical points are unstable.

So we should not expect to see them (frequently) in natural systems that are not under some kind of feedback control.

The sandpile model provides an example of a system where the critical state is an attractor.

Quoth Wikipedia:

"In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system."

In steady-state, the system exhibits:

1) Heavy tailed distributions (Exercise 8.2).

2) Fractal geometry (maybe; see Exercise 8.3)

3) Pink noise

What are the essential model features that yield SOC?

1) Slowly driven

2) non-equilibrium systems

3) with extended degrees of freedom

4) and a high level of nonlinearity.

Sandpile model

Initial conditions

1) As in Bak Tang Wiesenfeld (BTW), initially all cells are piled high. The system runs until no more avalanches.

2) Single source: one cell piled very high, run until no more avalanches

Pink noise

Crash course in DSP.

Read Section 8.6: what is a power spectrum? How it is computed?

The power spectrum of a noise signal is generally noisy. To make a claim about the long-run average power at each frequency, you can take a long signal

1) Break it into segments

2) Compute the power spectrum of each segment

3) Take the average of the power spectrums.

That's pretty much what Welch's algorithm is.

SOC, holism, and causation

1) Do Exercise 8.6 at your table, then let's discuss

In The Fractal Geometry of Nature, Benoit Mandelbrot proposes what he calls a “heretical” explanation for the prevalence of heavy-tailed distributions in natural systems. It may not be, as Bak suggests, that many systems can generate this behavior in isolation. Instead, there may be only a few, but interactions between systems might cause the behavior to propagate.

To support this argument, Mandelbrot points out:

• • The distribution of observed data is often “the joint effect of a fixed underlying true distribution and a highly variable filter”.

• Heavy-tailed distributions are robust to filtering; that is, “a wide variety of filters leave their asymptotic behavior unchanged”.

What do you think of this argument? Would you characterize it as reductionist or holist?

2) Do Exercise 8.7 at your table, then let's discuss

Read about the “Great Man” theory of history at http://thinkcomplex.com/great. What implication does self-organized criticality have for this theory?

Project 1 Draft Review

Your draft review should include the following elements:

Question: What is your understanding of the experiment the team is replicating? What question does it answer? How clear is the team's explanation?

Methodology: Do you understand the methodology? Does it make sense for the question? Are there limitations you see that the team did not address?

Results: Do you understand what the results are (not yet considering their interpretation)? If they are presented graphically, are the visualizations effective? Do all figures have labels on the axes and captions?

Interpretation: Does the draft report interpret the results as an answer to the motivating question? Does the argument hold water?

Replication: Are the results in the report consistent with the results from the original paper? If so, how did the authors demonstrate that consistency. Is it quantitative or qualitative?

Extension: Does the report explain an extension to the original experiment clearly? Is it a sensible extension in the sense that it has the potential to answer an interesting question that the original experiment did not answer?

Progress: Is the team roughly where they should be at this point, with a replication that is substantially complete and an extension that is clearly defined and either complete or nearly so?

Presentation: Is the report written in clear, concise, correct language? Is it consistent with the audience and goals of the report? Does it violate any of the recommendations in my style guide?

Mechanics: Is the report in the right directory with the right file name? Is it formatted professionally in Markdown? Does it include a meaningful title and the full names of the authors? Is the bibliography in an acceptable style?

Whose report should you read?

Elias <-> Danny

Shirin <-> Nathan

Erika <-> Henry

Jeremy <-> Adam

Nick <-> Jon

Please arrange to get access to each other's draft.