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### Model: Power Law Relationships

*Strategies and Rubrics for Teaching Chaos and Complex Systems Theories as Elaborating, Self-Organizing, and Fractionating Evolutionary Systems, *

Fichter, Lynn S., Pyle, E.J., and Whitmeyer, S.J., 2010, Journal of Geoscience Education (in press)

**Description:** A power law relationship exists when the events in a complex system plot as a straight line with a negative slope on a log-log graph (see for example,

Barton, 2001, or if that data plots as an asymptote on a linear graph. Power law relationships are ubiquitous in complex systems.

**Presentation:** Most students are probably familiar with log-log graphs, but not by the name “power law relationship.” We do a short introduction to the mathematics of power laws, both on a linear graph and a log-log graph, and then give several examples, including stock market prices, earthquake distributions, flood/storm/hurricane intensities, extinction sizes, and Zipf’s law as it applies to word usage, the size of cities, and music. If we go into greater depth it is to demonstrate the relationship between fractal geometry and power law distributions.

The deeper significance of power laws is that they are generated by—are the result of—an evolutionary process. This is best demonstrated in self-organizing evolutionary systems, such as self-organized criticality or cellular automata, but shows up in all evolutionary systems.

**Anticipated Learning Outcome: **
13. Small—low energy—events are very common, but do very little work. Large—high energy—events are very rare, but do most of the work in a system.

Once we have established this concept we use it in many different situations. In addition to the examples listed above the senior author uses power laws in a stratigraphy course to discuss preservation potential and time representation of events in the stratigraphic record.

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