- One of the simplest imaginable probabilistic cellular automata is the
Stepping Stone or Multitype Voter Model. Sites of the lattice
are populated with different colors - representing genetic types or
individuals of differing political persuasions, say. Repeatedly, with
probability p, each cell adopts the state of a random one of its
4 nearest neighbors. Here p is a parameter governing the
synchronicity of the system: for p = 1 all sites update in parallel,
whereas for small p the system is almost asynchronous (and individual
sites change much more slowly).
- Nearly 15 years ago, my first serious computer simulation of an
interacting system implemented the Stepping Stone Model on a Commodore 64.
About 10 years ago I chose the same process, started from a completely
random arrangement of 32K colors, for the first in a series of
Particle Postcards many of which now
adorn the main pages here in the Kitchen. Appropriately, this week's soup comes
from the Shelf.
- So as the PSK approaches its second birthday, it seemed
fitting to launch our forthcoming series of in-house Java CA
simulations with a little applet of the same model. Many thanks to
Bruce Christenson for the most excellent hacking on this project.
The interface below should be self-
explanatory. Load either a random initial state or one of the 6 preset
initial configurations and then hit Start to run the movie or
Step to advance one frame at a time. Save and restore
act like the Memory buttons on a pocket calculator.
- Let us briefly discuss the preset experiments, with a timely electoral spin.
First, Eye Chart loads various enclaves of Republicans (red) in a sea of
Democrats (blue). Run the dynamics to watch an assortment of randomly
evolving lattice animals (polyominoes) until, most likely, the GOP is
completely eliminated. But note how long the reactionary pockets persist.
Next, Rings shows a multicolor initial state, perhaps corresponding to an
Italian election. For Multitype Voter dynamics on our finite torroidal world,
regardless of the starting state, it is easy to see that some single party
eventually takes over completely. Thus, in the Rings experiment, I pose the following PUZZLE:
What is the EXACT chance that yellow wins ?
- If you think you know, send email to the
Chef.
I'll let you know how you did, and award a prize for the correct answer.
- One nice feature of the Stepping Stone rule is that its updates occur
without a priori knowledge of the total number of colors involved. Thus one
can feed any bitmap to the rule. Our last four preset configurations illustrate this
twisted form of image processing. Alas, Bruce! is not The Boss, but
rather our peerless Java programmer. We offer Jack and Jill
for gender parity. Personally, I get a special satisfaction watching Jack
melt.
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