Recipe for the week of December 19 - 25
Soap Bubble clustering of a Plurality Vote Rule
Some of the simplest self-organizing cellular automata can be conceptualized as voter models. For instance, in the two-color CA known as majority vote, with each synchronous update the opinion at x is replaced by the majority opinion within a neighborhood of x. The special case p(1) = p(2) = 0 of last week's non-linear voter model is a nearest-neighbor majority vote CA. As the size of the neighbor set increases, such systems approximate a nonlinear partial differential equation called motion by mean curvature (mmc) that exhibits surface-tension clustering over time.
This week's soup illustrates an intriguing multi-color variant of the same idea that we call plurality vote. The process started from a uniform random configuration of 16 pastels over the array of about 3/4 of a million sites. Think of an election, Italian say, with 16 political parties instead of 2. Each individual changes allegiance in an attempt to conform to the most popular local faction, adopting the plurality opinion if there is a clear favorite. (In case of a tie, no change occurs). In the present example the neighbor set for site x is a 9 by 9 box centered at x.
As time goes on, consensus forms on many droplets of increasing size, while other small cohorts are swallowed by larger clusters that surround them. The emergent dynamics are again driven by a digital variant of motion by mean curvature, with surface tension effects reminiscent of soap bubble interactions. Note in particular the charactristic, nearly stable vertex angles of approximately 120 degrees. One hopes that CA experiments such as this will shed light on the nonlinear behavior of multitype mmc systems, which are poorly understood at present.