Thanks for all the interesting comments, everyone. Lee's comments mirror what someone else told me, re: the random walk. They also said that populations of agents/particles - such as in Brownian motion - tend to exhibit clusters in 2D, but not in 3D. (I would assume - but haven't explored - that particles in 3D could still cluster under the influence of attractors. Thoughts?)
Steve pointed out a couple of things that I'll have to keep in mind for future models - the local minima in 2D vs. 3D, and free edges in a >2D network structure - but I don't think apply to the current models I'm exploring. But I'm definitely going to read up on the equivalence of CA models. Jochen raised the important point that most social science models tend to have an abstract relationship between nodes. I agree that the dimensionality probably isn't meaningful here ... with abstract concepts for the edges, I reckon each additional node would probably add another dimension to the model (if one were concerned with mapping the model to Euclidean space.) And since I'm not using a clustering algorithm in >10D space, I won't worry about Robert's point (but it's a very cool concept. Again, I'll keep that tucked away for future reference.) Let's look back at Jochen's comment, in regards to spatial patterns in 2D vs. 3D. (Can't seem to access Lee's link at the moment, so I'll stay away from the math for now.) I'm thinking about a pred-prey model (in 2D) similar to Ken Hawick's found here <http://www.massey.ac.nz/~kahawick/cstn/015/cstn-015.pdf>, such that the prey reproduce, forming a blob or circle. Then the predators come in, and the circle degenerates into a crescent shape. (We replicated these type of clusters and their movements in our model, although with somewhat different agents and rules.) Since the spatial relationships matter, I assumed that a 3D model of the same agents would tend to form spheres rather than circles; and when the predators come in, they dig out a bowl shape rather than a crescent shape ... although for the same general reasons. But I'm just guessing here. That covers the spatial effects. In terms of populations, I assumed that either the pred or the prey would be more efficient and therefore have different numbers, but that these would scale up or down, and thus show the same general dynamics (a la the Lotka-Volterra equations). If I had to guess, I would say the prey population would tend to be larger (relative to the pred population) than in the 2D model. But I wouldn't really count on that until I ran the simulation. Does that sound about right? Has anyone played around with other spatial effects, moving from 2D to 3D? Lee - on a side note, how familiar are you with the Lotka-Volterra dynamics? Particularly with more than two populations (i.e., 3 or more strictly defined tropic levels). We've found some interesting results, but I don't yet know if they are interesting to just me, or would be to anyone. -Ted On Mon, Mar 1, 2010 at 5:49 PM, Robert Holmes <[email protected]>wrote: > Once you get past about 10 dimensions you hit problems of "distance > concentration" (as it's known in the machine learning community). Basically, > all distances between pairs of points for D>10 are pretty much the same. > That impacts any distance-based clustering or visualization techniques that > you are trying to use. > > -- R > > > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org >
============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
