Experiments in Behavioral Network Science: Brief Coloring and Consensus Postmortem (Revised and Updated 4/2/07) Networked Life CSE 112 Spring 2007 Michael Kearns & Stephen Judd

2 Summary of Events Held 18 coloring and 18 consensus experiments Each one had a different network structure (details withheld for now) All 18 coloring experiments globally solved –average duration ~ 35 seconds 17/18 consensus experiments globally solved –average duration ~ 62 seconds So everyone made $70 or $72 –not bad for less than a couple of hours of “work” –no promises for next round… Recall (worst-case) status of problems for centralized computation Seems it is easier to get people to disagree than to agree Let’s look at the Gallery of Consensus Art –[1] [2] [3] [4] [5][1] [2] [3] [4] [5]

3 Network Formation Model (added 4/2/07) Single parameter p (a probability) p=0: a chain of 6 cliques of size 6 each (see figure) p>0: each intra-clique edge is “rewired” with probability p: –first flip p-biased coin to decide whether to rewire –if rewiring, choose one of the endpoints to “keep” the edge –then choose new random destination vertex from entire graph Values of p used: 0, 0.1, 0.2, 0.4, Three trials for each value of p; different random network for each trial

4 S, I attach a revised figure showing how a number of macroscopic graph quantities vary as a function of p. In order to plot them all on a common scale and to compare their rates of change w.r.t. p, for each quantity I have normalized it so that the minimum value is always 0 (i.e. subtracted off the min value) and the max is always 1 (i.e. divided by the max value, after subtracting off the min). So the y-values are always between 0 and 1 and thus you can't infer absolute values, only relative. The quantities are: 1. average-case diameter (decreases with p) 2. clustering coeff (decreases with p) 3. standard deviation of the degree distribution (increases with p) 4. maximum degree (increases with p) 5. minimum degree (decreases with p) 6. ratio of the stationary random-walk probabilities of the most visited and least visited states (by standard theory, this is just the ratio of max to min degrees; increases with p) Regarding 6., note that we can think of the stationary probability of a vertex as a measure of its "centrality", so 6. is a measure of how much spread there is between the most and least central vertices. As observed before, both 1. and 2. fall off pretty rapidly with p, which argues for sampling more finely at small p. On the other hand, 3-6 have a much wider "range of response".

5 Sticking with just the consensus games for now, the attached consplot.jpeg shows a single curve for each consensus game. The curve shows what fraction of players at any given time are playing the color that ended up being the "winning" color. You can see that many of the games have extremely rapid adoption of the winning color, while others involve quite a long time and lots of "wandering".

6 S., I am attaching a revision/revisitation of a plot I sent you in the very first results following the experiments. It shows the average experiment duration (so exactly one of these is at 180 secs, all others completed before) vs. p for both coloring and consensus on the same plots… …Digging a little deeper, I compared the completion times of coloring experiments for p = 0, 0.1, 0.2 to those for p = 0.4, 0.6, 1.0. These two sets of completion times pass a two-sample, unpaired, unequal variance t-test for different means at the P = level of statistical significance, beating the magic standard of 0.05 (i.e. it is a reportable result by scientific conventions). So there are significant structural effects on coloring performance. By the way, note that the "flattening" of both curves beyond p=0.4 nicely justifies our spacing of p-values, which recall we did based on most of the "action" in various structural measures (diameter, clustering coeff, etc.) was at smaller values of p. Nice to see that thesame seems to hold true of the collective performance.