I hate when computers don't work.

NK Networks Lab Abstract
We performed these experiments to help us better understand how we get order for free: “self-organization that arises naturally,” according to Kauffman. In examining NK networks, order for free is how the random initial state creates order by flowing into a state cycle. What is interesting about NK networks is that when K = 2, we get order for free. When K is greater than 2, the network is chaotic, and when K is less than 2, the network is static. So, what’s so special about K = 2 giving us order for free?
We used the computer to simulate NK networks. Using the first program we found the size of the frozen components with N = 16 and K = 1, 2, 3, 4. In the second program, we looked at how the number of basins of attraction, size of the largest basin, length of the state cycles and length of trajectories changed with variations on K when N =9. We kept N constant and varied K because we wanted to compare and contrast what made K = 2 so special, relative to other K values.
The result of the frozen component experiment is that the lower K is, the greater the number of frozen components. So they are inversely proportional. When K = 1, we averaged 15 frozen components. In fact, in 9 of our 10 trials for K = 1, there were 16 frozen components. When K = 2, 3 and 4, the averages were 8.5, 4.5 and 4.1, respectively. It seems that as K approaches 0 from the right, the number of frozen components approaches N, but as K approaches infinity, the number of frozen components for a given N is .

Results for last part: For a given N, as K increases, the number of cycle states 100 arbitrary starting states “drains” into increases, as does cycle length and the trajectory from initial state to cycle state. We have no conclusive evidence for the largest basin: with N=9, the size of the largest basin seems unrelated to K (or maybe a slight negative trend), however, when N=36, there is a distinct negative trend (as K increases, basin size decreases).

Discussion: We need to answer “what’s so special about K = 2 giving us order for free?” from our introduction. Examining the two experiments, K=1 has many frozen components, and has very large basins (which lead to the same behavior), has very short cycle lengths (which are boring), and has short trajectories (which means that the initial states are very close to cycle state, which were boring), and are therefore relatively static and uninteresting.

When K=3 and K=4, the cycle lengths and trajectories have gone up, which makes the behavior more interesting. The size of the largest basin maybe going down: if this is true, that means that there are more cycles of behavior (because the number of initial state cycles is going up as K becomes larger, and since the basins are becoming smaller, there must be many, many more patterns of behavior). Let’s look at a fraction:

(number of state cycles) / (size of largest basin of attraction) approximates number of cycles of behavior.

We know that the number of state cycles is becoming bigger and the size of the largest basin of attraction is becoming smaller (which maybe proportional to the average size of a basin of attraction). That means the number of cycles of behavior will increase. It may increase beyond are ability to count: I suspect that the reason the computers fail from time to time is that the math of the network when K=3 or K=4 is just too big for the computer to handle.

So what’s special about K=2? Maybe its that we can do the math for it.