The outcome is startling ... ... the total system is dominated, not by the forward network, but by the feedback network ... The output signals will be of greater 'purity' than we had any right to expect.
The word feedback caught my eye. Feedback is a key principle of complex adaptive systems. Feedback is also one of the 4 values of eXtreme Programming.
So I've been playing around with Feller's walk some more...
Each walk is 1000 steps long, and at each step you flip a fair coin. As you walk you keep a running total (starting at zero), adding 1 to your total if you flip a head, subtracting 1 from your total if you flip a tail.
What will the cumulative total look like as the walk progresses?
Here's the plot of the total during a single walk (ending at about -25).
Apparently, most people's intuition is that the total will hover around zero. But it doesn't. You cannot rely on randomness to correct the problems that randomness creates. Without constraints variance will accumulate. To see this I simulated 5000 walks and counted the total at various points along the walk:
- after 10 flips gold
- after 100 flips blue
- after 1000 flips red
I've chopped the graph's x-axis to [-50,+50], but you can see that at 1000 flips the total ranged from -114 to +140. One walk finished with a total of +140! That means it had 570 heads and 430 tails! You can also see that after 10 flips along the walk, the total was zero 1204 times (out of 5000).
I've modified my simulation to allow feedback. Every Nth step, instead of flipping the coin, I can nudge the total back towards zero. If I set N==25 then one in 25 steps (on average) will be a nudge towards zero, and 24 will still be a random flip.
How will the total vary as N varies?
I've put my simulation up on rawgithub so you can try out various values of N if you want to before reading on...
Here's 5000 simulations of a 1000 flip walk with N==25.
The effectiveness of a one feedback nudge every 25 steps amazes me. Startling! Just as Stafford Beer said. The number of times the total equalled zero at the end of the walk tripled, the range dropped from 254 to 166, and perhaps most impressive of all, the variance halved.
Here's 5000 simulations of a 1000 flip walk with N==10.
Look how similar the lines and stats for blue-100 and red-1000 have become!
Let's hear it for feedback!