The paradox of order and randomness

Consider the following two images:




First view each image as is, and then click on them to see larger versions. Ignore for a moment the different sizes, and the copyright notice in the second picture (hope I'm not breaking any laws!) What's going on in these pictures?

The first is a randomly generated image, in which a computer essentially flipped a coin to decide the color (black or white) of each pixel. The second is composed of alternating black and white pixels in a checkered pattern (click on the image to see this clearly.)

At this resolution, the first picture still has some texture to it. But zoom out a bit more and it would reduce to a uniform grey, just like the second.

This highlights something of a paradox in complex systems theory: complete randomness is actually pretty boring. Sure, it's unpredictable, but because it has no structure, there's not much else you can say about it. And if you squint at it, it all averages out to grey. Contrast this to the following fractal image:



Now this picture has a lot of interesting structure to describe, like most complex systems.

Why is this a paradox? Because according to the defintions of complexity we discussed some months ago, a completely random system is more complex than anything else! Any order or structure in a system makes it easier to describe, thereby reducing complexity according to conventional definitions. So the fractal is actually less complex than the random image.

Complex systems researchers have recognized this problem for a long time, but there's no consensus on how to resolve it. Some have suggested adopting a different definition of complexity that behaves something like this:



That is, complexity is greatest somewhere between total order and complete randomness. But this is unsatisfying; complexity is not a mere mixture between order and randomness, but a delicate balance combining features of the two.

Of course, I have my own opinion as to how this paradox should be resolved. But that's a tale for another time.