There is something intensely interesting here from an information theory perspective. I need to dive into this deeper, but I wrote a paper in 2008 (cited 110 times) called Survival of the Sparsest. I showed that when computational networks were permitted to evolve their connectivity (under a selective regime) they would evolve to a kind of minimum energy state with minimal network complexity (economic with no spurious connectivity). Looking at biological gene networks, I showed that this pattern of sparse connectivity showed up again and again.

For a given function, this suggests that there may only be one or two network topologies that satisfy the conditions of being both efficient and functional. This makes a good null hypothesis if you can see the input and output states and need to guess the structure of that network. If you don't see that structure, then maybe this suggests that there is some other confounding variable lurking out there. For example, maybe the network has extra connectivity because redundancy is a functional requirement and not just nice to have.

I'm guessing that there's some relationship here between root(2N) and the minimal complexity networks I was evolving. Haven't had time to digest this, but does anyone know if N refers to the number of elements N in a system, or the number of connections in an NxN matrix?

Paper http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2538912/

Lot's of meat in the supplementary http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2538912/bin/msb2...