Here's a puzzle that I'm curious if anyone can attack with a simple method without resorting to simulation. I can tell you the answer (well, an answer) from running some programs. But I'm writing a blog post about it and would kind of like something "proof-y" to say about it.
Let's imagine you have a network of nodes, each holding a real number value. This network is SIMD ("single instruction, multiple data"), and on each clock cycle any node can examine its own value V and the value held by its immediate neighbors. After the examination it can decide to send some portion of its value to its neighbors, and some to keep. The total amount must not exceed the value it had been allocated.
The goal of the game I posit is to stabilize the network such that the values on all nodes are as close to equal as possible in the fewest number of clock cycles. The algorithm chosen to run by all nodes must not have any knowledge of the topography of the network as a whole.
As per Kenny Rogers, "Every gambler knows, the secret to surviving, is knowing what to throw away, and knowing what to keep..." So what area of theory would one use, if the same rule is to be used on each node, to come up with how much to give to each neighbor or keep if the goal is minimizing the cycles to reach equilibrium/diffusion (within a chosen tolerance, as they're real numbers, call it T=0.001 or somesuch)?
I did some simulations taking the maximum of the number of connections you and your immediate neighbors have, call that M. I got the fastest convergence by giving each neighbor V/(M-1) and give yourself -V/(M-1). The sum of the value you distributed is still V.
What makes sense is that in a network whose topology you don't know, the local neighbor values (beyond just the number of neighbors and the number of neighbors they have) is a red herring and you might as well ignore it. What also makes sense is that creating a "dip" which will be "filled in" by collaborators will reach equilibrium quicker. But it destabilizes if you ever go past the -V/(M-1) point of keeping this "negative value". Cross the line and fibrillates the network without ever reaching equilibrium, even if what you give out sums to V.
I'm looking for something like a Pole-Zero plot that I can directly correspond to this problem to explain it. My CS knowledge is deep, but my math skills top out at linear algebra and differential equations...so put it in "not a total idiot, but still a layman"'s terms. :-)