Wednesday: onwards with the evolutionary graph theory!

Publish Date: 
Wed, 03/30/2011 - 21:03

Ahoy! Look at this!

ω0,1 vs. ω-1,0 in EGT simulation, on a power-law directed network of 1000 vertices, starting with the mutant type on all vertices of in degree 10 or greater.

To fill in some details: ω0,1 is the count of vertices bearing the mutant type, weighted by each vertex's out degree, while ω-1,0 is the count weighted by inverse of the vertex's in degree. So when we initialize the vertices with high in degree, ω0,1 is at some intermediate value because initialized vertices have any old out degree, but ω-1,0 is very low.

My key result is that in the EGT process, ω-1,0 changes slowly and in no particular direction, while the other moments including ω0,1 change more quickly and approach equality with ω-1,0. This is exactly what we see in this simulation result, and we'll see the same thing in the other 3 cases - high out degree, low in degree, low out degree. The trace will start on different sides of the square, but will always go vertically to the diagonal and stay near the diagonal. Then when I redo them for the VM process, it will be the same, but they will all go horizontally to the diagonal, because in that case it's ω0,1 that changes slowly and the other measures converge on it.

To do: those graphs! And, quit doing these figures cowboy style and get them into makefile rules and onto the wiki. And maybe another question that came up yesterday: there's an exponential function of ω-1,0 that is a martingale for the approximation to the EGT process, but the approximation doesn't consider the unchanging structure of the network, so that long-term correlations between neighbors could cause the actual process to differ from the approximation. Explore whether the martingale quantity is not one in the original process, and if not how does it diverge and in which direction. This might shed some light on why there's so much variance in last Thursday's results.

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