Thursday: Evolutionary Graph Theory

Publish Date: 
Thu, 03/24/2011 - 15:43

A long time ago I redid Sood, Antal, and Redner's diffusion equations for the evolutionary graph theory process on complex networks, generalizing it from undirected to directed graphs. It's been basically ready to submit for the last couple years, except for the figures and a bit of filling out, which is frustrating. I'm making a push to get the figures together and get it out the door.

The key figure is fixation probability as a function of the initial vertex's degree (given a mutation that begins on a single vertex). Like SAR, I find that the in EGT process fixation depends inversely on the degree, and in the voter model it depends directly on the degree, but unlike them I can show that it's the in degree in the EGT case and the out degree in the VM case. I don't like that the relationship is not nearly as clean as it looks in the SAR paper. This is the fixation probability vs. in degree (in the EGT case):

Fixation probability vs. in degree for EGT process in a small sample of directed Molloy-Reed power-law networks. A much bigger data set will be finished running sometime soon.

I'm confident the straight line with slope -1 is there when I take the mean probability as a function of in degree (it'll fit better as soon as I collect a bigger data set):

Mean fixation probability vs. in degree in same data as above.

But the variances are huge! But I guess that's just the way it works...

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