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If I have the fitness of each genome, how do I determine which genome will crossover with which, and so on, so that I get a new population?

Unfortunately, I can't find anything about it in the original paper, so I ask here?

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The good thing about genetic algorithms is that they are exchangeable. If you have the fitness of each individual, any algorithm (i.e. roulette, rank, tournament) will do.

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The original work on NEAT(Neuroevolution of augmenting topologies) was by Ken Stanley in 2002 at The University of Texas at Austin. The web page for the project is here I suggest you download and read the paper linked from that page. As for selection of genome pairs, NEAT makes use of a speciation model so the selection of such pairs is constrained to at least prefer pairs from the same 'species', on the assumption that the species has evolved such that species population is isolated under reproduction. The innovation that has been 'bred' into the species is thus preserved under reproduction. Selection by fitness alone is insufficient in such models. This differs from the simple GA where pair selection is unconstrained.

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