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I'm learning about NEAT from the paper Evolving Neural Networks through Augmenting Topologies.

I'm having trouble understanding how adjusted fitness penalizes large species and prevents them from dominating the population, I'll demonstrate my current understanding through an example and, hopefully, someone will correct my understanding.

Let's say we have two species, $A$ and $B$, species $A$ did really well the last generation and were given more children, this generation they have $4$ children and their fitnesses are $[8,10,10,12]$, while $B$ has $2$ children and their fitnesses are $[9,9]$. The adjusted fitnesses for $A$ will be $[2, 2.5, 2.5, 3]$ and for B will be $[4.5, 4.5]$.

Now, onto distributing children, the paper states:

Every species is assigned a potentially different number of offspring in proportion to the sum of adjusted fitnesses $f'_i$ of its member organisms.

So, the sum of adjusted fitnesses is $10$ for $A$ and $9$ for $B$, thus $A$ gets more children and keeps growing.

How does this process penalize large species and prevent them from dominating the population?

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Consider what would the outcome be if offspring was assigned on the basis of fitness (unadjusted). Sum of A fitnesses would be 40 and B=18. Fitness ratio for both species would be 2.(2):1. In case of adjusted fitness the numbers are A=15 and B=9, which gives ratio of 1.(6):1, thus A is assigned less offspring based on adjusted fitness then unadjusted.

Also note that every new genome assigned to a species decreases all it's members adjusted fitness. In your case members of species A are more successful than members of B, so it should grow. The mechanism is designed to hinder growth of successful species, not to block it entirely. This allows more diversity, which is important when a successful species reaches a dead end and a previously less successful one can take over.

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The reason the adjusted fitness prevents species from growing to big is do to the fact that the summation that determines the divisor in the adjusted fitness function reduces to the number of genomes in the species fi belongs to, so as species grow the adjusted fitness for every genome belonging to it is divided by a larger number and thus receives a lower adjusted fitness value which in turn will also reduce the summed adjusted fitness of the species. This smaller adjusted fitness then affects the number of offspring the species will get to create after elitism reduces the population.

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