Editing Stuart Kauffman (section)

==Fitness landscapes==
[[File:Visualization of two dimensions of a NK fitness landscape.png|thumb|Visualization of two dimensions of a NK fitness landscape. The arrows represent various mutational paths that the population could follow while evolving on the fitness landscape.]]
{{main|Fitness landscape}}

Kauffman's NK model defines a [[combinatorial]] [[phase space]], consisting of every string (chosen from a given alphabet) of length <math>N</math>. For each string in this search space, a [[scalar (mathematics)|scalar]] value (called the ''[[fitness function|fitness]]'') is defined. If a distance [[metric (mathematics)|metric]] is defined between strings, the resulting structure is a ''landscape''.

Fitness values are defined according to the specific incarnation of the model, but the key feature of the NK model is that the fitness of a given string <math>S</math> is the sum of contributions from each locus <math>S_i</math> in the string:

:<math>F(S) = \sum_i f(S_i),</math>

and the contribution from each locus in general depends on the value of <math>K</math> other loci:

:<math>f(S_i) = f(S_i, S^i_1, \dots, S^i_K), \, </math>

where <math>S^i_j</math> are the other loci upon which the fitness of <math>S_i</math> depends.

Hence, the fitness function <math>f(S_i, S^i_1, \dots, S^i_K)</math> is a [[Map (mathematics)|mapping]] between strings of length ''K''&nbsp;+&nbsp;1 and scalars, which Weinberger's later work calls "fitness contributions". Such fitness contributions are often chosen randomly from some specified probability distribution.

In 1991, Weinberger published a detailed analysis<ref name="AnalyticOptima">{{cite journal|last=Weinberger|first=Edward|journal=Physical Review A|date=November 15, 1991|volume=44|issue=10|series=10|pages=6399–6413|doi=10.1103/physreva.44.6399|title=Local properties of Kauffman's N-k model: A tunably rugged energy landscape|pmid=9905770|bibcode=1991PhRvA..44.6399W}}</ref> of the case in which <math>1 << k \le N</math> and the fitness contributions are chosen randomly. His analytical estimate of the number of local optima was later shown to be flawed.{{citation needed|date=March 2018}} However, numerical experiments included in Weinberger's analysis support his analytical result that the expected fitness of a string is normally distributed with a mean of approximately
<math> \mu + \sigma \sqrt{{2 \ln (k+1)} \over {k+1}}</math>
and a variance of approximately
<math> {{(k+1)\sigma^2} \over {N[k+1 + 2(k+2)\ln(k+1)]}}</math>.