NK model

The NK model defines a combinatorial phase space, consisting of every string (chosen from a given alphabet) of length ${\displaystyle N}$. For each string in this search space, a scalar value (called the fitness) is defined. If a distance 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 ${\displaystyle S}$ is the sum of contributions from each locus ${\displaystyle f_{i}(S)}$ in the string:

${\displaystyle F(S)=\sum _{i}{\tilde {f}}_{i}(S),}$

and the contribution from each locus in general depends on its state and the state of ${\displaystyle K}$ other loci,:

${\displaystyle {\tilde {f}}_{i}(S)=f_{i}(S_{i},S_{k_{i1}},\dots ,S_{k_{iK}}),}$

where ${\displaystyle k_{ij}}$ is the index of the ${\displaystyle j}$th neighbor of locus ${\displaystyle i}$.

Hence, the fitness function ${\displaystyle f_{i}}$ is a mapping between strings of length K + 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[6] of the case in which ${\displaystyle 1<<k\leq N}$ and the fitness contributions are chosen randomly. His analytical estimate of the number of local optima was later shown to be flawed . 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

${\displaystyle \mu +\sigma {\sqrt {{2\ln(k+1)} \over {k+1}}}}$

and a variance of approximately

${\displaystyle {{(k+1)\sigma ^{2}} \over {N[k+1+2(k+2)\ln(k+1)]}}}$.

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.

For simplicity, we will work with binary strings. Consider an NK model with N = 5, K = 1. Here, the fitness of a string is given by the sum of individual fitness contributions from each of 5 loci. Each fitness contribution depends on the local locus value and one other. We will employ the convention that ${\displaystyle f(S_{i})=f(S_{i},S_{i+1})}$, so that each locus is affected by its neighbour, and ${\displaystyle f(S_{5})=f(S_{5},S_{1})}$ for cyclicity. If we choose, for example, the fitness function f(0, 0) = 0; f(0, 1) = 1; f(1, 0) = 2; f(1, 1) = 0, the fitness values of two example strings are:

${\displaystyle F(00101)=f(0,0)+f(0,1)+f(1,0)+f(0,1)+f(1,0)=0+1+2+1+2=6.}$

${\displaystyle F(11100)=f(1,1)+f(1,1)+f(1,0)+f(0,0)+f(0,1)=0+0+2+0+1=3.}$

Illustration of tunable topology in the NK model. Nodes are individual binary strings, edges connect strings with a Hamming distance of exactly one. (left) N = 5, K = 0. (centre) N = 5, K = 1. (right) N = 5, K = 2. The colour of a node denotes its fitness, with redder values having higher fitness. The embedding of the hypercube is chosen so that the fitness maximum is at the centre. Notice that the K = 0 landscape appears smoother than the higher-K cases.

The value of K controls the degree of epistasis in the NK model, or how much other loci affect the fitness contribution of a given locus. With K = 0, the fitness of a given string is a simple sum of individual contributions of loci: for nontrivial fitness functions, a global optimum is present and easy to locate (the genome of all 0s if f(0) > f(1), or all 1s if f(1) > f(0)). For nonzero K, the fitness of a string is a sum of fitnesses of substrings, which may interact to frustrate the system (consider how to achieve optimal fitness in the example above). Increasing K thus increases the ruggedness of the fitness landscape.

The NK model has found use in many fields, including in the study of spin glasses, epistasis and pleiotropy in evolutionary biology, and combinatorial optimisation.