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NK model : ウィキペディア英語版
NK model
The NK model is a mathematical model described by its primary inventor Stuart Kauffman as a "tunably rugged" fitness landscape. "Tunable ruggedness" captures the intuition that both the overall size of the landscape and the number of its local "hills and valleys" can be adjusted via changes to its two parameters, N and K, defined below. The NK model has found application in a wide variety of fields, including the theoretical study of evolutionary biology, immunology, optimisation and complex systems. The model was also adopted in organizational theory, where it is used to describe the way an agent may search a landscape by manipulating various characteristics of itself. For example, an agent can be an organization, the hills and valleys represent profit (or changes thereof), and movement on the landscape necessitates organizational decisions (such as adding product lines or altering the organizational structure), which tend to interact with each other and affect profit in a complex fashion.
An early version of the model, which considered only the smoothest (K=0) and most rugged (K=N) landscapes, was presented in Kauffman and Levin (1987). The model as it is currently known first appeared in Kauffman and Weinberger (1989).
One of the reasons why the model has attracted wide attention in optimisation is that it is a particularly simple instance of a so-called NP-complete problem〔Weinberger, E. (1996), "NP-completeness of Kauffman's N-k model, a Tuneably Rugged Fitness Landscape", Santa Fe Institute Working Paper, 96-02-003.〕
== Mathematical details ==
The NK model defines a combinatorial phase space, consisting of every string (chosen from a given alphabet) of length 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 S is the sum of contributions from each locus S_i in the string:
:F(S) = \sum_i f(S_i),
and the contribution from each locus in general depends on the value of K other loci:
:f(S_i) = f(S_i, S^i_1, \dots, S^i_K), \,
where S^i_j are the other loci upon which the fitness of S_i depends.
Hence, the fitness function f(S_i, S^i_1, \dots, S^i_K) 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〔 of the case in which 1 << k \le 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
\mu + \sigma \sqrt
and a variance of approximately
.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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