On the Number of NK-Kauffman Networks Mapped into a Functional Graph
Federico Zertuche
Instituto de Matemáticas
Unidad Cuernavaca, Universidad Nacional Autónoma de México
Avenida Universidad 1001
Cuernavaca, Morelos 62210, Mexico
Abstract
NK-Kauffman networks 
, where N is the number of Boolean variables and K the average number of connections, are studied. The K connections are random and chosen with equal probability, while the Boolean functions are randomly chosen with a bias p. The injectivity of the map 
, where 
is the set of functional graphs with 
nodes, is studied. In the asymptotic regime N ≫ 1, it is found that a critical connectivity 
exists such that ψ is many-to-one for 
and injective for 
. The analysis is extended when the tautology and contradiction Boolean functions are excluded from the construction of 
. For such a case, it is found that ψ always remains injective.
