Complex Systems

On the Number of NK-Kauffman Networks Mapped into a Functional Graph Download PDF

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.