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

Abstract

NK-Kauffman networks ℒ K N , 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 Ψ : ℒ K N →  2 N , where  2 N is the set of functional graphs with 2 N nodes, is studied. In the asymptotic regime N ≫ 1, it is found that a critical connectivity K c~ ln ln N exists such that Ψ is many-to-one for K < K c and injective for K > K c. The analysis is extended when the tautology and contradiction Boolean functions are excluded from the construction of ℒ K N. For such a case, it is found that Ψ always remains injective.