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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.