The modular structure of Kauffman networks

This is the second paper of two about the structural properties that influence the asymptotic dynamics of random boolean networks. Here we study the functionally independent clusters in which the relevant elements, introduced and studied in our first paper [U. Bastolla, G. Parisi, Physica D 115 (1998) 203-218], are subdivided. We show that the phase transition in random boolean networks can also be described as a percolation transition. The statistical properties of the clusters of relevant elements (that we call modules) give an insight on the scaling behavior of the attractors of the critical networks that, according to Kauffman, have a biological analogy as a model of genetic regulatory systems.