Kauffman Boolean model in undirected scale-free networks

Abstract

We investigate analytically and numerically the critical line in undirected random Boolean networks with arbitrary degree distributions, including the scale-free topology of connections P(k) ~ k(-gamma). We explain that the unattainability of the critical line in numerical simulations of classical random graphs is due to percolation phenomena. We suggest that recent findings of discrepancy between simulations and theory in directed random Boolean networks might have the same reason. We also show that in infinite scale-free networks the transition between frozen and chaotic phases occurs for 3<gamma<3.5 . Since most critical phenomena in scale-free networks reveal their nontrivial character for gamma<3 , the position of the critical line in the Kauffman model seems to be an important exception to the rule.