On the dynamics of scale-free Boolean networks

The dynamical features of Random Boolean Networks (RBN) are examined, in the case where a scale-free distribution of outgoing connectivities is introduced. RBN are known to display two major dynamical behaviours, depending upon the value of some model parameters, In the "ordered" regime the number of attractors is a growing polynomial function of the number of nodes N, while in the "chaotic" regime the growth is exponential. We present here a modification of the classical way of building a RBN, which maintains the property that all the nodes have the same number of incoming links, but which gives rise to a scale-free distribution of outgoing connectivities. Because of this modification, the dynamical properties are deeply modified: the number of attractors is much smaller than in classical RBN, their length and the duration of the transients are shorter. Perhaps more surprising, the number of different attractors is almost independent of the network size, over almost three order of magnitudes. Besides pertaining to the study of the dynamics of nonlinear networks, these results may have interesting biological implications.