On the dynamics of random Boolean networks with scale-free outgoing connections

In the classical model of Random Boolean Networks (RBN) the number of incoming connections is the same for every node, while the distribution of outgoing links is Poissonian. These RBN are known to display two major dynamical behaviours, depending upon the value of some model parameters: an “ordered” and a “chaotic” regime. We introduce 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 connections. 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. Moreover, the number of different attractors is almost independent of the network size, over almost three orders of magnitudes (while in classical RBN this number grows with the size of the network). These results are based upon a detailed study of networks where each node has two input connections. A limited study of networks with three input connections per node shows that also in this case the number of attractors is almost independent of the network size.