  We study two types of simplified Boolean dynamics over scale-free networks,
both with synchronous update. Assigning only Boolean functions AND and XOR to
the nodes with probability $1-p$ and $p$, respectively, we are able to analyze
the density of 1's and the Hamming distance on the network by numerical
simulations and by a mean-field approximation (annealed approximation). We show
that the behavior is quite different if the node always enters in the dynamic
as its own input (self-regulation) or not. The same conclusion holds for the
Kauffman KN model. Moreover, the simulation results and the mean-field ones (i)
agree well when there is no self-regulation, and (ii) disagree for small $p$
when self-regulation is present in the model.
