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

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... . 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.