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Activity or spin patterns on a random scale-free network are studied using mean field analysis and computer simulations. These activity patterns evolve in time according to local majority rule dynamics which is implemented using (i) parallel or synchronous updating and (ii) random sequential or asynchronous updating. Our mean field calculations predict that the relaxation processes of disordered activity patterns become much more efficient as the scaling exponent γ of the scale-free degree distribution changes from γ> 5/2 to γ< 5/2. For γ> 5/2, the corresponding decay times increase as ln (N) with increasing network size N whereas they are independent of N for γ< 5/2. In order to check these mean field predictions, extensive simulations of the pattern dynamics have been performed using two different ensembles of random scale-free networks:(A) multi-networks as generated by the configuration method, which …