  Activity or spin patterns on random scale-free network are studied by 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
$\gamma$ of the scale-free degree distribution changes from $\gamma >5/2$ to
$\gamma < 5/2$. For $\gamma > 5/2$, the corresponding decay times increase as
$\ln(N)$ with increasing network size $N$ whereas they are independent of $N$
for $\gamma < 5/2$. 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 typically leads to many self-connections and
multiple edges, and (B) simple-networks without self-connections and multiple
edges.
