  While degree correlations are known to play a crucial role for spreading
phenomena in networks, their impact on the propagation speed has hardly been
understood. Here we investigate a tunable spreading model on scale-free
networks and show that the propagation becomes slow in positively (negatively)
correlated networks if nodes with a high connectivity locally accelerate
(decelerate) the propagation. Examining the efficient paths offers a coherent
explanation for this result, while the $k$-core decomposition reveals the
dependence of the nodal spreading efficiency on the correlation. Our findings
should open new pathways to delicately control real-world spreading processes.
