Recent developments of the multilayer paradigm include efforts to understand the role played by the presence of several layers on the dynamics of processes running on the networks. The possible existence of new phenomena associated to the richer multilayer topology has been discussed and examples of these differences have been systematically searched for. Here, we show that the interconnectivity of the layers may have an important impact on the speed of the dynamics run in the network and that microscopic changes such as the addition of one single inter-layer link can notably affect the arrival at a global stationary state. As a practical testbed, these results obtained with spectral techniques are confirmed with a Kuramoto dynamics for which the synchronization consistently accelerates after the addition of single inter-layer links.
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