  Multiplex networks are convenient mathematical representations for many
real-world -- biological, social, and technological -- systems of interacting
elements, where pairwise interactions among elements have different flavors.
Previous studies pointed out that real-world multiplex networks display
significant inter-layer correlations -- degree-degree correlation, edge
overlap, node similarities -- able to make them robust against random and
targeted failures of their individual components. Here, we show that
inter-layer correlations are important also in the characterization of their
$\mathbf{k}$-core structure, namely the organization in shells of nodes with
increasingly high degree. Understanding $k$-core structures is important in the
study of spreading processes taking place on networks, as for example in the
identification of influential spreaders and the emergence of localization
phenomena. We find that, if the degree distribution of the network is
heterogeneous, then a strong $\mathbf{k}$-core structure is well predicted by
significantly positive degree-degree correlations. However, if the network
degree distribution is homogeneous, then strong $\mathbf{k}$-core structure is
due to positive correlations at the level of node similarities. We reach our
conclusions by analyzing different real-world multiplex networks, introducing
novel techniques for controlling inter-layer correlations of networks without
changing their structure, and taking advantage of synthetic network models with
tunable levels of inter-layer correlations.
