  Centrality measures identify the most important nodes in a complex network.
In recent years, multilayer networks have emerged as a flexible tool to create
increasingly realistic models of complex systems. In this paper, we generalize
matrix function-based centrality and communicability measures to the case of
layer-coupled multiplex networks. We use the supra-adjacency matrix as the
network representation, which has already been used to generalize eigenvector
centrality to temporal and multiplex networks. With this representation, the
definition of single-layer matrix function-based centrality measures in terms
of walks on the networks carries over naturally to the multilayer case. Several
aggregation techniques allow the ranking of nodes, layers, as well as
node-layer pairs in terms of their importance in the network. We present
efficient and scalable numerical methods based on Krylov subspace techniques
and Gauss quadrature rules, which provide a high accuracy in only a few
iterations and which scale linearly in the network size under the assumption of
sparsity in the supra-adjacency matrix. Finally, we present extensive numerical
studies for both directed and undirected as well as weighted and unweighted
multiplex networks. While we focus on social and transportation applications
the networks' size ranges between $89$ and $2.28 \cdot 10^6$ nodes and between
$3$ and $37$ layers.
