  Shortest paths are representative of discrete geodesic distances in graphs,
and many descriptors of networks depend on their counting. In multiplex
networks, this counting is radically important to quantify the switch between
layers and it has crucial implications in the transportation efficiency and
congestion processes. Here we present a mathematical approach to the
computation of the joint distribution of distance and multiplicity
(degeneration) of shortest paths in multiplex networks, and exploit its
relation to congestion processes. The results allow to approximate
semi-analytically the onset of congestion in multiplex networks as a function
of the congestion of its layers.
