  In multiplex networks, cycles cannot be characterized only by their length,
as edges may occur in different layers in different combinations. We define a
classification of cycles by the number of edges in each layer and the number of
switches between layers. We calculate the expected number of cycles of each
type in the configuration model of a large sparse multiplex network. Our method
accounts for the full degree distribution including correlations between
degrees in different layers. In particular, we obtain the numbers of cycles of
length 3 of all possible types. Using these, we give a complete set of
clustering coefficients and their expected values. We show that correlations
between the degrees of a vertex in different layers strongly affect the number
of cycles of a given type, and the number of switches between layers. Both
increase with assortative correlations and are strongly decreased by
disassortative correlations. The effect of correlations on clustering
coefficients is equally pronounced.
