  Clustering and closure coefficients are among the most widely applied
indicators in the description of the topological structure of a network. Many
distinct definitions have been proposed over time, particularly in the case of
weighted networks, where the choice of the weight attributed to the triangles
is a crucial aspect. In the present work, in the framework of weighted directed
multilayer networks, we extend the classical clustering and closure
coefficients through the introduction of the clumping coefficient, which
generalizes them to incomplete triangles of any type. We then organize the
class of these coefficients in a systematic taxonomy in the more general
context of weighted directed multilayer networks. Such cohesion coefficients
have also been adapted to the different scales that characterize a multilayer
network, in order to grasp their structure from different perspectives. We also
show how the tensor formalism allows incorporating the new definitions, as well
as all those existing in the literature, in a single unified writing, in such a
way that a suitable choice of the involved adjacency tensors allows obtaining
each of them. Finally, through some applications to simulated networks, we show
the effectiveness of the proposed coefficients in capturing different
peculiarities of the network structure on different scales.
