  Networks are often made up of several layers that exhibit diverse degrees of
interdependencies. A multilayer interdependent network consists of a set of
graphs $G$ that are interconnected through a weighted interconnection matrix $
B $, where the weight of each inter-graph link is a non-negative real number $
p $. Various dynamical processes, such as synchronization, cascading failures
in power grids, and diffusion processes, are described by the Laplacian matrix
$ Q $ characterizing the whole system. For the case in which the multilayer
graph is a multiplex, where the number of nodes in each layer is the same and
the interconnection matrix $ B=pI $, being $ I $ the identity matrix, it has
been shown that there exists a structural transition at some critical coupling,
$ p^* $. This transition is such that dynamical processes are separated into
two regimes: if $ p > p^* $, the network acts as a whole; whereas when $ p<p^*
$, the network operates as if the graphs encoding the layers were isolated. In
this paper, we extend and generalize the structural transition threshold $ p^*
$ to a regular interconnection matrix $ B $ (constant row and column sum).
Specifically, we provide upper and lower bounds for the transition threshold $
p^* $ in interdependent networks with a regular interconnection matrix $ B $
and derive the exact transition threshold for special scenarios using the
formalism of quotient graphs. Additionally, we discuss the physical meaning of
the transition threshold $ p^* $ in terms of the minimum cut and show, through
a counter-example, that the structural transition does not always exist. Our
results are one step forward on the characterization of more realistic
multilayer networks and might be relevant for systems that deviate from the
topological constrains imposed by multiplex networks.
