  Networks in the real world do not exist as isolated entities, but they are
often part of more complicated structures composed of many interconnected
network layers. Recent studies have shown that such mutual dependence makes
real networked systems potentially exposed to atypical structural and dynamical
behaviors, and thus there is a urgent necessity to better understand the
mechanisms at the basis of these anomalies. Previous research has mainly
focused on the emergence of atypical properties in relation with the moments of
the intra- and inter-layer degree distributions. In this paper, we show that an
additional ingredient plays a fundamental role for the possible scenario that
an interconnected network can face: the correlation between intra- and
inter-layer degrees. For sufficiently high amounts of correlation, an
interconnected network can be tuned, by varying the moments of the intra- and
inter-layer degree distributions, in distinct topological and dynamical
regimes. When instead the correlation between intra- and inter-layer degrees is
lower than a critical value, the system enters in a supercricritical regime
where dynamical and topological phases are not longer distinguishable.
