  Interdependence is a fundamental ingredient to analyze the stability of many
real-world complex systems featuring functional liasons. Yet, physical
realizations of this coupling are still unknown, due to the lack of a
theoretical framework for their study. To address this gap, we develop an
interdependent magnetization framework and show that dependency links between
$K-1$ pairwise networks of Ising spins can be rigorously mapped to directed
$K$-spin interactions or to adaptive thermal couplings. We adopt the thermal
portrait to determine analytically the phase diagram of the model under
different structural configurations and we corroborate our results by extensive
simulations. We find that interdependence acts like an entropic force that
amplifies site-to-site thermal fluctuations, yielding unusual forms of
vulnerability and making the system's functioning often unrecoverable. Finally,
we discover an isomorphism between the ground state of random multi-spin models
and interdependent percolation on randomly coupled networks. This connection
raises new perspectives of cross-fertilization, providing unfamiliar methods
with relevant implications in the study of constraint satisfaction as well as
to the functional robustness of interdependent systems.
