  The functionality of nodes in a network is often described by the structural
feature of belonging to the giant component. However, when dealing with
problems like transport, a more appropriate functionality criterion is for a
node to belong to the network's backbone, where the flow of information and of
other physical quantities (such as current) occurs. Here we study percolation
in a model of interdependent resistor networks and show the effect of
spatiality on their coupled functioning. We do this on a realistic model of
spatial networks, featuring a Poisson distribution of link-lengths. We find
that interdependent resistor networks are significantly more vulnerable than
their percolation-based counterparts, featuring first-order phase transitions
at link-lengths where the mutual giant component still emerges continuously. We
explain this apparent contradiction by tracing the origin of the increased
vulnerability of interdependent transport to the crucial role played by the
dandling ends. Moreover, we interpret these differences by considering an
heterogeneous $k$-core percolation process which enables to define a
one-parameter family of functionality criteria whose constraints become more
and more stringent. Our results highlight the importance that different
definitions of nodes functionality have on the collective properties of coupled
processes, and provide better understanding of the problem of interdependent
transport in many real-world networks.
