Recently, it has been shown that the removal of a random fraction of nodes from a system of interdependent spatial networks can lead to cascading failures which amplify the original damage and destroy the entire system, often via abrupt first-order transitions. For these distinctive phenomena to emerge, the interdependence between networks need not be total. We consider here a system of partially interdependent spatial networks (modelled as lattices) with a fraction |q| of the nodes interdependent and the remaining |$1-q$| autonomous. In our model, the dependency links between networks are of geometric length less than |r|. Under full dependency (|$q=1$|), this system was shown to have a first-order percolation transition if |r>rc≈ 8|. Here, we generalize this result and show that for all |$q>0$|, there will be a first-order transition if |r>rc(q)|. We show that |rc(q)| increases monotonically with decreasing |q| and |lim _q→ 0⁺rc(q)=∞|. Additionally, we present a detailed description and explanation of the cascading failures in spatially embedded interdependent networks near the percolation threshold |pc|. These failures follow three mechanisms depending on the value of |r|. Below |rc| the system undergoes a continuous transition similar to standard percolation on a lattice. Above |rc| there are two distinct first-order transitions for finite and infinite |r|, respectively. The cascading failure for finite |r| is characterized by the emergence of a critical hole which then spreads through the system while the infinite |r| transition is more similar to the case of random networks. Surprisingly, we find that this spreading transition can still occur even if |p<pc|. We present measurements of cascade dynamics which differentiate between these phase transitions and elucidate their mechanisms. These results extend previous research on spatial networks to the more realistic case of partial dependency and shed new light on the specific dynamics of dependency-driven cascading failures.
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Danziger et al. (2014) studied this question.