  The increasing complexity and interdependency of today's networks highlight
the importance of studying network robustness to failure and attacks. Many
large-scale networks are prone to cascading effects where a limited number of
initial failures (due to attacks, natural hazards or resource depletion)
propagate through a dependent mechanism, ultimately leading to a global failure
scenario where a substantial fraction of the network loses its functionality.
These cascading failure scenarios often take place in networks which are
embedded in space and constrained by geometry. Building on previous results on
cascading failure in random geometric networks, we introduce and analyze a
continuous cascading failure model where a node has an initial
continuously-valued state, and fails if the aggregate state of its neighbors
fall below a threshold. Within this model, we derive analytical conditions for
the occurrence and non-occurrence of cascading node failure, respectively.
