  In many cases of attacks or failures, memory effects play a significant role.
Therefore, we present a model that not only considers the dependencies between
nodes but also incorporates the memory effects of attacks. Our research
demonstrates that the survival probability of a random node reached by a random
edge surpasses the inverse of the average degree ($1/{\langle k \rangle}$), and
a giant component emerges regardless of the strength of dependencies. Moreover,
if the dependency strength exceeds $1/{\langle k \rangle}$, the network
experiences an abrupt collapse when an infinitesimally small fraction of nodes
is removed, irrespective of the memory effect. Our proposed model provides
insights into the interplay between dependencies between nodes, memory effects,
and the network structures under attacks or failures. By considering these
factors, we can better assess the vulnerability of complex systems and develop
strategies to mitigate cascading failures.
