  The rapid advancement of technology underscores the critical importance of
robustness in complex network systems. This paper presents a framework for
investigating the structural robustness of interconnected network models. This
paper presents a framework for investigating the structural robustness of
interconnected network models. In this context, we define functional nodes
within interconnected networks as those belonging to clusters of size greater
than or equal to $s$ in the local network, while maintaining at least $M$
significant dependency links. This model presents precise analytical
expressions for the cascading failure process, the proportion of functional
nodes in the stable state, and a methodology for calculating the critical
threshold. The findings reveal an abrupt phase transition behavior in the
system following the initial failure. Additionally, we observe that the system
necessitates higher internal connection densities to avert collapse, especially
when more effective support links are required. These results are validated
through simulations using both Poisson and power-law network models, which
align closely with the theoretical outcomes. The method proposed in this study
can assist decision-makers in designing more resilient reality-dependent
systems and formulating optimal protection strategies.
