We analyze the stability of the network's giant connected component under
impact of adverse events, which we model through the link percolation.
Specifically, we quantify the extent to which the largest connected component
of a network consists of the same nodes, regardless of the specific set of
deactivated links. Our results are intuitive in the case of single-layered
systems: the presence of large degree nodes in a single-layered network ensures
both its robustness and stability. In contrast, we find that interdependent
networks that are robust to adverse events have unstable connected components.
Our results bring novel insights to the design of resilient network topologies
and the reinforcement of existing networked systems