  Network structures in a wide array of systems such as social networks,
transportation, power and water distribution infrastructures, and biological
and ecological systems can exhibit critical thresholds or tipping points beyond
which there are disproportionate losses in the system functionality. There is
growing concern over tipping points and failure tolerance of such systems as
tipping points can lead to an abrupt loss of intended functionality and
possibly non-recoverable states. While attack tolerance of networked systems
has been intensively studied for the disruptions originating from a single
point of failure, there have been instances where real-world systems are
subject to simultaneous or sudden onset of concurrent disruption at multiple
locations. Using open-source data from the United States Airspace Airport
network and Indian Railways Network, and random networks as prototype class of
systems, we study their responses to synthetic attack strategies of varying
sizes. For both types of networks, we observe the presence of warning regions,
which serve as a precursor to the tipping point. Further, we observe the
statistically significant relationships between network robustness and size of
simultaneous distribution, which generalizes to the networks with different
topological attributes for random failures and targeted attacks. We show that
our approach can determine the entire robustness characteristics of networks of
disparate architecture subject to disruptions of varying sizes. Our approach
can serve as a paradigm to understand the tipping point in real-world systems,
and the principle can be extended to other disciplines to address critical
issues of risk management and resilience.
