  When assessing spatially-extended complex systems, one can rarely sample the
states of all components. We show that this spatial subsampling typically leads
to severe underestimation of the risk of instability in systems with
propagating events. We derive a subsampling-invariant estimator, and
demonstrate that it correctly infers the infectiousness of various diseases
under subsampling, making it particularly useful in countries with unreliable
case reports. In neuroscience, recordings are strongly limited by subsampling.
Here, the subsampling-invariant estimator allows to revisit two prominent
hypotheses about the brain's collective spiking dynamics:
asynchronous-irregular or critical. We identify consistently for rat, cat and
monkey a state that combines features of both and allows input to reverberate
in the network for hundreds of milliseconds. Overall, owing to its ready
applicability, the novel estimator paves the way to novel insight for the study
of spatially-extended dynamical systems.
