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In real-world applications, observations are often constrained to a smallfraction of a system. Such spatial subsampling can be caused by theinaccessibility or the sheer size of the system, and cannot be overcome bylonger sampling. Spatial subsampling can strongly bias inferences about asystem's aggregated properties. To overcome the bias, we derive analytically asubsampling scaling framework that is applicable to different observables,including distributions of neuronal avalanches, of number of people infectedduring an epidemic outbreak, and of node degrees. We demonstrate how to inferthe correct distributions of the underlying full system, how to apply it todistinguish critical from subcritical systems, and how to disentanglesubsampling and finite size effects. Lastly, we apply subsampling scaling toneuronal avalanche models and to recordings from developing neural networks. Weshow that only mature, but not young networks follow power-law scaling,indicating self-organization to criticality during development.

Subsampling scaling: a theory about inference from partly observed systems