  Criticality and symmetry, studied by the renormalization groups, lie at the
heart of modern physics theories of matters and complex systems. However,
surveying these properties with massive experimental data is bottlenecked by
the intolerable costs of computing renormalization groups on real systems.
Here, we develop a time- and memory-efficient framework, termed as the random
renormalization group, for renormalizing ultra-large systems (e.g., with
millions of units) within minutes. This framework is based on random
projections, hashing techniques, and kernel representations, which support the
renormalization governed by linear and non-linear correlations. For system
structures, it exploits the correlations among local topology in kernel spaces
to unfold the connectivity of units, identify intrinsic system scales, and
verify the existences of symmetries under scale transformation. For system
dynamics, it renormalizes units into correlated clusters to analyze scaling
behaviours, validate scaling relations, and investigate potential criticality.
Benefiting from hashing-function-based designs, our framework significantly
reduces computational complexity compared with classic renormalization groups,
realizing a single-step acceleration of two orders of magnitude. Meanwhile, the
efficient representation of different kinds of correlations in kernel spaces
realized by random projections ensures the capacity of our framework to capture
diverse unit relations. As shown by our experiments, the random renormalization
group helps identify non-equilibrium phase transitions, criticality, and
symmetry in diverse large-scale genetic, neural, material, social, and
cosmological systems.
