Multifractality in Complex Systems: From Scaling Structure and Long-Range Organization to Learning Systems

Multifractal analysis has evolved from a formalism for heterogeneous scaling into a broader framework for investigating structure, dynamics, representation, computation, and learning systems. This critical integrative review synthesizes methodological foundations, evidence criteria, complex-system applications, network methods, computational uses, and multifractal analyses of learning systems. Particular attention is given to apparent versus supported multifractality and to finite size, temporal correlations, heavy-tailed distributions, estimator choices, surrogate testing, and mechanistic interpretation. To compare heterogeneous studies, we organize the literature through a three-dimensional framework in which locus identifies where multifractality is measured, role specifies how the resulting quantities are used, and evidence characterizes the support for their interpretation. The framework clarifies how multifractal information functions across description, diagnosis, representation, adaptive computation, trainable operators, and the analysis of learning systems. We further distinguish complexity in data, models, input-conditioned hidden representations, and outputs, identifying direct multifractal analysis of hidden representations as comparatively underexplored. Across domains, multifractal claims are treated as object-dependent and tied to the support, measure, scale, and estimator used to construct the analyzed object. Future progress will depend on stronger finite-size controls, cross-method comparability, causal validation, and explicit definitions of analytical objects.