Multifractality in Complex Systems: From Scaling Structure and Long-Range Organization to Learning Systems
- 1Department of Information Engineering, Electronics and Telecommunications, Sapienza University of Rome, Via Eudossiana 18, 00184 Rome, Italy
- 2Department of AI, Data and Decision Sciences, Luiss University of Rome, Viale Romania 32, 00197 Rome, Italy
Abstract
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.
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BibTeX
@article{santis2026multifractality,
title = {{Multifractality in Complex Systems: From Scaling Structure and Long-Range Organization to Learning Systems}},
author = {Enrico De Santis and Alessio Martino and Antonello Rizzi},
journal = {Fractal and Fractional},
year = {2026},
month = oct,
doi = {10.3390/fractalfract10110747},
url = {https://www.mdpi.com/2504-3110/10/11/747},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Multifractality in Complex Systems: From Scaling Structure and Long-Range Organization to Learning Systems
AU - Enrico De Santis
AU - Alessio Martino
AU - Antonello Rizzi
PY - 2026
DA - 2026/10/26
JO - Fractal and Fractional
DO - 10.3390/fractalfract10110747
UR - https://www.mdpi.com/2504-3110/10/11/747
AB - 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.
SN - 2504-3110
ER -