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Fractal Fract
Special Issues
Multifractal Analysis and Complex Systems
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Multifractal Analysis and Complex Systems

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: 15 September 2026 | Viewed by 2435

Special Issue Editors

Dr. Enrico De Santis

E-Mail Website
Guest Editor
Department of Information Engineering, Electronics and Telecommunications, University of Rome “La Sapienza”, 00184 Rome, Italy
Interests: complex system modeling; soft and granular computing techniques with applications to artificial intelligence and natural language processing; green transition (smart grids); predictive maintenance; multifractal analysis and fractal-based methods
Dr. Alessio Martino

E-Mail Website
Guest Editor
Department of AI, Data and Decision Sciences, LUISS University, 00197 Rome, Italy
Interests: machine learning and pattern recognition in non-metric spaces; graph theory, large-scale, and distributed computing; applications to bioinformatics, NLP, and smart energy networks
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to provide a forum for the latest research and advances in multifractal analysis and its applications to complex systems, including artificial intelligence and neural network models. Multifractal analysis has emerged as a powerful mathematical and computational framework to characterize scaling behaviors, hierarchical structures, and self-similar patterns in real-world phenomena. These methods are increasingly relevant for analyzing the internal dynamics, learning behaviors, and emergent complexity of AI systems.

We welcome original research and comprehensive reviews that address both theoretical developments and practical applications of multifractal analysis and affine methods. Topics of interest include, but are not limited to, new algorithms and models for multifractal analysis; multifractal properties in natural and engineered systems; connections with statistical physics, information theory, and network science; and innovative uses in smart grids, predictive maintenance, finance, climate modeling, biological systems, and artificial intelligence. In particular, we encourage contributions exploring how multifractal techniques can reveal hidden structures and behaviors in general complex systems and also in deep learning architectures, time series generated by AI agents, or complex data-driven systems.

This Special Issue aims to foster interdisciplinary collaboration and highlight emerging directions in the interplay between multifractality, complexity science, and artificial intelligence.

Dr. Enrico De Santis
Dr. Alessio Martino
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • multifractal analysis
  • fractal geometry
  • self-affine systems
  • complex systems
  • scaling laws
  • nonlinear dynamics
  • neural networks
  • network science
  • applied mathematics

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Published Papers (3 papers)

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Research

40 pages, 6580 KB  
Article
Self-Organized Criticality and Multifractal Characteristics of Power-System Blackouts: A Long-Term Empirical Study of China’s Power System
by Qun Yu, Zhiyi Zhou, Jiongcheng Yan, Weimin Sun and Yuqing Qu
Fractal Fract. 2026, 10(4), 239; https://doi.org/10.3390/fractalfract10040239 - 3 Apr 2026
Viewed by 458
Abstract
Power system blackouts represent typical manifestations of instability in complex systems, whose evolution often exhibits non-stationarity, long-range correlations, and nonlinear scaling behavior. Most reliability assessment methods widely used in engineering practice are built on the core assumptions of event independence and light-tailed distribution, [...] Read more.
Power system blackouts represent typical manifestations of instability in complex systems, whose evolution often exhibits non-stationarity, long-range correlations, and nonlinear scaling behavior. Most reliability assessment methods widely used in engineering practice are built on the core assumptions of event independence and light-tailed distribution, which will inevitably lead to systematic underestimation of extreme tail risks when blackouts actually present long-range memory and power-law heavy-tailed characteristics. Based on long-cycle historical blackout records of China’s power grid spanning 1981–2025, this paper develops an integrated framework combining Self-Organized Criticality (SOC) theory, Hurst exponent analysis, symbolic time-series methods, and Multifractal Detrended Fluctuation Analysis (MFDFA). This study systematically characterizes the evolution law and inherent dependence structure of blackout events from four dimensions: statistical scaling, temporal correlation, nonlinear structure, and multi-scale fractal spectrum. The results show that both the load-loss magnitudes and inter-event intervals of blackouts follow strict power-law distributions, with the system exhibiting scaling behavior consistent with SOC theory. The blackout event sequence presents significant long-range positive correlation and self-similarity, confirming a persistent long-term memory effect in the system evolution. Symbolic analysis further reveals the nonlinear fluctuation patterns and burst clustering behavior of the blackout process, reflecting the intermittency and complexity of blackout risks. MFDFA results verify that the blackout sequence has a broad-spectrum multifractal structure across different temporal scales, and Monte Carlo shuffle tests demonstrate that this multifractality mainly arises from intrinsic long-range temporal correlations, rather than being driven solely by heavy-tailed distribution. This study confirms that blackouts in China’s power grid are not random independent events, but present fractal statistical characteristics consistent with the self-organized critical mechanism. The findings provide a novel fractal perspective and quantitative framework for the statistical characterization, operational security assessment, and multi-scale early-warning modeling of blackout risks in China’s large-scale power systems. Full article
(This article belongs to the Special Issue Multifractal Analysis and Complex Systems)
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19 pages, 2342 KB  
Article
An Improved GRU Financial Time Series Prediction Model
by Yong Li
Fractal Fract. 2026, 10(4), 227; https://doi.org/10.3390/fractalfract10040227 - 28 Mar 2026
Viewed by 768
Abstract
Forecasting financial time series (FTS) is essential for analyzing and understanding the dynamics of financial markets. Traditional recurrent neural network (RNN) models often suffer from low prediction accuracy on non-stationary and abruptly changing data, as their gating mechanisms struggle to capture evolving trends [...] Read more.
Forecasting financial time series (FTS) is essential for analyzing and understanding the dynamics of financial markets. Traditional recurrent neural network (RNN) models often suffer from low prediction accuracy on non-stationary and abruptly changing data, as their gating mechanisms struggle to capture evolving trends in FTS. This paper introduces variational mode decomposition (VMD) and multifractal analysis to enhance the gating mechanism of the gated recurrent unit (GRU). By quantifying the changing characteristics of FTS, the proposed model dynamically adjusts the gating weights. In addition, a state fusion strategy is employed to improve the utilization efficiency of historical information. Experiments are conducted using daily data of the SSE 50, CSI 300, and CSI 1000 indices, spanning from 4 January 2002, to 26 December 2025. The results demonstrate that, compared to traditional models, the proposed model better captures the evolving characteristics of FTS and achieves higher prediction accuracy. Full article
(This article belongs to the Special Issue Multifractal Analysis and Complex Systems)
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25 pages, 14211 KB  
Article
Constructing New Fractal-like Particle Aggregates Using Seeded DLA and Concentration Gradient Diffusion Approaches
by Sancho Salcedo-Sanz, Pablo Álvarez-Couso, Luis Castelo-Sardina and Jorge Pérez-Aracil
Fractal Fract. 2026, 10(1), 68; https://doi.org/10.3390/fractalfract10010068 - 19 Jan 2026
Viewed by 543
Abstract
Hybridization of existing fractal aggregate construction methods has been used to obtain new fractal-like structures, with different properties and fractal dimensions to aggregates obtained using the hybridized methods alone. In this paper we propose the hybridization of the Diffusion-Limited Aggregation (DLA) approach with [...] Read more.
Hybridization of existing fractal aggregate construction methods has been used to obtain new fractal-like structures, with different properties and fractal dimensions to aggregates obtained using the hybridized methods alone. In this paper we propose the hybridization of the Diffusion-Limited Aggregation (DLA) approach with other methods for constructing fractal-like aggregates, such as Iterated Function Systems (IFSs), Lindenmayer systems (L-Systems), Strange Attractors (SAs) or Percolation-based fractal construction approaches. The proposed approach is a variation of the seeded DLA algorithm used previously in the literature, which consists of considering existing fractal aggregates as condensation nuclei before the DLA simulation. In this case, we revisit the seeded DLA scheme and test different existing fractals as nuclei, such as Strange Attractors or different IFS fractals. We also introduce a simple algorithm for simulating the diffusion of particle aggregate structures, based on concentration gradient diffusion. We show how different fractal aggregates diffuse using this model, and how the diffused versions of the fractal aggregates can then be used themselves as condensation nuclei for the seeded DLA algorithm, obtaining new fractal aggregates. We characterize the new fractal-like aggregates constructed by means of their fractal dimensions, calculated by using the box-counting approach. The obtained fractal-like aggregates have potential applications in computer graphics and multi-media art, due to their esthetic and visually attractive structures based on particles. Applications of the aggregates in statistical and material physics, as well as the modeling of new aggregate types using condensation nuclei and their applications in the development of algorithms, mathematical operators or antenna design, are also reported. Full article
(This article belongs to the Special Issue Multifractal Analysis and Complex Systems)
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