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Fractal Fract
Special Issues
Fractal Analysis and Data-Driven Complex Systems
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Fractal Analysis and Data-Driven Complex Systems

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

Deadline for manuscript submissions: 31 July 2026 | Viewed by 1518

Special Issue Editor

Prof. Dr. J. Alberto Conejero

E-Mail Website
Guest Editor
Institute of Pure and Applied Mathematics (IUMPA-UPV), Polytechnic University of Valencia, E-46022 Valencia, Spain
Interests: applied mathematics; graph theory; data science; interdisciplinary applications of mathematics to computer science, engineering and biology
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Complex systems arising in natural, engineered, and social domains often exhibit multiscale, nonlinear, and self-similar structures that cannot be fully described by traditional analytical tools, and require the use of networks, automata, or agent systems. When describing its behaviour, fractal phenomena usually arise, such as scale invariance, irregularity, and long-range dependence in such systems.

In parallel, the existence of vast data availability, data centers and computational resources have permitted the eclosion of artificial intelligence, which is now considered as a research tool in nearly all science fields. In the present context, data-driven methodologies have enabled the extraction of hidden patterns and governing principles from large-scale and high-dimensional data.

This Special Issue aims to bring together fractal theory, computational methods, and data-driven approaches to advance the modeling, analysis, and understanding of complex systems. The focus is on both theoretical developments and applications, emphasizing how fractal concepts can be integrated with modern data-driven techniques to analyze real-world complex phenomena.

The Special Issue welcomes original research articles and review papers on topics including, but not limited to, the following:

Fractal Theory and Methods

  • Fractal geometry and multifractal analysis;
  • Fractional calculus and fractal operators;
  • Scaling laws and self-similarity in complex systems;
  • Fractal measures, dimensions, and entropy-based metrics.

Data-Driven and Computational Approaches

  • Machine learning and deep learning for fractal feature extraction;
  • Data-driven discovery of scaling laws and fractal structures;
  • Sparse modeling and reduced-order modeling of fractal systems;
  • Anomalous diffusion.

Complex Systems Applications

  • Fractal analysis in physics, materials science, and turbulence;
  • Biological and physiological signal analysis (e.g., EEG, ECG, genomics);
  • Financial markets and economic complexity;
  • Climate, geophysical, and environmental systems;
  • Urban systems, networks, and social dynamics.

Networks and Dynamical Systems

  • Fractal and multifractal properties of complex networks;
  • Data-driven modeling of nonlinear and chaotic dynamics;
  • Fractional Brownian motion;
  • Time-series analysis using fractal-based methods.

Prof. Dr. J. Alberto Conejero
Guest Editor

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

  • fractal geometry
  • multifractal analysis
  • fractional brownian motion
  • time-series analysis
  • scaling laws and self-similarity

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (3 papers)

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Research

17 pages, 573 KB  
Article
PID Control of α-Order Systems in Fractal Time
by Alireza Khalili Golmankhaneh, Inés Tejado, Delfim F. M. Torres, Rawid Banchuin and Hamdullah Şevli
Fractal Fract. 2026, 10(5), 300; https://doi.org/10.3390/fractalfract10050300 - 29 Apr 2026
Viewed by 370
Abstract
This paper presents a novel proportional–integral–derivative (PID) control framework for first α-order systems evolving in fractal time. The main contribution is the extension of classical control theory to systems exhibiting anomalous temporal scaling by employing local fractal derivatives. In contrast to fractional-order [...] Read more.
This paper presents a novel proportional–integral–derivative (PID) control framework for first α-order systems evolving in fractal time. The main contribution is the extension of classical control theory to systems exhibiting anomalous temporal scaling by employing local fractal derivatives. In contrast to fractional-order PID (FOPID) approaches, which primarily model memory effects, the proposed fractal PID framework captures time-scaling behavior arising in non-smooth environments, such as viscoelastic friction and irregular contact surfaces. The closed-loop dynamics are formulated as a second α-order fractal differential equation, from which a characteristic equation is derived to establish conditions for asymptotic stability. It is shown that, for a constant reference input and positive controller gains, the tracking error converges to zero as t→∞. In addition, a quantitative performance analysis demonstrates that the fractal-order α governs temporal stretching: smaller values of α lead to increased rise and settling times and reduced oscillation frequency. The effectiveness of the proposed approach is illustrated through applications to a thermal system with fractal heat input and robotic actuators operating in irregular environments. These results highlight the potential of fractal-time control as a systematic framework for modeling and controlling dynamical systems with non-integer temporal structure. Full article
(This article belongs to the Special Issue Fractal Analysis and Data-Driven Complex Systems)
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22 pages, 25614 KB  
Article
Fractal Modeling and Coordinated Evolution of Railway Networks in China’s Urban Systems: A Dual Perspective of Spatial Distribution and Temporal Accessibility
by Meng Fu, Hexuan Zhang and Yanguang Chen
Fractal Fract. 2026, 10(5), 283; https://doi.org/10.3390/fractalfract10050283 - 24 Apr 2026
Viewed by 347
Abstract
Railways constitute a core component of China’s national comprehensive transportation network, and their spatial organization and temporal accessibility jointly shape transport integration and system efficiency. Identifying their evolution from the dual perspectives of spatial expansion and time compression is therefore of both theoretical [...] Read more.
Railways constitute a core component of China’s national comprehensive transportation network, and their spatial organization and temporal accessibility jointly shape transport integration and system efficiency. Identifying their evolution from the dual perspectives of spatial expansion and time compression is therefore of both theoretical and practical significance. Drawing on fractal theory, this study examines the structural characteristics, evolutionary trends, and driving factors of railway networks in China’s five major urban systems from 2014 to 2024 from a “space–time” dual perspective. The results show that railway networks exhibit a staged pattern of “spatial filling preceding temporal correlation”, with a lag of approximately 1–8 years—about 1 year in the Guangdong–Hong Kong–Macao Greater Bay Area (GBA), 5 years in the Middle Yangtze River (MYR) region and Beijing–Tianjin–Hebei (BTH), and up to 8 years in the Chengdu–Chongqing (CC) region. In addition, clear regional differences are observed: the Yangtze River Delta (YRD) is polycentric, with the greatest potential, projected to continue rapid spatial growth until 2027 and to remain in a fast-growth phase of temporal correlation; GBA is highly coordinated; BTH is developed but characterized by dual-core agglomeration; CC grows rapidly with lagging functionality; and MYR is corridor-dependent with limited potential. These findings indicate that network functionality does not emerge synchronously with infrastructure expansion, but depends on subsequent improvements in operational organization and service capacity. Compared with single-scale-based indicators, the “spatial distribution–temporal correlation” framework more effectively captures network performance and provides quantitative support for transport optimization and coordinated regional development. Full article
(This article belongs to the Special Issue Fractal Analysis and Data-Driven Complex Systems)
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42 pages, 4491 KB  
Article
Fractional Diffusion on Graphs: Superposition of Laplacian Semigroups Incorporating Memory
by Nikita Deniskin and Ernesto Estrada
Fractal Fract. 2026, 10(4), 273; https://doi.org/10.3390/fractalfract10040273 - 21 Apr 2026
Viewed by 427
Abstract
Subdiffusion on graphs is often modeled by time-fractional diffusion equations; yet, its structural and dynamical consequences remain unclear. We show that subdiffusive transport on graphs is a memory-driven process generated by a random time change that compresses operational time, produces long-tailed waiting times, [...] Read more.
Subdiffusion on graphs is often modeled by time-fractional diffusion equations; yet, its structural and dynamical consequences remain unclear. We show that subdiffusive transport on graphs is a memory-driven process generated by a random time change that compresses operational time, produces long-tailed waiting times, and breaks Markovianity while preserving linearity and mass conservation. While the subordination representation and complete monotonicity properties of the Mittag-Leffler function are classical, we develop a graph-based synthesis in which Mittag-Leffler dynamics admit an exact convex, mass-preserving representation as a superposition of Laplacian semigroups evaluated at rescaled times. This perspective reveals fractional diffusion as ordinary diffusion acting across multiple intrinsic time scales and enables new structural and dynamical interpretations of graphs. This framework uncovers heterogeneous, vertex-dependent memory effects and induces transport biases absent in classical diffusion, including algebraic relaxation, degree-dependent waiting times, and early-time asymmetries between sources and neighbors. These features define a subdiffusive geometry on graphs, enabling the recovery of global shortest paths, in contrast to the graph exploration of diffusive geometry, while simultaneously favoring high-degree regions. Finally, we show that time-fractional diffusion can be interpreted as a singular limit of multi-rate diffusion, in an appropriate asymptotic sense. Full article
(This article belongs to the Special Issue Fractal Analysis and Data-Driven Complex Systems)
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