Introduction to Focus Issue: Data-driven models and analysis of complex systems

The 2021 Nobel Prize in Physics put complex systems^1–7 in the limelight.^8,9 It also emphasized the importance of studying natural and man-made non-linear systems to understand their emerging behaviors, such as synchronization, chaos, intermittence, extreme events, and tipping points, to name a few. These behaviors have been researched not only due to the fascinating spatiotemporal patterns they can create but also because of their possible catastrophic consequences. Despite coming from disparate systems—including physical, biological, ecological, atmospheric, technological, social, and financial systems—and having a broad range of temporal and spatial scales, the emergent behaviors require no external or central controller and show universal properties. For example, they strongly depend on the structure of the interactions between the components of a complex system, which are typically represented as a network. Consequently, in order to study the emerging collective behaviors, the complex network structures, to increase forecasting accuracy, and to advance our general understanding of complex systems, we rely on the analysis of numerical and mathematical models, (big) data recordings, and innovative machine learning methods, which is what the works within this Focus Issue put forward.

We organize the 47 published works in our Focus Issue into five sections: innovations in system characterization (Sec. II),^10–25 new analyses of synchronous dynamics and the relevance of interactions in complex systems (Sec. III),^26–37 forecasting and predictability of complex systems (Sec. IV),^38–44 detection of extreme events and emergent behavior (Sec. V),^45–50 and structural and dynamical stability analyses (Sec. VI).^51–55 

The following articles introduce methods to characterize the brain,^10–15 the heart,¹⁶ neuronal models,^17,18 the dynamics of variable stars,¹⁹ semiconductor lasers,²⁰ the climate system,²¹ nonlinear dynamical systems,²² quantifying effects on social systems,^23,24 and network structures.²⁵ 

Mannone et al.¹⁰ propose a general and abstract definition of a neurological disorder. They consider such a disease as an operator that alters the weights of the connections between neural agglomerates, considered the nodes of a brain network. These alterations change communication pathways in the brain and are demonstrated in the examples of epilepsy, Alzheimer–Perusini’s disease, and schizophrenia.

Llonghena et al.¹¹ introduce hyperbolic embedding methods for detecting and localizing perturbations in networks, demonstrating their effectiveness on both simulated and real-world data. Applied to brain networks, the approach quantifies the impact of epilepsy surgery, marking the first use of geometric network embedding in epilepsy research.

The work by Granado et al.¹² employs a multiscale symbolic approach to characterize the complex brain dynamics in the case of refractory temporal lobe epilepsy. The authors utilize various metrics from information theory to quantify the information content of brain dynamics filtered into different frequency bands. Analyzing high-resolution intracranial electroencephalograms recorded prior to and far away from seizure activity, they attain new insights into the relationship between very low (the so-called delta band) and very high-frequent oscillations (so-called HFO ripples) that highlights their potential as biomarkers.

To detect determinism and nonlinear features in time series without phase space reconstruction or surrogate data, Aguilar-Hernández et al.¹³ introduce a robust, noise-resistant index based on Fourier phases. It provides time-frequency resolution, distinguishes regular from chaotic dynamics, and is validated using dynamical models and real-world data, including epileptic brain recordings and paleolake sediment variations.

The work of Makarov et al.¹⁴ introduces a tractable method to estimate the correlation dimension of a signal—which quantifies its complexity—from linear operations, which is particularly relevant to analyze electrophysiological signals (such as local field potentials). The method is shown to provide an improvement of the time resolution and a significant reduction in the number of data needed in the classical estimation of the correlation dimension, which is based on the state-space reconstruction that uses the Takens theorem and then applies the Grassberger–Procaccia method.

Gancio et al.¹⁵ compare different permutation-entropy-based features from electroencephalogram (EEG) recordings in machine learning classification between eyes-open and eyes-closed brain states. The authors contrast temporal and spatial encoding schemes (by averaging over time, by averaging over EEG channels, or by pooling in time and space) to, for example, show that ordinal pattern symbols constructed in the lateral–medial direction provide a better discriminant power than symbols in the anterior–posterior direction.

Andrzejewska et al.¹⁶ employ various uni-variate and bi-variate time-series-analysis techniques to characterize various physiological states of the human organism. They investigate electrocardiographic time series recorded in healthy subjects during a baseline condition, during exercise, and during a recovery phase employing analysis techniques from the information theory. The authors observe that uni-variate analysis techniques that characterize the complexity of a time series as less significant, in contrast to bi-variate techniques that characterize interactions between different aspects of the dynamics of the heart (such as changing inter-beat intervals and changing electrical properties).

The work of Souza et al.¹⁷ introduces an extension to the adaptive-exponential (AdEx) integrate-and-fire neuronal model. The extension is done through the incorporation of non-integer derivatives in a specific biological range, where the authors analyze how changing the fractal dimension influences the firing patterns of a single neuron, including adaptation, tonic spiking, initial, irregular, and regular bursting. Their conclusion is that these patterns depend on the reset parameters and the non-integer order of the derivative, showing that the AdEx model with fractal derivatives enables a better representation of the adaptability, variability, and frequency shown in experiments.

Menesse et al.¹⁸ decomposes transfer entropy into unique, redundant, and synergistic modes of information transfer by using the integrated information decomposition framework. They show that unique information transfer best reveals structural topology in silico and in vitro data of neuronal assemblies, while redundancy obscures direct interactions. This approach improves connectivity mapping by minimizing the influence of higher-order interactions and redundancy.

Thakur et al.¹⁹ combine machine learning approaches with recurrence quantification analysis and recurrence networks for a classification of dynamical states based on time series of system observables. They perform a comprehensive study on the application of analysis techniques to distinguish periodic, chaotic, and hyper-chaotic dynamics and to discern it from noise. Eventually, the authors demonstrate applicability to real world data from the astronomy domain and provide evidence for their approach to achieve comparable classification accuracy in the case of discrete systems.

Boaretto et al.²⁰ investigate the nonlinear dynamics of semiconductor lasers with optical feedback. These systems may qualify as novel photonic information processing systems, but fundamental mechanisms for implementing photonic neurons require further exploration. The authors combine ordinal analysis with machine learning to characterize the spike timing of a laser operating in a chaotic regime. They observe statistical ordinal properties of inter-spike interval sequences that are similar to Flicker noise signals.

Teruya et al.²¹ presents a comprehensive analysis of the equatorial region of Earth’s atmosphere, focusing on the Madden–Julian Oscillation (MJO) and its associated phenomena. The authors employ a normal mode decomposition of atmospheric reanalysis ERA-5 datasets and Sparse Identification of Nonlinear Dynamics (SINDy) to investigate the influence of nonlinearity and moisture on amplitude growth, propagation speed, and coupling mode of equatorially trapped waves. The results show that the MJO is characterized by a combination of Rossby and Kelvin modes, with negative correlations in the amplitude and propagation frequency.

Yılmaz Bingöl et al.²² introduce a method for dimensionality reduction. They combine Reservoir Computing (RC), which efficiently approximates the Koopman operator, with the Hankel Alternative View of Koopman (HAVOK), this way, capturing system dynamics from time-series data. This novel data-driven approach provides a powerful tool for analyzing and predicting the behavior of complex dynamical systems.

Krawczyk and Malarz²³ investigate scientific cooperation on the intercontinental level by creating a collection of about 14 million publications around the papers by one of the highly cited authors working in complex networks and their applications. The authors observe a rank-frequency distribution of the probability of sequences describing continents and number of countries to follow a power law reminiscent of Zipf’s law. Moreover, the number of various continent sequences in a function of the number of analyzed publications grows according to Heap’s law.

Rosillo-Rodes et al.²⁴ study a novel mathematical model for language shifts. Their model describes a group of speakers where two languages are present and is based on the Abrams–Strogatz language competition model. The system studied here is composed of two subgroups of people in which individuals have different perceptions about the languages, taking into account personal preferences and societal opinions. Their results show that when subgroups with opposing language preferences interact strongly, the coexistence of different languages becomes threatened.

The work of Ha et al.²⁵ proposes a clustering coefficient for hypergraphs that can characterize the microscopic organization of higher-order structures. The authors focus on “quads,” the minimal cycle in bipartite graphs consisting of two nodes and two hyper-edges that conforms a square to alternate vertices. They obtain analytical expressions for the quad clustering coefficient (QCC) in random hypergraph models and in regular configurations. When examining real-world hypergraphs, they discover that a large fraction of nodes have a maximal QCC value, which is larger than in the corresponding random synthetic model and is something not seen in classical networks with binary interactions.

The following articles introduce synchronization analyses^26–31 and the relevance of characterizing interactions in different complex systems.^32–37 

Pikovsky and Rosenblum²⁶ propose the so-called Wiener order parameter to quantify synchrony in globally coupled populations of units. The approach is based on characterizing regularity of the mean fields in a population of globally coupled units. The authors discuss pros and cons of their approach adding examples from various paradigmatic coupled systems.

The analytical and numerical work by Jain et al.²⁷ analyzes the synchronization of Kuramoto oscillators that interact through multiplex networks. It proposes a novel type of mathematical solution that explains the emergence of synchronization in these networks and agrees with the numerical experiments. The solution is obtained from reduced models—capturing solely the intra- or inter-layer connectivity of the multiplex network—and is linearly stable if and only if both the intra-layer and inter-layer solutions are stable.

Potratzki et al.²⁸ study the temporal evolution of the complex-valued order parameter for the Kuramoto model with heterogeneous natural frequencies in power grid networks of different sizes, which are partly synthetic or derived from real-world power grid structures. Their results show that the synchronous dynamics depends strongly on the initial conditions and parameter heterogeneity (that is, the disorder of the system), although resemble the synchronization dynamics observed in paradigmatic networks (such as small-world, scale-free, and random models). The authors conjecture that this dependence can be partly explained by the topological and spectral properties of the networks.

The work of Camargo et al.²⁹ analyzes the dynamics and synchronization properties of a ring of linearly coupled chaotic Colpitts oscillators. They find that this network of oscillators shows a phase in which the system is highly sensitive to changes in the coupling parameter, achieving complete synchronization or desynchronization in an irregular or random way as the parameter is changed. They term this behavior as intermittent-like. This intermittent-like phase precedes the completely synchronous phase, which is achieved for stronger coupling strengths and is analyzed using the master stability function formalism.

Zabaleta-Ortega et al.³⁰ analyze chaotic electronic oscillators in a network using normalized persistent entropy (NPE) from persistent homology. NPE effectively characterizes coupling strength and oscillator distance, capturing richer dynamics than the phase-locking value. It identifies transitions to synchronization and differentiates strong coupling (near neighbors) from weak coupling (distant neighbors), making it a valuable tool for analyzing oscillator networks.

Ma et al.³¹ introduce an automated algorithm for detecting cardiorespiratory coordination (CRC), comparing CRC and cardiorespiratory phase synchronization (CRPS) during sleep. CRPS is more sensitive to sleep-stage transitions, and a negative correlation between CRPS and CRC emerges with high breathing frequency fluctuations. These findings enhance our understanding of cardiorespiratory interactions and highlight the clinical potential of the proposed algorithm.

Porta et al.³² re-investigate the state space correspondence strategy based on $k$-nearest neighbor cross-predictability to assess the directionality of cardiorespiratory couplings. The authors investigate stochastic nonlinear bivariate autoregressive (NBAR) processes interacting in a closed loop, as well as recordings of spontaneous variability of heart period and respiration in healthy subjects under different experimental situations. The approach is found to be suitable for identifying the correct directionality in all possible configurations of the complexity of two NBAR processes, and it is useful to disentangle closed-loop cardiorespiratory interactions.

The work of Trebbien et al.³³ provides a statistical analysis based on the principal component analysis of European day-ahead electricity prices between 2019 and 2023. The main contribution of the article is its novel analysis of the spatial pattern of price dynamics in Europe and its correlation with events that impact price dynamics, such as the conflict in Ukraine. In particular, the results of this work explain several discrepancies in how electricity prices increased from the biding zone to the biding zone.

The work of Rath et al.³⁴ is focused on nonlinear causalities in financial markets, which are critical to understanding complex systems, surpassing the limitations of linear correlation. The study introduces a framework that combines transfer entropy and convergent cross-mapping to quantify and interpret linear and non-linear dependencies, offering tools for market analysis, trading strategies, and risk management.

The study by Barnes and coauthors³⁵ reviews the evolution of the phase coherence measure and discusses various numerical quantification methods. The authors then demonstrate that time-localized phase coherence is more robust to noise and artifacts, highlighting its importance for accurate analysis and interpretation of complex physical systems.

The study by Xia and Johnson³⁶ models the spread of harmful online content across interconnected social media communities using a nonlinear dynamical framework. It derives an analytic condition for system-wide outbreaks, showing that even with low infection rates, high inter-community link creation, and low link loss can trigger widespread spreading. Policymakers should address these dynamics when designing strategies to curb harmful content dissemination.

In the work by Gallarta-Sáenz et al.,³⁷ the authors examine how reputation-based interactions affect innovation. Using a dynamic model of knowledge sharing, it shows that moderate reputation influence accelerates discovery compared to random interactions. However, excessive reliance on reputation concentrates knowledge among a few, reducing technological diversity and leaving most individuals without technical skills.

The following articles introduce forecasting analyses,^38–40 network inference methods,^41,42 predictability quantification,⁴³ and control of complex systems.⁴⁴ 

In their perspective, Panahi and Lai³⁸ address the problem of forecasting the occurrence of a critical transition based solely on data without knowledge about the system equations. The authors critically review adaptable reservoir computing as a machine-learning-based approach tailored to solving this prediction problem. They showcase the approach for the prediction of a crisis and amplitude death as well as for creating digital twins of nonlinear dynamical systems and discuss limitations and future perspectives.

Malpica-Morales et al.³⁹ address the problem of accurately predicting electricity day-ahead prices, a prerequisite for competitive electricity markets. To account for concurrent multiple non-stationary effects, the authors develop a data-driven model that combines an $N$-dimensional Langevin equation with a neural-ordinary differential equation. The effectiveness of the proposed framework is exemplified using the Spanish electricity day-ahead market as a prototypical case study.

Duque et al.⁴⁰ introduces a statistical model based on artificial neural networks for the streamflow variability of the Rio Negro, a river in Uruguay. The scope of the model is to forecast flood events one-day ahead, using precipitation data and water flow levels at certain hydrological stations in the Rio Negro Basin. Results reveal stochastic characteristics in the temporal dynamics and chaotic behaviors near dams.

Ding et al.⁴¹ tackle the challenge of reconstructing complex networks and predicting the dynamics from incomplete data. They propose a deep learning framework consisting of network inference, state estimation, and dynamical learning. The suitability of the framework is tested and demonstrated on a large number of synthetic and real-world networks as well as on various discrete and continuous dynamical processes.

Acharya et al.⁵⁶ use the U.S. air transport system, where the connectivity is known and used as a benchmark to evaluate functional network reconstruction methods. It highlights challenges such as nonstationarities, noise, and limited data, exploring the effects of functional measures, detrending methods, and system size on reconstruction accuracy. The findings provide insights that could improve functional network analyses in fields like neuroscience.

Singhal et al.⁴² introduce a framework, sequential inference of network connectivity, to effectively determine network connectivity. The framework, which considers network inference as a bilinear optimization problem, does not require assumptions on network topology or prior knowledge of network dynamics. The authors demonstrate the superior inference accuracy of their iterative technique compared to existing methods by investigating experimental and simulated datasets that comprise oscillatory, non-oscillatory, and chaotic dynamics.

The work by Tarigo et al.⁴³ tackles the problem of quantifying the predictability of a multi-stable system (i.e., one with multiple coexisting attractors) using basin entropy.⁵⁷ Predictability is understood as the probability of ending in a particular equilibrium solution when starting with a given initial condition. Hence, the basin entropy of a multi-stable system measures the average uncertainty in the final states, which is affected by the structure of the boundaries between the attractors. Here, the authors show that basin entropy captures the complexity of a simple multi-stable delayed system, being sensitive to the proximity of a Hopf bifurcation but insensitive to a pitchfork bifurcation.

The paper by Mou et al.⁴⁴ addresses the challenge of controlling complex systems with limited intervention data. It introduces a novel two-stage framework that leverages abundant observational data to pre-train a model of system dynamics using action-extended graph neural networks. This pre-trained model is fine-tuned using a variant of model predictive control while minimizing intervention costs and combining observational data with limited interventions.

The following articles improve the detection of extreme events in the climate system⁴⁵ and financial markets⁴⁶ and analyze the emergence of different collective behaviors.^47–50 

Sonone and Gupta⁴⁵ construct climate networks based on surface air temperature data to investigate tropical cyclones in the Indian Ocean. By combining techniques from the percolation theory with a topological characterization of climate networks, they identify distinct signatures of these cyclones. Importantly, the derived quantities also carry information predictive of an upcoming climate extreme and provide an estimate of cyclone severity.

Rabindrajit et al.⁴⁶ analyze high-frequency stock market order data using a Markov chain model. They reveal active trader participation during high volatility days, where traders use limit orders primarily to influence the market. Similar trading strategies were observed on both high and low-volatility days, with banking stocks demonstrating resilience through recurring full execution orders. These insights can aid in understanding order dynamics and developing trading strategies during extreme macroeconomic events.

Díaz-Guilera et al.⁴⁷ explore the Kuramoto model on a circular topology with excitatory and inhibitory interactions, uncovering new attractor states beyond synchronization. Analytical and computational analysis shows that inhibitory interactions significantly alter the system’s attractor structure, even for identical oscillators.

The numerical work done by Crnkić and Jaćimović⁴⁸ focuses on chimeras and traveling waves of two variants of the Kuramoto model (the one by Abrams et al.⁵⁸ and the conformist–contrarian⁵⁹) with non-uniform coupling in the thermodynamic limit near to and far away from the Poisson manifold. The authors show that chimeras and traveling waves always “breathe” away from the Poisson manifold. Also, they show that the stationary chimera and traveling waves in the conformist–contrarian model are essentially manifestations of the same phenomenon.

Rabus et al.⁴⁹ address the important issue of structure/function relationship in networked dynamical systems. They investigate the relationship between internal dynamics and output statistics in a recurrent neural network trained to sustain behavioral states for durations following a power-law distribution. The authors observe that the relationship between neural activity and behavior is not one-to-one or trivial in complex networks.

The work by Gao et al.⁵⁰ proposes a realistic model of animal motion that incorporates visual lateralization (i.e., distinguishing the involvement of the left and right brain hemisphere) and single information channel, integrating recent experimental observations and focusing on inter-individual attractions (i.e., discarding repulsion). This model explains why some birds with eyes in the front of the head (such as owls) have no emerging collective behavior and suggests that including lateralization allows us to explain the positioning and cohesiveness of animal collective behaviors.

The following articles perform dynamical stability analyses of power-grids,⁵¹ particle systems,⁵² networked dynamical systems,⁵³ and structural stability of networks.^54,55

Titz et al.⁵¹ address the problem of predicting the dynamic stability of power grids that would help to prevent the catastrophic impact of large-scale outages. They improve upon known ways of predicting the dynamic stability of a power grid to line failures by combining classical power system tools, graph-theory-inspired feature-engineering, and interpretable machine learning models. The computational efficiency improvement makes their method well-suited to real-world applications, being tested using a second-order Kuramoto model to simulate line failures for eight classes of synthetic grid models.

The work by Kumar⁵² presents a topological method based on persistent homology for characterizing the dynamical stability of particle ensemble systems. The characterization is done during the system’s evolution by comparing the persistence diagrams of “particle holes” in phase space at different times. The results show that, for the systems considered, the rate of (maximum particle) energy change can be captured by a distance metric between persistence diagrams, as well as transitions from linear to non-linear growth phases. Results are illustrated by applying the method to a dataset of an exponentially expanding circle of points (representing an unstable system) and two particle ensemble systems: a Hamiltonian flow and an electron two-stream instability.

Skardal and Restrepo⁵³ investigate a reservoir computing approach that detects externally provided disturbances in networked dynamical systems. This model-free approach allows to detect disturbances by taking the state of the system as an input to the reservoir, and its training only requires observations of the network under the influence of a known forcing function. The success of this approach is illustrated using numerical simulations of food webs (Lotka–Volterra systems) and neuronal activity (Wilson–Cowan), namely, linearly and nonlinearly forced systems.

Martínez-Martínez et al.⁵⁴ examine bipartite tight-binding graphs with $N$ nodes divided equally into two sets: one with on-site gain and the other with on-site loss, connected randomly with probability $p$⁠. The connectivity is quantified by $α$⁠, the ratio of existing connections to possible connections. The system’s non-Hermitian Hamiltonian H(⁠ $γ$⁠, $α$⁠, $N$⁠), where $γ$ is the gain/loss strength, exhibits pseudo-Hermiticity and, in certain cases, PT-symmetry. In both cases, specific parameter combinations yield a predominantly real spectrum despite non-Hermiticity. The work contributes to stability analysis by exploring how the interplay between $γ$ and connectivity $α$ affects the spectral properties of non-Hermitian bipartite tight-binding graphs.

The study by Dong et al.⁵⁵ examines the robustness of coupled networks, where node functionality depends on internal connectivity and support from another network. A theoretical framework is developed to analyze cascading failures due to attacks, deriving conditions for system stability and collapse thresholds. Their results reveal an abrupt phase transition and show that higher intra- and inter-network link density are required to prevent collapse under stricter coupling conditions. These findings offer insights into designing more resilient infrastructure systems.

One of the key challenges in complex systems research is understanding how individual components interact to produce emergent properties that cannot be predicted from the behavior of individual parts alone. This requires developing well-characterized models that can capture the collective behavior of the system such that we can identify the underlying mechanisms driving the emergent phenomena and their stability. For example, since changes in a network structure can affect the emerging behavior in a complex system, we still need a theory that can quantify perturbations to a network structure analogous to Lyapunov’s stability theory.⁶⁰ 

Complex systems often exhibit dynamic (time-varying) interactions; namely, the components/units of a complex system influence each other in nonlinear and time-dependent ways. Accurately modeling these interactions is challenging due to the complexity and interconnectedness of these systems. For example, recent research has revealed the importance of considering interactions that go beyond the classical pairwise type,^61–63 known as higher-order interactions. Hence, we need to start thinking of dynamic models that can capture the evolving (higher-order) relationships between components and correctly forecast the behavior of the system over time.

Another challenge is that complex systems research often requires analyzing large and heterogeneous data sets to discover patterns, relationships, and trends. The handling and interpreting of big data require advanced data analysis techniques such as topological data analysis (TDA) or machine learning algorithms (ML), among others,^64–67 along with visualization tools to extract meaningful insights and make informed decisions. However, on the one hand, TDA approaches still have several challenges to deal with, e.g., the large computational complexity, interpretability for non-experts, and noise sensitivity. On the other hand, some of the ML and AI tools available still lack transparency and can carry biases due to weak testing, validation, or monitoring.

Our Focus Issue presents cutting-edge results in relation to system characterization and stability analyses, as well as advances in understanding the role of interactions in complex systems and their emergent phenomena, including synchronization, chaos, chimeras, and extreme events. Moreover, this Focus Issue highlights the growing potential of applying data-driven models to understand complex systems and be able to forecast their behavior.

We acknowledge the novel and insightful contributions of the authors of the articles included in this Focus Issue as well as the referees, who made numerous helpful comments and suggestions that strengthened all the articles. We also appreciate the professionalism of the Chaos editorial staff, who provided unfailing support through every step of the process. Eventually, we thank Jürgen Kurths for editing Ref. 23, Elizabeth Cherry for editing Ref. 32, Michael Small for editing Ref. 22, and Ulrich Parlitz for editing Ref. 44. J. H. Martínez acknowledges the support by the Ministerio de Ciencia e Innovación, Spain under Grant Nos. PID2020-113737GB-I00 and PID2023-147827NB-I00. J.H.M. would like to thank J. A. Martínez for valuable conversations.

The authors have no conflicts to disclose.

Johann H. Martínez: Conceptualization (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Klaus Lehnertz: Conceptualization (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Nicolás Rubido: Conceptualization (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this work.