Reconstructing causal dynamic networks from multivariate time series is a foundational problem in complex systems science. Yet, the key scientific issue is not simply causality detection, but causal interpretability. Interpretable causality is the foundation for testable mechanistic hypotheses, transferable forecasting, and principled decision-making for intervention and control. In real-world complex systems, causal inference is often compromised by noise, missing data, high dimensionality, nonlinearity, time delays, heterogeneity, and partial observability. Classic approaches to interpretable causality yield explicit, inspectable quantities such as causal graphs, coefficients, and governing equations. However, this methodological shift has heightened expectations: AI-extended approaches are increasingly required to recover explicit causal mechanisms rather than opaque predictive dependencies, thereby preserving interpretability. This review summarizes four classic methods and their AI-extended counterparts based on time series data, including Granger frameworks, information-theoretic measures, nonlinear state–space/manifold reconstruction, and mechanistic differential-equation learning. Next, we elucidate their motivations, core principles, the origins of interpretability, and the assumptions required for meaningful conclusions. Finally, we highlight representative applications across climate studies, neuroscience, epidemiology, finance, social science, ecology and molecular biology, followed by a discussion of comparative analysis, open challenges, and future research directions.

Abrupt maritime-energy disruption can generate system-wide production losses before firms and policymakers can adjust. Existing assessments usually emphasize direct exposure or long-run equilibrium responses, which makes them less suitable for short-run risk assessment in energy-dependent production systems. We develop a threshold-cascade framework that combines dual-track dependence topology, edge-level inventories, smooth operability bands, and a separate price-validation step to identify the blockade intensity at which a localized chokepoint shock becomes systemic production loss. The framework is evaluated against the March 2021 Suez blockage and the 2022 Russia-Ukraine producer-price episode, and then applied to a 2026 Strait of Hormuz stress scenario using the Organisation for Economic Co-operation and Development (OECD) Inter-Country Input-Output (ICIO) tables, 2025 edition, with the 2022 benchmark year. Under the baseline 150-day horizon, terminal loss first reaches 50% at about 32% blockade intensity, with a broader calibrated threshold band of 32-46%. Losses spread beyond the point of origin and become concentrated in East and Southeast Asian manufacturing supply chains and in downstream consumer markets after inventories at connected hubs are depleted. Policy experiments show that single-channel interventions shift the threshold only modestly, whereas an integrated package that relaxes logistics, inventories, and upstream scarcity moves the threshold to about 46% in this calibration. The analysis targets the weeks-to-months interval before substitution, contract renegotiation, and broader market adjustments dominate. Within that interval, the model identifies when buffers fail, how production losses spread, and which intervention packages delay systemic disruption.

A maritime chokepoint shock can spread beyond the countries directly connected to the affected route. We develop a monthly resilience screening model for a hypothetical Strait of Hormuz closure that separates direct disruption, scarcity-driven trade squeeze, household access pressure and delayed production risk. Hormuz severity variants and related Red Sea, Black Sea and fertilizer node scenarios test the stability of the modeled mechanism ordering. A three-event historical panel provides a limited external consistency check, not full model validation. In the baseline Hormuz stress test, mean modeled import availability pressure ranges from 0.0023 to 0.0076 on the normalized scale during March–August 2026. Trade squeeze accounts for 92.1–94.5% of that pressure, compared with 5.5–7.9% for direct disruption. Of 161 countries entering model-Alert, 39 later enter model-Crisis or a higher internal state. Crossings concentrate where reallocation pressure coincides with weak household access and delayed production risk. Historical estimates are directionally consistent for some events, although the Suez window exhibits non-flat pre-event coefficients and is not interpreted causally. The state labels are internal thresholds, not humanitarian classifications. The model identifies which propagation layer is binding and how quickly the associated intervention window closes.

A blockade or severe disruption in the Strait of Hormuz would test energy supply resilience by reducing crude oil and LNG availability and by raising routing, freight, insurance, port-handling, warehousing, and transport-support costs. This paper develops a short-run multi-regional input–output stress test to assess where such an energy-route shock enters the production system, how reserves and inventories reduce pass-through, which cross-border links carry residual costs, and where final demand absorbs them. Using the OECD ICIO 2025 edition for 2022, we map the shock to oil and gas extraction, refining, utilities, transport, and transport-support sectors, with an additional premium for major Gulf energy exporters. We propagate the shock for seven input–output rounds under inventory damping. First-round exposure and later-round burden do not coincide, as energy-intensive materials, aviation services, chemicals, minerals, metals, electronics, and machinery face higher downstream costs through material and logistics purchases. With 30% inventory absorption, the upstream energy shock needed for downstream manufacturing to reach a 10% added-cost threshold rises from 73.6% to 85.3%. The results support targeted reserve release, coordinated rerouting, port-logistics priority, inventory management around high-value links, and continuity protection for vulnerable sectors.

The temporal rich-club (TRC) phenomenon, in which a tight and persistent collection of key nodes is formed, is widely observed in real-world settings. However, the underlying mechanisms that drive the formation of TRC structures remain insufficiently understood. This study investigates the TRC phenomenon in international trade through an analysis of 30 time-evolving trade networks of new energy minerals (cobalt, lithium, nickel, and copper) from 1994 to 2023. We select and weight the features in network evolution using the differentiable information imbalance (DII) method and develop an analytical framework to analyze TRC formation and elucidate its evolutionary mechanism. The empirical results demonstrate statistically significant TRC characteristics in these trade networks. Mathematical modeling reveals that TRC emergence is driven by three coexisting mechanisms, including path dependence, degree of homophily, and intrinsic national attributes such as economic development and resource endowment. These findings provide additional insights into the stability and evolution of global energy mineral trade networks.

Early warning of regime switching in a complex financial system is a critical and challenging issue in risk management. Previous research has examined regime switching through analyzing the fluctuation features in a single point in time series; however, it has rarely examined the dynamic spillovers across multivariable time series. This paper develops an early warning model of regime switching that incorporates a spillover network model and a machine learning model. Typical energy prices and stock market indices are selected as the sample data. The key spillover networks can be detected according to the distribution of the network indicators. The early warning signals can be captured by six typical machine learning models, and the random forest model has better performance. The robustness of the model is also discussed. Our study enriches regime switching research and provides important early warning signals for policymakers and market investors.

Reconstructing a dynamic network evolving over time is a fundamental scientific problem for understanding the propagation and diffusion of risk in complex systems. In this paper, we reconstruct the dynamic network of the global stock market at 4927 time points using selfdynamics and inter-dynamics model while considering time factors. Through orthogonal basis decomposition, the dynamic interactions within the stock market are transformed into a linear inverse problem that can be solved using parallel computing. We analyse the evolution of the dynamic interaction among the stock indices and characterize the systemic risk of stock market from a network topology perspective. The results suggest that the occurrence of unexpected events enhances the dynamic interactions among stock indices. The self-dynamics and interdynamics model is capable of identifying stock indices that are more significantly affected by such events. Furthermore, these stock indices tend to be concentrated in countries or regions that are highly correlated with unexpected events. Additionally, the average path weight and clustering coefficient are effective indicators of systemic risk in the stock market. This paper also compares the self-dynamics and inter-dynamics model with Granger test, GARCH-BEKK and DY framework methods and finds that the results are still robust. This method offers distinct advantages in characterizing systemic risk.

To explore the impact of charging infrastructure on electric vehicles (EVs) diffusion, a multi-agent model of EVs-charging infrastructure construction (EV-CIC) is established based on complex adaptive system (CAS). The simulation examines four infrastructure factors and two policy interventions. The results show that the installation rate of private charging piles has the greatest impact, increasing market share by 4% for every 10% rise, followed by the number of public charging piles with 1.58% per 100 units, failure rate with 1.3% per 10% reduction, and charging price with 1% per 10 yuan. High subsidy rates show strong effects in the early stages, while sharing policies for private charging piles show better long-term benefits, increasing market share by 13% compared to non-sharing scenarios. In conclusion, private charging piles, whether through increasing installation rates or enhancing sharing policies, could lead to significant breakthroughs in promoting the development of the EV market.

This paper investigates China's coal price volatility spreaders (CPVSs) from the supply side to locate the volatility source since coal price volatility may destabilize many downstream products' prices or even bring uncertainties to macroeconomic output. Especially in the carbon neutrality context, China's coal market is being reconstructed and responding to imbalances between supply and demand; identifying the CPVSs helps alleviate rising market instability and prevent energy-induced system risk. To achieve this objective, we explore causalities among 938 weekly coal prices reported by different coal-producing areas of China from 2006.9.4 to 2021.7.12 using the transfer entropy method. Then, coal price volatility influence is quantified to identify the CPVSs by conjointly using complex network theory and a rank aggregation method. The validity test demonstrates that the proposed hybrid method efficiently identifies the CPVSs as it correlates to many price determinants, e.g., electricity and coal consumption and generation. The empirical results show that causalities among coal prices changed dramatically in 2016, 2018, and 2020, affected by coal decapacity and carbon neutrality policies. Before 2018, coal-producing provinces with strong demand for coal and electricity, e.g., Jiangxi, Chongqing, and Sichuan, were CPVSs; after 2019, those with comparative advantages in coal supply, e.g., Gansu and Ningxia, were CPVSs. Overall, the coal market is unstable and sensitive to energy policy and external shocks. Policymakers and market participants are recommended to monitor and manage the CPVSs to improve energy security, avoid policy-induced instability and prevent risks caused by coal price fluctuations.

Forecasting all components in complex systems is an open and challenging task, possibly due to high dimensionality and undesirable predictors. We bridge this gap by proposing a data-driven and model-free framework, namely, feature-and-reconstructed manifold mapping (FRMM), which is a combination of feature embedding and delay embedding. For a high-dimensional dynamical system, FRMM finds its topologically equivalent manifolds with low dimensions from feature embedding and delay embedding and then sets the low-dimensional feature manifold as a generalized predictor to achieve predictions of all components. The substantial potential of FRMM is shown for both representative models and real-world data involving Indian monsoon, electroencephalogram (EEG) signals, foreign exchange market, and traffic speed in Los Angeles Country. FRMM overcomes the curse of dimensionality and finds a generalized predictor, and thus has potential for applications in many other real-world systems.

Investigating the correlations between time series is a fundamental approach to reveal the hidden mechanisms in complex systems. However, the estimated correlations often show time-dependent behaviors, which may create uncertainty for decision-making in various scenarios. Thus, forecasting the evolution of these varying correlations may be helpful, but it is still unsolved entirely. We bridge this gap by proposing a data-driven framework: (a) we first embed all the pairwise correlations within a complex system into multivariate correlation-based series by sliding windows; (b) we then identify two different low-dimensional representations of multivariate correlation-based series through delay embedding and dimensionality reduction; (c) finally, multistep ahead predictions of varying correlations can be achieved by training a mapping between two low-dimensional representations. Both model and real-world systems are used to illustrate our framework, including finance, neuroscience, and climate. Our framework is robust and has the potential to be used for other complex systems. Hopefully, forecasting the evolution of correlations in complex systems can be a useful complementary, since existing works mainly focus on the predictions of components within the systems.

Critical transitions in the crude oil price are important in market management. Previous methods have focused on generic early warning signals in complex systems if a critical transition is approaching and ignored the specific feature of the oil price. This paper proposes a heteroscedastic network model in which early warnings of critical transitions are identified based on the community structure of the network representing the dynamic process of a time series. Using WTI crude oil price data, we detect early warning signals of critical transitions. Our findings indicate that major switches exist between early warnings and critical transitions in different periods, and the corresponding features can be analyzed based on the fundamentals and expectations of traders. Importantly, based on the network indicators associated with early warnings, the fundamental features may be similar during certain periods, and the changes in fundamentals and expectations before and after critical transitions are not random. A new complex system perspective is used to explore early warnings for critical transitions, and useful implications for energy-related market investors and policy-makers are provided.

It is widely reported that the functional connectivity estimated by statistical correlations is often varying within nonlinear systems. Generally, these varying correlations between time series are detected by sliding windows. Still, it is unclear how these correlations evolve within a chaotic system. This work intends to give a quantitative framework to identify the dynamics of correlations within chaotic systems. To this end, we embed the pairwise statistical correlations (from time series within a system) into a correlation-based system by sliding windows. This allows for detecting the dynamics of correlations within a complex system through the embedded correlation-based system. Three chaotic systems (i.e., the Lorenz, the Rossler, and the Chen systems) are employed as benchmark examples. We find that both linear and nonlinear correlations within three chaotic systems show chaotic behaviors on some short window sizes, then transit to non-chaotic states with window size increasing. Moreover, the chaotic dynamics of nonlinear correlations exhibit higher uncertainty than the linear one and the original chaotic systems. The chaotic behaviors of correlations within chaotic systems give another evidence of the difficulty of prediction for chaotic systems. Meanwhile, the identified state transitions (concerning the window size) of correlations may provide a quantitative rule to select an appropriate window size for sliding windows.

Multistep prediction is an open challenge in many real-world systems for a long time. Despite the advantages of previous approaches, e.g., step-by-step iteration, they have some shortcomings, such as accumulated errors, high cost, and low interpretation. To this end, Gaussian process regression and delay embedding are used to create a combination framework, namely spatial–temporal mapping (STM). Delay embedding is employed to reconstruct an isomorphic dynamical structure with the original system through a single time series, which provides the fundamental architecture for multistep predictions (interpretation). Gaussian process regression is used to achieve predictions by identifying a mapping between the reconstructed dynamical structure and the original structure. This combination framework outputs multistep ahead predictions in a single step (low cost). We test the feasibility of STM for both model systems, including the 3-species ecology system, the Lorenz chaotic system, and the Rossler chaotic system, and several real-world systems, involving energy, finance, life science, and climate. STM framework outperforms traditional iterative approaches and has the potential for many other real-world systems.

This work investigates the interactions between oil prices and exchange rates of 6 typical oil importers (China, Japan, and India) and exporters (Canada, Russia, and Saudi Arabia) from 2006 to 2022. We employ a novel method to capture their causal interactions, namely pattern causality, and compare the results to that based on the volatility spillover method. The empirical analysis supports most existing findings that oil prices are bidirectional correlated with exchange rates. However, unlike previous studies that only investigate positive and negative causalities, we highlight dark causality as a more complex interaction. Moreover, dark causality suggests that successive increases (decreases) in oil prices tend to drive the exchange rates of oil exporters to act in an oscillatory manner rather than in a purely positive or opposite trend, and vice versa. Furthermore, we also reveal that dark causality shows dominance during crises, e.g., the global financial crisis, the European debt crisis, the epidemic of COVID-19, and the Russia-Ukraine conflict. Revealing three types of causalities between oil prices and exchange rates helps policymakers develop more diversified macroeconomic policies. Moreover, the newly identified dark causality can be a useful indicator for investors to risk management.

The outbreak of the 2022 Russia-Ukraine conflict exacerbated the natural gas supply shortage in European countries. European countries restarted coal-fired power plants to maintain economic and social operations. The uneven distribution of coal resources in the world makes coal international trade inevitable. The intricate trade relations between trading countries have formed a coal trade network. When a country’s coal exports are limited due to geopolitical factors, it will cause coal supply risks. The risk will spread to more countries along the trade network, eventually leading to the collapse of the trade network. This paper builds a crisis propagation model of the coal supply under the Russia-Ukraine conflict using the cascading failure model. The results showed that the Czech Republic, Ireland, Portugal, and Bulgaria become abnormal as the proportion of coal exports β increases. When the Russian Federation reduced its coal exports by 80% and countries maintained only 10% coal exports against crisis, 23 European countries were the worst. Iceland, Ireland, Turkey and other countries were spread by the indirect risk and became abnormal countries. The Czech Republic and Bulgaria were spread by multiple risk and became abnormal countries.

Correlation analysis serves as an easy-to-implement estimation approach for the quantification of the interaction or connectivity between different units. Often, pairwise correlations estimated by sliding windows are time-varying (on different window segments) and window size-dependent (on different window sizes). Still, how to choose an appropriate window size remains unclear. This paper offers a framework for studying this fundamental question by observing a critical transition from a chaotic-like state to a nonchaotic state. Specifically, given two time series and a fixed window size, we create a correlation-based series based on nonlinear correlation measurement and sliding windows as an approximation of the time-varying correlations between the original time series. We find that the varying correlations yield a state transition from a chaotic-like state to a nonchaotic state with increasing window size. This window size-dependent transition is analyzed as a universal phenomenon in both model and real-world systems (e.g., climate, financial, and neural systems). More importantly, the transition point provides a quantitative rule for the selection of window sizes. That is, the nonchaotic correlation better allows for many regression-based predictions.

This paper focuses the various impact of the severity of COVID-19 development on air quality in different types of cities. We analyze the different degrees of improvement of concentrations of six air pollutants (PM2.5, PM10, SO2, NO2, CO and O3) in different types of Chinese cities with difference method and ensemble empirical mode decomposition (EEMD), and then adopt the recursive plots (RPs) and recursive quantitative analysis (RQA) to discuss whether air quality is more difficult to predict during the outbreak. The empirical results indicate that: (1) After the initial outbreak, only the emissions of NO2, CO and PM2.5 declined for the first 1-3 months, and during the fourth to fifth months the emissions of six air pollutants were elevated in most cities; (2) For the cities with serious epidemic situations in Hubei, the air quality is improved significantly, but for the cities experiencing a second outbreak, the air quality was first enhanced and then deteriorated, and the sensitivity of air quality to COVID-19 re-outbreak is decreasing; (3) In comparison, the predictability of AQI has declined in cities with serious epidemic situations in Hubei, but AQI achieves a stable state sooner in cities with mild epidemic.

Early warning is an important and challenging issue in governmental policy-making.This study proposes a skillful spillover network-based machine learning model to provide early warnings of critical transition in energy and stock markets.First, the critical transition of stock and energy time series can be detected using a hidden Markov model.Second, a dynamic spillover network is established, which can help to understand the characteristics of return volatility from the perspective of the time-varying structure of spillover relationships.A machine learning algorithm is employed to model the early warning of critical transition based on the topological structures of the network.The results demonstrated that the proposed model can identify the early warning of critical transition with the warning day, e.g., one day or thirty days, with a high generalization ability.Our study enriches critical transition research and can offer important warning signals for policy-makers and market investors.

Under the influence of national policies, development plans, and subsidies, industries from the same nation perform similarly and inherit a part of their nations' characteristics. Unlike the one-layer input-output network that treats industries as isolated, our model treats industries from the same nation as a relative tight community. Thus, the characteristics of industries are inherited from their nation, which is defined as the nation-based characteristics. To accurately account for the nation-based contributions of each industry, we construct a global input-output NoN (network of networks) model. The global input-output NoN is transformed from the international input-output tables based on hypermatrices and the network of networks model. Based on the global input-output NoN, we propose six nation-based indexes to assess different aspects of industries' nation-based contributions. Our network model and nation-based indexes are applied on the international input-output tables from 2010 to 2015. We confirm the applicability of our model and discover several interesting findings. Overall, the most important one is that some nonmanufacturing industries (e.g., health, education, and public administration) may be more critical than some significant manufacturing industries. Copyright (C) 2022 EPLA