Forecasting early warning signals for abrupt transitions in oil systems: A deep learning-based network and regime switching model

Abstract

Abrupt transitions that are associated with global crisis events can lead to systemic collapse. Previous models for investigating early warning signals for time series–based oil systems focused on the fluctuation characteristics of single points and ignored the dynamic process in the oil series, which is characterized both by historical changes in the oil series and by its nonlinear fluctuations coupled with related variables. In this study, an early warning model that combines a deep learning model, a Markov regime-switching model and a reconstructed hybrid network, including a self-dynamic network and a relationship-dynamic network, is proposed. West Texas Intermediate (WTI) crude oil and natural gas futures daily prices are selected as the sample data. Abrupt transitions are identified, and their characteristics that are influenced by typical events are investigated. The dynamic features can be measured through the structures of the hybrid network; the in-degrees, out-degrees, and weighted degrees of the nodes follow power-law distributions. Early warning signals for abrupt transitions can be effectively captured via the deep learning model. Importantly, the deep learning-based early warning model integrated with the hybrid network outperforms that integrated with the self-dynamic network, with the average accuracies of both models on the training and testing sets exceeding 90%. This work contributes to energy management research and provides early warning tools for policymakers and market investors.

Graphical abstract

Keywords

complex network time series / deep learning / early warning signals / abrupt transition / oil system / crude oil / natural gas

1 Introduction

Global crisis events, such as the Russia–Ukraine War, have become more frequent in recent years. As an important commodity and essential strategic resource, crude oil has drawn the interest of numerous academics and researchers, who have sought to understand the effects of such crises. From the perspective of complex systems, the concept that is associated with global crisis events is abrupt transition—a phenomenon that could lead to systemic collapse (Chen et al., 2025; Rietkerk et al., 2025; Scheffer, 2010; Yan et al., 2025). Specifically, the oil system suddenly shifts from its existing state or behavior to another state, where the system is represented by a time series. A notable example is the profound effect of the COVID-19 pandemic on the oil-related industrial chain, particularly the occurrence of negative prices, which was unprecedented in the history of WTI crude oil. This event signified a transition of the oil system from a stable, desirable state to an unstable, undesirable state, which raised concerns about potential systemic collapse. Consequently, the identification of early warning signals for abrupt transitions in oil systems has become a critical and pressing research challenge.

From the national security perspective, abrupt transitions in oil systems directly threaten energy security, which is a core pillar of national stability. Energy serves as a key leverage point in the geopolitical game; sudden increases in oil time series are often triggered by geopolitical conflicts or supply chain crises and significantly exacerbate inflationary pressures in oil-importing countries. Therefore, early warning signals can facilitate the release of reserves in advance and the diversification of energy imports or new energy dispatch to buffer risk. From the perspective of financial stability, crude oil plays a role as a physical commodity with strong financial attributes. Sudden changes in the oil system can trigger chain effects across related sectors, including energy stocks, futures and derivative markets. Early warning signals cannot only empower policymakers to prevent concentrated risk outbreaks and safeguard market stability but also assist market investors in making scientific asset allocation decisions and implementing robust risk management strategies.

A large body of research has been conducted on abrupt transitions in oil time series that are related to global crisis events because of their significance. Most economic studies have focused on using variable structural changes in the volatility of oil time series to detect such abrupt transitions, such as via a dependence-switching copula delta CoVaR model (Tiwari et al., 2022) or a regime-switching copula approach (Wang et al., 2024). However, these methods have difficulty explaining oil market crashes from a systemic perspective. Scholars in the field of complex systems have given much attention to capturing early warning signals for abrupt transitions in time series, such as critical slowing down (Guttal et al., 2016). Most of these studies have focused on using the fluctuation characteristics of single points in time series and other related factors to capture early warning signals; few have analyzed such signals from the perspective of dynamic processes, which is an approach that can reveal changeable hidden structures in the fluctuations of oil time series from a micro perspective.

Complex network time series play a key role in the study of dynamic processes in financial time series. The core idea of this framework is to transform univariate or multivariate financial time series into windowed time series, which are composed of a sequence of segments extracted from the original time series; then, a complex network or dynamic network is reconstructed from the windowed time series integrated into an economic or statistical model. By analyzing the topological structures of these networks, this framework can reveal the changeable structures of fluctuations in a univariate time series or the relationships across multivariate time series.

In this paper, an early warning model for abrupt transitions in an oil system represented by a univariable oil time series is proposed. The model integrates a complex network time series framework, a deep learning model, and a Markov regime-switching model. Previous studies have focused on the fluctuation characteristics of single points and other related factors in the investigation of early warning signals for abrupt transitions. Our main approach involves focusing on the dynamic process of the fluctuations in an oil time series, which is influenced both by the historical changes in the series and by the coupling of its nonlinear fluctuations with related variables. By integrating a deep learning model with these dynamic processes, the early warning signals for abrupt transitions in the oil system can be effectively detected. Daily futures prices of WTI crude oil and natural gas are used as sample data, and a deep neural network is constructed to evaluate the performance of the model. A robustness analysis is also conducted. This research advances the methodologies for analyzing abrupt transitions in the oil market and deepens the understanding of their early warning signals.

The remainder of this paper is organized as follows: In Section 2, a literature review is presented. In Section 3, the methodology is discussed. In Section 4, the results are reported. The conclusions are summarized in Section 5.

2 Literature review

In this section, the relevant literature is summarized in two parts. The first part focuses on research related to abrupt transitions in oil time series and encompasses two core themes: the detection of abrupt transitions and early warning of abrupt transitions. In the second part, studies of dynamic processes in financial markets that are based on complex network time series frameworks are discussed.

First, numerous studies have been conducted on the detection of abrupt transitions in oil and related oil markets because of their significance. For instance, Zhang and Zhang (2015) developed a Markov regime-switching model to examine behavioral shifts in WTI and Brent oil prices; their analysis identified three primary states of the oil market system, as reflected in the time series of oil price returns. Tiwari et al. (2022) used a dependence-switching copula delta CoVaR model to investigate the dependence structure between India’s sectoral equity markets and crude oil prices in relation to the state of the oil system. Wu et al. (2024) constructed a Markov regime-switching model to investigate the volatility characteristics of Chinese crude oil futures. Wang et al. (2024) established a regime-switching copula approach to explore the effects of global crisis events on the dependence and risk spillover between crude oil and gold. Hasanli (2024) developed time-varying regime-switching models to reanalyze the relationship between oil and natural gas prices and reported that this relationship is inherently asymmetric and prone to significant structural changes. Mari and Mari (2023) established deep learning-based regime-switching models for energy commodity prices (including crude oil prices) to describe the time evolution of market prices and capture the shapes of observed price time series.

Most of the aforementioned models have provided profound insights into the shift in the state (regime) of the oil market. However, the parameter limitations of these models make accurate detection of abrupt transitions difficult; furthermore, explaining oil market crashes or extreme events from a systemic perspective remains challenging. With the development of complex system theory, numerous scholars have focused on early warning signals for abrupt transitions in financial markets. Early warning signals for abrupt transitions represented by time series is regarded as a generic phenomenon that is characterized by an abrupt transition in a system’s state near a critical point, such as a phase transition (Gross et al., 2025; Song and Li, 2024) or a tipping point (Bian et al., 2025; Rietkerk et al., 2025). Indicators of early warning signals, including critical slowing down (Guttal et al., 2016), flickering (Gatfaoui and de Peretti, 2019), and skewness (Jian and Li, 2021), have been widely applied across various systems, such as climate systems (Masuda et al., 2024), ecosystems (O’Brien et al., 2023) and finance (Rye and Jackson, 2020) . For instance, Guttal et al. (2016) used critical slowing down to analyze the systematic risk of oil prices and reported that rising variability (rather than financial meltdowns) serves as a key indicator of early warning signals for abrupt transitions.

According to the literature, various methods for detecting early warning signals for abrupt transitions have demonstrated good applications, especially in the oil market. These models focus on the fluctuation characteristics of single points to capture early warning signals. However, several challenging issues remain in these early warning models. (1) There may be a small change in the state of an oil system when an abrupt transition occurs, which suggesting that capturing significant changes in the fluctuation characteristics of single points in the oil time series could be challenging (Scheffer et al., 2009) . (2) The fluctuation of an oil time series is influenced by historical changes in the oil series and by the coupling of its nonlinear fluctuation with related variables, such as natural gas. In addition, the fluctuations of an oil time series are influenced by many complex factors, such as supply and demand, climate risk, geopolitics, and typical global events (Kumar and Mallick, 2024; Mignon and Saadaoui, 2024; Zhang et al., 2023). This suggests that the characteristics of fluctuations in oil time series are nonlinear and uncertain. The dynamic process of fluctuations in oil time series can capture much more hidden information because the changeable hidden structure of fluctuations in a time series can be recovered from a micro perspective. Therefore, in this paper, an early warning model for abrupt transitions in a univariable oil time series is proposed from a dynamic process perspective.

Second, the dynamic process of fluctuations in oil time series, along with the relationships between oil markets and other markets, has been explored through complex network time series frameworks that integrate an economic model (or statistical model), time series analysis (Qiu et al., 2025), and complex network theory (Feng et al., 2025; Hellmann et al., 2020; Li et al., 2025; Zhong et al., 2025). For instance, Wang et al. (2018) developed an integrated data fluctuation network to characterize the dynamic process of fluctuations in a crude oil time series influenced by its historical fluctuations. An et al. (2023) constructed a complex network from a crude oil time series using the AR-GARCH model and investigated the dynamic process of fluctuations in an oil time series with heteroscedastic characteristics influenced by its historical fluctuation. This dynamic process can aid in the detection of abrupt transitions and early warning signals. To examine the time-varying relationship between the geopolitical risk index and both international and Chinese crude oil time series, Wang and Dong (2024) proposed a complex network in combination with the VAR-DY model, which characterizes the dynamic process of the relationship across these financial markets. Antonakakis et al. (2023) established a connectedness model that integrates network analysis with the TVP-VAR model; this framework was applied to examine the dynamic connectedness between the implied volatilities of oil prices and those of financial assets throughout the COVID-19 pandemic period. Lin et al. (2024) built a quantile-on-quantile connectedness model based on complex network theory, focusing on unraveling the dynamic transmission mechanism between oil price shocks and green bonds—with the specific goal of clarifying the potential of green bonds to act as a hedge against oil price shocks. The previous studies have investigated the dynamic process of a time series or the relationships across multivariable time series in financial markets, and some researches have focused on the early warning of critical transition (An et al., 2023). However, the dynamic process of fluctuations in financial time series, which is characterized both by the historical changes in the series and by the coupling of their nonlinear fluctuations with related variables, has rarely been considered. Therefore, investigating the early warning of abrupt transitions by reconstructing a hybrid network from the time series, where the network can characterize the dynamic process, is necessary.

3 Methodology

In this paper, an early warning model for abrupt transitions in a time series–based oil system is proposed, which integrates a deep learning model, a Markov regime-switching model and a reconstructed hybrid network, including a self-dynamic network and a relationship-dynamic network. The four steps of our proposed approach are illustrated in Fig. 1. First, a time series that represents an oil complex system that is influenced by another market can be transformed into a windowed time series and a bivariate windowed time series using a sliding window. Each windowed time series includes a sequence of segments that evolve into each other. Second, a Markov regime-switching model is established to identify abrupt transitions in the windowed time series, in which the complexity of each segment is measured using entropy theory. Third, a hybrid network is reconstructed from windowed time series using a complex network time series framework and economic models. The topological structure of the hybrid network can be used to characterize the dynamic features of fluctuations in the oil time series, which are characterized both by historical changes in the oil series and by the coupling of its nonlinear fluctuations with related variables, such as natural gas. Fourth, a deep learning-based early warning model that integrates the topological structure is established.

3.1 Windowed time series

The original time series is mapped into windowed time series to characterize the dynamic process of the original time series, as shown in Fig. 1(a). A target time series \{y_i{\}}_{i=1}^T and a related time series \{x_i{\}}_{i=1}^T are given. The target time series is the oil series in this study. A fixed-length sliding window is used to divide these time series into windowed time series {\mathit{W}}^y and {\mathit{W}}^x from left to right with a step. The window length is \omega and the step is τ. The mth segment in the windowed time series can be defined as follows:

(1){\mathit{W}}_m^y\text{=}\{y_i{\}}_{i=i_m}^{T_m},{\mathit{W}}_m^x\text{=}\{x_i{\}}_{i=i_m}^{T_m},

(2){\mathit{W}}^y=\{{\mathit{W}}_1^y,{\mathit{W}}_2^y,\dots ,{\mathit{W}}_m^y,\dots ,{\mathit{W}}_M^y\},

(3){\mathit{W}}^x=\{{\mathit{W}}_1^x,{\mathit{W}}_2^x,\dots ,{\mathit{W}}_m^x,\dots ,{\mathit{W}}_M^x\},

where i_m=\tau \left(m-1\right)+1, T_m=\tau \left(m-1\right)+\omega, and \tau \le \omega (Zhen et al., 2019). Thus, each windowed time series is a sequence of M segments that evolve into each other, which can characterize the dynamic process of fluctuations in the original time series. A bivariate windowed time series is the union of the windowed time series {\mathit{W}}^x and {\mathit{W}}^y.

An important indicator in a windowed time series is the window length. On the one hand, there is only one segment if the window length is equal to the total number of data points in the time series, which is the simplest windowed time series. On the other hand, if the step \tau is one, the windowed time series has the maximum number of segments, which evolve into each other. The maximum amount of overlapping information is found in the adjacent segments.

3.2 Abrupt transition of a target time series

First, for assessing the complexity of a time series, sample entropy is a useful technique that is strongly consistent and is unaffected by the length of the data. It provides a more accurate representation of the entropy in signal analysis than the approximate entropy does (Widodo et al., 2011). The vector of the m-th segment \{y_1^m,y_2^m,\dots ,y_{\omega }^m\} in a windowed time series {\mathit{W}}^y can be formed as follows:

(4){\mathit{Y}}_m\left(i\right)=[y_i^m,y_{i+1}^m,\dots ,y_{i+m-1}^m],i=1\mathrm{t}\mathrm{o}\omega -m+1.

The distance between two segments can be calculated as follows:

(5)\begin{array}{c}{d}_m{[\mathit{Y}}_m\left(i\right),{\mathit{Y}}_m\left(j\right)]=max{[\mathit{Y}}_m\left(i+k\right)-{\mathit{Y}}_m\left(j+k\right)],\\ 0\le k\le m-1.\end{array}

We suppose that r represents the tolerance for accepting matrices, {\mathit{v}}_m\left(i\right) represents the number of d_m{[\mathit{Y}}_m\left(i\right),{\mathit{Y}}_m\left(j\right)] \le r, and {\mathit{\mu }}_m\left(i\right) represents the number of d_{m+1}{[\mathit{Y}}_{m+1}\left(i\right),{\mathit{Y}}_{m+1}\left(j\right)] \le r; for i\ne j, the functions B_i^m(r) and A_i^m(r) can be constructed as follows:

(6)B_i^m(r)={\mathit{v}}_m\left(i\right)/(\omega -m+1),i=1\text{to}\omega -m+1,

(7)A_i^m(r)={\mathit{\mu }}_m\left(i\right)/(\omega -m+1),i=1\text{to}\omega -m+1.

The function B^m\left(r\right) represents the probability that two sequences match at m points, and A^m\left(r\right) represents the probability that two sequences match at m+1 points; they are expressed as follows:

(8)B^m\left(r\right)=({\sum }_{i=1}^{\omega -m}{B}_i^m\left(r\right))/(\omega -m),

(9)A^m\left(r\right)=({\sum }_{i=1}^{\omega -m}{A}_i^m\left(r\right))/\left(\omega -m\right).

On the basis of the above description, the sample entropy can be defined as follows:

(10)SamE\left(m,r\right)=\underset{\omega \to \mathrm{\infty }}{lim}\{-\ln \frac{A^m\left(r\right)}{B^m\left(r\right)}\}.

The sample entropy is estimated as follows:

(11)SamE\left(m,r,\omega \right)=-\mathrm{l}\mathrm{n}[\frac{A^m\left(r\right)}{B^m\left(r\right)}].

On the basis of the above definition, the sample entropy can be used to evaluate the complexity of a segment in the windowed time series {\mathit{W}}^y. A sequence of complexities \{SE\left(1\right),SE\left(2\right),\dots ,SE\left(M\right)\} can be obtained to characterize the dynamic process of the complexity of a segment in {\mathit{W}}^y.

Second, the Markov regime-switching model is a typical abrupt transition model that has been widely applied in various fields, such as energy (de Castro Matias and Tabak, 2025), stocks (Segnon et al., 2024), and climate science (Benkraiem et al., 2025). Its core idea is that a time series operates in two or more regimes (or states), each with a distinct probability distribution, and the switches between regimes are governed by an underlying process or variable. In the Markov regime-switching model (Doornik, 2013), which is a nonlinear time series model, the unobserved random variable s_t is assumed to follow a Markov process, which is defined by the transition probabilities from one regime i to another j:

(12)p_{j|i}=\mathrm{P}\left[s_{t+1}=j|s_t=i\right],i,j=0,1,\dots ,s-1.

Notably, the probability of a system switching from one regime to another depends on the current regime. An example of a first-order Markov regime-switching dynamic regressive model with k regimes is presented in (Kim and Nelson, 2017):

(13)y_t={\alpha }_{s_t}+x_t^{\prime }{\beta }_{s_t}+{\epsilon }_t,{\epsilon }_t\sim N(0,{\sigma }_{s_t}^2),

where y_t denotes the dependent variable, {\alpha }_{s_t} represents a state-dependent intercept term, {\beta }_{s_t} represents a state-dependent regression coefficient, x_t^{\prime } represents the independent variable, and {\epsilon }_t represents a state-dependent error term that is independent and identically distributed. If the dependent variable can be modeled as noise around a constant intercept that changes across different regimes, the model simplifies to y_t={\alpha }_{s_t}+{\epsilon }_t, {\epsilon }_t\sim N(0,{\sigma }_{s_t}^2). The model parameters are estimated via the maximum likelihood method. For instance, given two regimes = \{0,1\}, a segment in the windowed time series {\mathit{W}}^y can be mapped into a regime R\left(m\right)\in \{0,1\}. Importantly, the regime means the low or the high one.

On the basis of the above model, a sequence of regimes \{R\left(1\right),R\left(2\right),\dots ,R\left(M\right)\} is obtained from the sequence \{SE\left(1\right),SE\left(2\right),\dots ,SE\left(M\right)\}. The switches between two different regimes are subsequently detected; thus, an abrupt transition is identified.

3.3 Reconstruction of the hybrid network

According to dynamics theory, a complex system evolves in accordance with dynamical laws that are governed by the self-dynamics of the agents and the interactions among the agents. We consider a dynamical complex system that comprises N agents, which are denoted as x_1,x_2,\dots ,x_N. We let f\left(x_i\right) represent the baseline dynamics of agent x_i. The coupling function K_{ij}(x_i,x_j) represents the interaction between agents x_i and x_j, and the vector \dot{x_i}\left(t\right)\in {\mathbb{R}}^n represents the state of agent x_i at time t. The dynamic behavior of the complex system can be described as follows:

(14)\dot{x_i}\left(t\right)=f\left(x_i\right)+{\sum }_{i\ne j}{K}_{ij}(x_i,x_j).

In this study, a hybrid network including a self-dynamic network and a relationship-dynamic network is reconstructed from the windowed time series, where the model integrated the dynamical theory and a complex network time series framework. The self-dynamic network is reconstructed from a windowed time series, which characterizes the dynamic process of the fluctuation in the target time series influenced by its historical fluctuation. The relationship-dynamic network is reconstructed from a bivariate windowed time series, which can reflect the dynamic process of nonlinear relationship between the target time series and the related time series.

3.3.1 Reconstruction of the self-dynamic network

Step 1: Fluctuation in a segment via GARCH model

Generalized autoregressive conditional heteroscedasticity (GARCH)-type models are the important ones to measure the fluctuation feature of a time series with heteroscedastic (Horváth et al., 2025; Nelson, 1990). In this section, the GARCH (1,1) model is established to measure the fluctuation feature of a segment {\mathit{W}}_m^y in the windowed time series {\mathit{W}}^y, which is defined as follows:

(15)y_t^m=\mu +{ϵ}_t,

(16){ϵ}_t={\sigma }_t{\epsilon }_t,{\epsilon }_t\sim \mathrm{N}(0,1),

(17){\sigma }_t^2=\lambda +\alpha {ϵ}_{t-1}^2+\beta {\sigma }_{t-1}^2,

where y_t^m denotes the value of the tth data point within this segment. The mean equation is constant, where \mu serves as the parameter. The term {\epsilon }_t represents the residual, which is assumed to follow a normal distribution. The parameter \lambda corresponds to the long-term average value. The coefficient \alpha characterizes the ARCH effect, whereas the coefficient \beta describes the GARCH effect. A stationarity test is conducted in each segment. Additionally, the ARCH effect of each segment is examined using the Ljung–Box test, with the significance level set to 5%. If the segment passes either the stationarity test or the ARCH effect test, it is labeled “T”; otherwise, it is labeled “F.”

Step 2: Extraction of the GARCH-based fluctuation patterns by coarse-graining

The GARCH-based fluctuation pattern that corresponds to each segment is derived using the equal-width coarse-graining model. Specifically, the four parameters {\mu , \lambda ,\alpha ,\beta} are assigned to distinct intervals through the coarse-graining process. These intervals are then combined with the results of the ARCH effect test and the stationarity test to generate the GARCH-based fluctuation pattern. For instance, we consider a segment that passes both tests, with the parameters \mu =0.01, \lambda =0.03, \alpha =0.72, and \beta = 0.15. The corresponding GARCH-based fluctuation pattern is expressed as \mu (0,08]\lambda (0,0.05]\alpha (0.7,0.8]\beta (0.1,0.2]TT. This pattern suggests the main heteroscedastic characteristics of the segment during a short-term period. The windowed time series can subsequently be transformed into a sequence of GARCH-based fluctuation patterns that evolve into each other.

Step 3: Construction of a self-dynamic network

The sequence of the GARCH-based fluctuation patterns is transformed into a self-dynamic network. Each node represents the type of GARCH-based fluctuation pattern, each directed edge represents the evolution from one GARCH-based pattern to another, and each weight represents the frequency of the evolution between two patterns.

3.3.2 Reconstruction of the relationship-dynamic network

Step 1: Nonlinear relationship in a segment via a polynomial regression model

A polynomial regression model is used to quantify the nonlinear relationship between two variables in each segment of a bivariate windowed time series. A univariate polynomial regression model that describes the relationship between variables y and x is defined as follows:

(18)y={\beta }_0+{\beta }_{1}x+\dots +{\beta }_n{x}^n+\epsilon ,

where {\beta }_0 denotes the constant term, {{\beta }_1,\dots ,\beta }_n represent the regression coefficients, and \epsilon represents the error term. We assume that the segment contains \omega data points. We let Y denote the matrix that is associated with the segment {\mathit{W}}_m^y in the windowed time series {\mathit{W}}_{}^y and X denote the matrix that is associated with the segment {\mathit{W}}_m^x in the windowed time series {\mathit{W}}_{}^x. This model is described as follows:

(19)Y=\beta X+\epsilon

(20)\begin{array}{l}Y=\left[\begin{array}{c}{y}_1^m\\ ⋮\\ y_{\omega }^m\end{array}\right],\\ X=\left[\begin{array}{cccc}1& x_1^m& \dots & {\left(x_1^m\right)}^n\\ \dots & \dots & \dots & \dots \\ 1& x_{\omega }^m& \dots & {\left(x_{\omega }^m\right)}^n\end{array}\right],\\ \beta =\left[\begin{array}{c}{\beta }_0\\ ⋮\\ {\beta }_n\end{array}\right],\\ \epsilon =\left[\begin{array}{c}{\epsilon }_0\\ ⋮\\ {\epsilon }_{\omega }\end{array}\right].\end{array}

Furthermore, a stationarity test is also performed for each segment in the two windowed time series. A segment is labeled “T” if it passes the stationarity test and “F” if it fails the test.

Step 2: Extraction of the relationship-based fluctuation pattern by coarse-graining

Following the quantification of segments using the polynomial regression model, an equal-width coarse-graining model is established to partition the regression parameters { {\beta }_0,…,{\beta }_n} into distinct intervals. By integrating these parameter intervals with the stationarity test results of the corresponding segments, a relationship-based fluctuation pattern is generated. This pattern captures the main characteristics of the nonlinear relationship between two variables in a segment. Afterward, a sequence of relationship-based fluctuation patterns that evolve into each other is obtained.

Step 3: Construction of a relationship-dynamic network

The sequence of relationship-based fluctuation patterns is transformed into a relationship-dynamic network. Each node is defined as a unique type of relationship-based fluctuation pattern; each directed edge represents the evolution from one pattern to another; and the weight of each edge represents the frequency of the evolution between the corresponding two patterns.

3.3.3 Topological structure of the hybrid network

The hybrid network includes a self-dynamic network and a relationship-dynamic network. In each subnetwork of the hybrid network, the key network indicators of the nodes are analyzed, including the in-degree, out-degree, weighted degree, betweenness centrality, closeness centrality, and clustering coefficient. These indicators are defined as follows:

The in-degree (ID) of a node represents the number of neighboring nodes that evolve into it. The out-degree (OD) of a node represents the number of neighboring nodes that it evolves into. The weighted degree (WD) of a node is defined as the sum of the frequencies of evolution between it and its adjacent nodes. The betweenness centrality (BC) of a node reflects the degree to which it acts as a “bridge” for evolution between nonadjacent nodes in the network. The closeness centrality (CC) of a node quantifies the average shortest path distance from it to all the other nodes in the network. The clustering coefficient (LC) of a node characterizes the degree of interconnection between its direct neighboring nodes.

A segment is associated with the combined topological structures of the self-dynamic network and the relationship-dynamic network. We define the network indicators of the mth segment that correspond to the self-dynamic network as I{D}_{\mathrm{o}}^m, O{D}_{\mathrm{o}}^m, W{D}_{\mathrm{o}}^m, B{C}_{\mathrm{o}}^m, C{C}_{\mathrm{o}}^m, and L{C}_{\mathrm{o}}^m. Similarly, the network indictors of the mth segment that correspond to the relationship-dynamic network are denoted as I{D}_{\mathrm{o}\mathrm{n}}^m, O{D}_{\mathrm{o}\mathrm{n}}^m, W{D}_{\mathrm{o}\mathrm{n}}^m, B{C}_{\mathrm{o}\mathrm{n}}^m, C{C}_{\mathrm{o}\mathrm{n}}^m and L{C}_{\mathrm{o}\mathrm{n}}^m. A topological structure matrix {\mathit{F}\mathit{e}\mathit{a}\mathit{t}\mathit{u}\mathit{r}\mathit{e}(\mathit{W}}^y)can be constructed, which reflects the dynamic feature of the fluctuation of a segment in the windowed time series:

(21)\left[\begin{array}{c}I{D}_{\mathrm{o}}^1,O{D}_{\mathrm{o}}^1,W{D}_{\mathrm{o}}^1,B{C}_{\mathrm{o}}^1,C{C}_{\mathrm{o}}^1,L{C}_{\mathrm{o}}^1,I{D}_{\mathrm{o}\mathrm{n}}^1,O{D}_{\mathrm{o}\mathrm{n}}^1,W{D}_{\mathrm{o}\mathrm{n}}^1,B{C}_{\mathrm{o}\mathrm{n}}^1,C{C}_{\mathrm{o}\mathrm{n}}^1,L{C}_{\mathrm{o}\mathrm{n}}^1\\ \dots \\ I{D}_{\mathrm{o}}^m,O{D}_{\mathrm{o}}^m,W{D}_{\mathrm{o}}^m,B{C}_{\mathrm{o}}^m,C{C}_{\mathrm{o}}^m,L{C}_{\mathrm{o}}^m,I{D}_{\mathrm{o}\mathrm{n}}^m,O{D}_{\mathrm{o}\mathrm{n}}^m,W{D}_{\mathrm{o}\mathrm{n}}^m,B{C}_{\mathrm{o}\mathrm{n}}^m,C{C}_{\mathrm{o}\mathrm{n}}^m,L{C}_{\mathrm{o}\mathrm{n}}^m\\ \dots \\ I{D}_{\mathrm{o}}^M,O{D}_{\mathrm{o}}^M,W{D}_{\mathrm{o}}^M,B{C}_{\mathrm{o}}^M,C{C}_{\mathrm{o}}^M,L{C}_{\mathrm{o}}^M,I{D}_{\mathrm{o}\mathrm{n}}^M,O{D}_{\mathrm{o}\mathrm{n}}^M,W{D}_{\mathrm{o}\mathrm{n}}^M,B{C}_{\mathrm{o}\mathrm{n}}^M,C{C}_{\mathrm{o}\mathrm{n}}^M,L{C}_{\mathrm{o}\mathrm{n}}^M\end{array}\right].

Notably, when the coupling of nonlinear relationships with the related variables in the fluctuations in the target time series is ignored, the hybrid network simplifies to a self-dynamic network. Correspondingly, the topological structure matrix {\mathit{F}\mathit{e}\mathit{a}\mathit{t}\mathit{u}\mathit{r}\mathit{e}(\mathit{W}}^y) can be simplified as follows:

(22)\left[\begin{array}{c}I{D}_{\mathrm{o}}^1,O{D}_{\mathrm{o}}^1,W{D}_{\mathrm{o}}^1,B{C}_{\mathrm{o}}^1,C{C}_{\mathrm{o}}^1,L{C}_{\mathrm{o}}^1\\ \dots \\ I{D}_{\mathrm{o}}^m,O{D}_{\mathrm{o}}^m,W{D}_{\mathrm{o}}^m,B{C}_{\mathrm{o}}^m,C{C}_{\mathrm{o}}^m,L{C}_{\mathrm{o}}^m\\ \dots \\ I{D}_{\mathrm{o}}^M,O{D}_{\mathrm{o}}^M,W{D}_{\mathrm{o}}^M,B{C}_{\mathrm{o}}^M,C{C}_{\mathrm{o}}^M,L{C}_{\mathrm{o}}^M\end{array}\right].

3.4 Deep learning-based early warning model

In this section, a deep learning-based early warning model that integrates the topological structure matrix is proposed. Deep learning is a popular subset of machine learning, and deep learning frameworks are advanced versions of artificial neural networks (ANNs) (Dang et al., 2024; Peng et al., 2025; Shen et al., 2025; Yu and Wang, 2024; Zhi et al., 2024). As a common type of deep learning algorithm, deep neural networks are also known as multilayer perceptrons or feedforward artificial neural networks. The core idea of deep neural networks is to establish multilayer neural networks to simulate the hierarchical learning method of the human brain. A deep neural network is trained via stochastic gradient descent using back-propagation, which performs well on tabular data. The deep learning-based early warning model that is proposed in this paper can be described as follows:

First, early warning signals for abrupt transitions are extracted. In a sequence of regimes \{R\left(1\right),R\left(2\right),\dots ,R\left(M\right)\}, where R\left(m\right)\in \{0,1\}, abrupt transitions occur when the regimes change between adjacent time steps. An abrupt transition {SM}_m is expressed as follows:

(23)SM(m)=\{\begin{array}{l}1,R\left(m\right)\ne R\left(m-1\right)\\ 0,R\left(m\right)=R(m-1)\end{array}.

We let k denote the number of early warning days. If an abrupt transition occurs at time m (i.e., SM(m)=1), the early warning signals for this switch are set to 1 for the k days that precede the transitions: EM(m-i) = 1 for i=1,2,\dots ,k. For all the other time steps, the early warning signals are set to 0. The initial value of all the early warning signals is 0. Via this process, a sequence of early warning signals \{EM\left(1\right),EM\left(2\right),\dots ,EM\left(M\right)\}, where EM\left(m\right)\in \{0,1\}, is obtained.

We consider an example sequence and identify the corresponding sequence of early warning signals for an abrupt transition. For a given sequence of regimes with two types {0,0,0,0,0,0,1,1,1,1}, the switching of regimes is from 0 to 1. Thus, the sequence of the abrupt transition is {0,0,0,0,0,0,1,0,0,0}. If the number of early warning days is four, the corresponding sequence of early warning signals is {0,0,1,1,1,1,0,0,0,0}.

Second, a deep neural network–based early warning model is constructed. It consists of three key steps: (1) The input data of the model are the topological structure of the hybrid network (the topological structure matrix), which reflects the dynamic characteristics of fluctuations in the target time series. The output data are the sequence of early warning signals. (2) The data set is randomly partitioned into training and testing sets. A deep neural network is established, and it is trained by using the training set to learn the nonlinear relationship between the topological structures of the hybrid network and early warning signals. (3) The trained deep neural network is validated by using a testing set to assess its performance and predict early warning signals. TP, TN, FP, and FN represent the numbers of true-positive, true-negative, false-positive, and false-negative results, respectively. The performance of the model is evaluated via the following parts:

(24)\mathit{\text{Precision}}=TP/(TP+FP),

(25)\mathit{\text{Recall}}=TP/(TP+FN),

(26)\text{F-measure}=2\times \mathrm{P}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\times \mathrm{R}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}/(\mathrm{P}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}+\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}),

(27)\text{Accuracy}=(TP+TN)/(TP+TN+FP+FN),

where the \text{F-measure} is a balance indicator for precision and recall. In addition, the performance of the model depends on the selected parameters of the model and the empirical data.

4 Results

4.1 Data

WTI crude oil futures prices are among the most representative global benchmark oil prices and are selected as the sample data. Natural gas futures prices are also selected because there are strong co-movements between natural gas and crude oil prices, including spillovers (Dai and Zhu, 2022), a nonlinear Granger causality link (Geng et al., 2017), and a symmetric and nonlinear relationship (Atil et al., 2014). Previous research has shown that the co-movements between natural gas and crude oil change significantly at intervals before and after both global and typical events (e.g., the Russia–Ukraine conflict), which are the key factors that affect abrupt transitions in oil systems. The time span of the selected daily futures prices ranges from January 20th, 2014, to April 17th, 2025, with the data sourced from the official website of the US Energy Information Administration. Specifically, the futures prices of WTI crude oil and natural gas are quoted in US dollars per barrel and US dollars per million British thermal units (Btu), respectively. The fluctuation trends of WTI crude oil and natural gas futures prices are illustrated in Fig. 2, which are more appropriate for characterizing the dynamic process of fluctuations in oil time series.

To evaluate the performance of the proposed model, simulation experiments are conducted by following the steps that are outlined in the Methodology section. First, oil and the bivariate windowed time series can be obtained from crude oil and natural gas log return time series. Second, a hybrid network is constructed from these windowed time series, which includes a self-dynamic network and a relationship-dynamic network. The self-dynamic network describes the dynamic process of the fluctuations in the oil time series that are influenced by its historical changes. The relationship-dynamic network characterizes the dynamic process of the nonlinear relationship between the oil and natural gas time series. The topological structures of these two subnetworks are analyzed. Third, the complexity of the oil windowed time series is quantified on the basis of entropy theory. Moreover, a Markov regime-switching model is used to detect abrupt transitions in the oil windowed time series. Finally, a deep neural network–based early warning model is constructed by integrating the topological structures of the hybrid network and the early warning signals for abrupt transitions. The performance of the model is discussed.

4.2 Distribution of nodes in the hybrid network

In this study, the topological structures of the two subnetworks in the hybrid network, which can help to capture the early warning signals for abrupt transitions in the oil time series, are explored. First, a sensitivity analysis of the window length is conducted, as this parameter serves as a key indicator in the reconstruction of the hybrid network. Second, the distributions of nodes in these two subnetworks are analyzed.

4.2.1 Sensitivity analysis of the window length

In this section, the sensitivity of the window length is examined. When the window includes all the sample data, there is only one window in the windowed time series. In this case, the model is the simplest, as it implies the presence of only one node in the hybrid network. Because the oil time series serves as the target time series, the window length is selected on the basis of two key factors: the diversity of GARCH-based fluctuation patterns in the self-dynamic network and its associated economic significance. The diversity of the fluctuation patterns is reflected by the numbers of nodes and edges because each node is a type of GARCH-based fluctuation pattern. The results are shown in Fig. 3. They indicate that both the number of nodes and the number of edges tend to decrease as the window length increases. Additionally, the reconstruction of the self-dynamic network is based on the GARCH model; thus, economic significance must be considered. For instance, if the window length is too small (e.g., 80), the establishment of a GARCH model in a segment in the oil windowed time series may have no economic significance. If the window length is too large (e.g., 1000), the numbers of nodes and edges are too small. This finding indicates that the diversity of GARCH-based fluctuation patterns is limited. In summary, the selection of an appropriate window length can be based on the actual situation; in this study, a window length of 260 days is used to reconstruct both the self-dynamic network and the relationship-dynamic network. This finding indicates that the diversity of GARCH-based fluctuation patterns is acceptable and that the window length corresponds approximately to an economic cycle of one year.

4.2.2 Distribution of the indicators in the self-dynamic network

To understand the characteristics of the dynamic process of the fluctuations in the oil time series that are influenced by its own historical fluctuations, in this section, the distribution of the indicators in the self-dynamic network are examined. Notably, a node is a type of GARCH-based fluctuation pattern that reflects the main heteroscedastic features of a segment in the oil windowed time series.

The distributions of the network indicators in the self-dynamic network are illustrated in Fig. 4. A double-logarithmic plot of the in-degree of each node versus its probability is presented in Fig. 4(a). The in-degree of a node refers to the number of incoming links, which quantifies how many different types of GARCH-based fluctuation patterns evolve into the target pattern. The results indicate that the in-degrees of the nodes follow a power-law distribution, which indicates that only a few types of GARCH-based fluctuation patterns with greater in-degrees play key roles in receiving incoming links. A double-logarithmic plot of the out-degree of each node versus its probability is shown in Fig. 4(b). The out-degree of a node corresponds to the number of outgoing links, which reflects the number of neighboring types of GARCH-based fluctuation patterns that evolve from the target pattern. The out-degrees of the nodes also follow a power-law distribution. This finding suggests that only a few types of GARCH-based fluctuation patterns with larger out-degrees play key roles in outgoing links.

The distribution of the weighted degree is illustrated in Fig. 4(c). The weighted degree of a node represents the sum of the frequencies of evolution between it and its neighboring nodes. The results demonstrate that the weighted degrees of the nodes follow a power-law distribution, which indicates that a few types of GARCH-based fluctuation patterns are associated with higher weighted degrees, which frequently appear throughout the dynamic process. The distribution of the betweenness centrality is presented in Fig. 4(d). The results reveal that the betweenness centralities of the nodes also follow a power-law distribution. This suggests that only a few types of GARCH-based fluctuation patterns have greater betweenness centralities and thus play key “bridging” roles in the dynamic process. Finally, the distributions of the closeness centrality and clustering coefficient are presented in Figs. 4(e) and (f), respectively. The results indicate that the distributions of these two metrics can facilitate the identification of the important types of GARCH-based fluctuation patterns: patterns that are associated with a higher closeness centrality or clustering coefficient have a greater probability of playing a key role in the dynamics process.

4.2.3 Distributions of the indicators in the relationship-dynamic network

In this section, the distributions of the indicators in the relationship-dynamic network are investigated. These distributions reflect the dynamic features of various types of relationship-based fluctuation patterns across the entire sample period. The types of relationship-based fluctuation patterns represent the key features of the nonlinear relationship between oil and natural gas in each segment of the bivariate windowed time series.

The distributions of the indicators of the relationship-dynamic network are presented in Fig. 5. Specifically, the distribution of the in-degree, where a node in the network represents a type of relationship-based fluctuation pattern, is presented in Fig. 5(a). The results indicate that the in-degrees of the nodes follow a power-law distribution. This suggests that there are only a few types of relationship-based fluctuation patterns with higher in-degrees, which play significant roles in receiving incoming links. The distribution of the out-degree is displayed in Fig. 5(b). The results show that the out-degrees of the nodes follow a power-law distribution. This implies that only a small number of relationship-based fluctuation patterns with larger out-degrees are critical for generating outgoing links.

The distribution of the weighted degree is shown in Fig. 5(c). The findings reveal that the weighted degrees of the nodes also follow a power-law distribution, which indicates that few relationship-based fluctuation patterns appear frequently throughout the dynamic process. Finally, the distributions of the betweenness centrality, closeness centrality, and clustering coefficient are presented in Figs. 5(d)−5(f), respectively. The results demonstrate that the most important types of relationship-based fluctuation patterns with higher probabilities of exhibiting high betweenness centrality, closeness centrality or clustering coefficient can be identified on the basis of these distributions.

4.3 Abrupt transition of the oil system

To identify abrupt transitions in the oil system from the time series data, we first calculate the sample entropy of each segment of the oil windowed time series, which quantifies the complexity of the segment. The dynamic feature of the complexity in a segment that is obtained using sample entropy is presented in Fig. 6, with the black line representing the mean value. The results demonstrate that the complexity of a segment in the oil windowed time series changes with time, with periods of significantly lower or higher complexity relative to the mean persisting for extended durations. For example, the segment maintained a low value consistently prior to June 2015. Second, we detect the abrupt transitions in the oil windowed time series. The probability distribution of the segment complexity across two distinct regimes, where Regime 1 and Regime 2 correspond to the low and high regimes, respectively, is presented in Fig. 7. The findings reveal that the complexity of the segment is highly likely to be categorized within one of the two regimes at any given time.

The switches between the two regimes are illustrated in Fig. 8, which captures multiple abrupt transitions between the high and low regimes. Initially, the segment remained entrenched in the low regime until June 2015, which indicated a sustained period of lower complexity. This phase coincided with the sharp slump in crude oil prices that started in the second half of 2014, which was driven primarily by the US shale oil revolution, which significantly increased the global oil supply. Subsequently, the high regime persisted for several years until March 2020. During this period, crude oil prices increased steadily from 2016 to October 2018. This volatile recovery was attributed mainly to the OPEC production cut agreements, which helped improve the balance between global oil supply and demand. However, concurrent with production cuts by OPEC, the United States actively increased its crude oil output. Additionally, prolonged China–US trade frictions, coupled with a gradual shift toward a generalized supply–demand balance in the global oil market amid slackening oil demand, contributed to the observed price volatility. Since 2019, the crude oil market has exhibited a moderately stagnant price trend.

Between March 2020 and April 2021, the series reverted to the low regime, which is reflected in the lower complexity of the oil windowed time series. This period was followed by a return to the high regime until December 2021, although the time spent in the high regime was shorter than that preceding March 2020. The series then transitioned back to the low regime, which persisted for a short period. These frequent shifts indicate increasing uncertainty in the complexity of the oil windowed time series from March 2020 to May 2022. The COVID-19 pandemic triggered a collapse in global oil demand during this period, which was most notably exemplified by the WTI crude oil price plunging below zero in April 2020. In early 2022, the outbreak of the Russia–Ukraine conflict led to a significant surge in oil prices, which prompted the complexity of the oil windowed time series to shift from the low to the high regime in May 2022. The key drivers of this shift included the embargo by the EU on Russian crude oil, which forced Russia to redirect its exports to Asia, thereby causing freight and insurance costs to increase, as well as the looming impact of US sanctions on Iran’s crude oil exports, which were set to take effect in 2025.

4.4 Deep learning-based early warning model

4.4.1 Model performance on the basis of the self-dynamic network

In this section, the performance of a deep learning-based early warning model integrated with the self-dynamic network is evaluated. We first construct a database that consists of input and output data sets. The input data set comprises the topological structure of the self-dynamic network, whereas the output data set contains early warning signals for abrupt transitions. The database is then randomly split into a training set (80% of the total data) and a testing set (20% of the total data). Notably, the minority-class samples of early warning signals can lead to a bias toward the majority-class data. A cross-sectional model is established by preprocessing training data (Menardi and Torelli, 2014). A deep neural network with three hidden layers, which is a widely used deep learning framework, is used to train the early warning model on the training set, which is followed by performance validation of the trained model on the testing set. The performance of the model depends mainly on the number of early warning days and the parameters of the deep neural network.

The performance of the model across different numbers of early warning days is illustrated in Fig. 9, and the maximum, minimum, and average values of key metrics for the early warning model are summarized in Table 1. The dynamic accuracy on both the training and testing sets is presented in Fig. 9(a), which shows how the accuracy varies with the number of early warning days. On the training set, the dynamic accuracy ranges from 0.851 to 0.947, with an average of 0.909. On the testing set, the dynamic accuracy is between 0.778 and 0.933, with an average of 0.854. The fluctuations in accuracy tend to decrease as the number of early warning days increased. The dynamic precision and recall on the training and testing sets, which reveal their time-varying characteristics, are shown in Figs. 9(b) and 9(c), respectively. The average precision is 0.968 on the training set and 0.996 on the testing set. In contrast, the average recall is 0.853 on the training set and 0.846 on the testing set. These findings show that the average precision is greater than the average recall.

As a balanced metric that integrates precision and recall, the dynamic F-measure of the training and testing sets is shown in Fig. 9(d), which reflects its variation with the number of early warning days. On the training set, the dynamic F-measure ranges from 0.841 to 0.974, with a mean of 0.906. On the testing set, it ranges from 0.863 to 0.964, with an average of 0.915. Like the fluctuations in accuracy, the fluctuations in the F-measure decreases as the number of early warning days increases. Importantly, when the number of early warning days is 24, the accuracies and F-measures on the training and testing sets are high. This finding indicates that 24 days is a suitable number of early warning days for the proposed model.

4.4.2 Model performance on the basis of the hybrid network

In this section, the performance of a deep learning-based early warning model integrated with the hybrid network, which includes the self-dynamic network and the relationship-dynamic network, is evaluated. First, we expand the original input database by incorporating the topological structure of the relationship-dynamic network. The updated input data set includes the topological features of both the self-dynamic network and the relationship-dynamic network, whereas the output data set includes the early warning signals for abrupt transitions. Second, the expanded database is randomly partitioned into a training set (80% of the total data) and a testing set (20% of the total data). A deep neural network is then constructed to train the early warning model on the training set, which is followed by performance validation on the testing set. The performance of the model for various numbers of early warning days is presented in Fig. 10 and summarized by key metrics in Table 2.

The dynamic accuracy on the training and testing sets, which indicates how the accuracy varies with the number of early warning days, is presented in Fig. 10(a). On the training set, the dynamic accuracy ranges from 0.929 to 0.991, with a mean of 0.977. On the testing set, the dynamic accuracy ranges from 0.874 to 0.964, with an average of 0.937. These results indicate that the model maintains high predictive accuracy across different numbers of early warning days. The dynamic precision and recall on the training and testing sets are presented in Figs. 10(b) and 10(c), respectively. The average precision is 0.993 on the training set and 0.998 on the testing set. In comparison, the average recall is 0.962 on the training set and 0.934 on the testing set. These results indicate that the average precision on the training and testing sets is greater than the corresponding average recall.

The dynamic F-measure on the training and testing sets, which demonstrates the variation of the F-measure with the number of early warning days, is presented in Fig. 10(d). The F-measure on the training set is between 0.927 and 0.991, with a mean of 0.977. On the testing set, the F-measure spans 0.930 to 0.981, with an average of 0.965. Notably, when the number of early warning days is 23, the accuracy and F-measure values on the training and testing sets are high, which suggests that 23 is a suitable number of early warning days.

When the hybrid network includes only the self-dynamic network, the input data include only the topological structure of the self-dynamic network. When the hybrid work integrates both the self-dynamic network and the relationship-dynamic network, the input data include the topological structures of both networks, which increases the dimensionality of the input data. The performance of the early warning model is improved. The improved performance is shown in Fig. 11. For instance, in Fig. 11(a), the dynamic improvements in accuracy on the training and testing sets are illustrated. In addition, the robustness of the proposed model is examined. We select the daily data of the S&P 500 and WTI crude oil futures prices as the sample data. A training set (70% of the total data) and a testing set (30% of the total data) are selected. The results are shown in the supplementary material. The results indicate that our model is robust.

5 Conclusions

In this paper, an early warning model for identifying early warning signals for abrupt transitions in time series–based oil systems is proposed. Previous models for investigating early warning signals focused primarily on the fluctuation characteristics of single points and other factors in oil time series and ignored the dynamic process in the oil series that is characterized both by historical changes in the oil series and by the coupling of nonlinear fluctuations with the related variables, such as natural gas. In this paper, a hybrid network that includes a self-dynamic network and a relationship-dynamic network is proposed for characterizing the dynamic process. Afterward, a deep learning-based early warning model that integrates the reconstructed hybrid network is proposed, where abrupt transitions of the oil time series are detected by the Markov regime-switching model and entropy theory. A deep neural network was established for capturing the early warning signals that correspond to the typical topological structure. Our findings can be summarized as follows:

(1) Abrupt transitions can be detected, and their features that are influenced by typical or global events can be investigated. The complexity of the oil time series is time-varying. Considering the high and low regimes of the time series, the series has a high probability of belonging to either regime at any given time. The durations of the regimes differ across periods, and the shifts between the two regimes can be identified. Notably, following the outbreak of the Russia–Ukraine War, the system remained in the high regime.

(2) A self-dynamic network is a simplified hybrid network that combines the GARCH model and complex network theory to characterize the dynamic process of fluctuations in the oil series that are influenced by historical changes. Network metrics (in-degree, out-degree, weighted degree, and betweenness centrality) of the nodes in the self-dynamic network follow power-law distributions. Nodes with higher values in these metrics (key nodes) can be identified according to their distributions. Furthermore, the performance of the early warning model varies with the number of early warning days. For example, the average accuracies on the training and testing sets were 0.909 and 0.854, respectively.

(3) The hybrid network includes the oil self-dynamic and relationship-dynamic networks. The relationship-dynamic network, which integrates nonlinear relationship models and complex network theory, captures the dynamic process of the nonlinear relationship between crude oil and natural gas time series. The network metrics (in-degree, out-degree, and weighted degree) of the nodes in the relationship-dynamic network follow power-law distributions, and the key nodes with greater network metrics are identified. The deep learning-based early warning model that is integrated with the hybrid network show improved performance, and all the indicators of the performance of the model change with the number of early warning days. Specifically, the average accuracy on the training set reached 0.977, and that on the testing set reached 0.937, which represented notable improvements over those of the deep learning-based early warning model integrated with the self-dynamic network. These improvements are attributed to the increased input dimensionality, which enables the model to capture more key features that are associated with early warning signals for abrupt transitions.

This study provides new insights for early warning of abrupt transitions in oil systems that are represented by time series and offers valuable guidance for policymakers and market investors for understanding the dynamic features of fluctuations in oil series that are characterized both by historical changes and by the coupling of nonlinear fluctuation with related variables, such as natural gas. On the one hand, when market investors consider fluctuations in other variables, they are advised to develop a comprehensive understanding of the dynamic fluctuations in crude oil prices and to use dynamic tools to capture early warning signals for abrupt transitions in oil systems, thereby increasing profitability. On the other hand, policymakers should deeply understand the fluctuations in crude oil prices as well as scenarios that are characterized by fluctuations in natural gas prices. They should rigorously monitor early warning signals for abrupt transitions in the movement of oil prices. By leveraging the early warning model, they can prevent substantial losses from being caused by sudden shifts in the fluctuations of oil prices and promote the stability of the oil market.

In this paper, an early warning model for abrupt transitions that combines a deep learning model, a Markov regime-switching model and a hybrid network that is reconstructed from time series is proposed. In future research, the effects of fluctuations in other systems (e.g., exchange rates and commodities) on the system of interest, along with multivariable relationships, should be considered. Other advanced economic models and deep learning models (e.g., graph neural networks) can be used to improve the performance of early warning signal models for abrupt transitions in the considered system.

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