Dynamic analysis and application of network structure control in risk conduction in the industrial chain

The study addresses how risk from stock price volatility propagates along the mining industry chain and how to control it within a complex financial network. Motivated by heightened uncertainty in China’s mining financial market amid carbon neutrality policies and recent volatility episodes, the authors integrate econometric risk measurement with network control theory to explore controllability and cost of control in risk transmission. The research aims to (i) measure volatility and systemic risk spillovers among mining stocks positioned upstream, midstream, and downstream in the industrial chain; (ii) construct a risk cascade network that accounts for industry driving effects; and (iii) design and simulate control strategies that reduce systemic risk with minimal energy, time, and node costs. The work is positioned as practically relevant for regulators seeking to move from “too big to fail” toward “too connected to fail” supervision.

Prior studies measure price co-movements and spillovers using Pearson correlation, Granger causality, entropy-based methods, copulas, DCC, and GARCH (including BEKK variants). CoVaR and ΔCoVaR are widely used for systemic risk contributions and tail spillovers. For risk control, approaches span macro indicators, firm financials, market information, machine learning prediction models, and policy levers (monetary/fiscal). With the rise of network perspectives, scholars have developed controllability concepts, exact and target control, and structure- and dynamics-based control strategies in complex networks, including economic and interbank contagion settings. Compared with quantile regression and copulas, GARCH better captures nonlinear, time-varying volatility structures aligned with this paper’s aims, motivating the choice of BEKK-GARCH and DCC-GARCH for spillover measurement and network construction, and structural controllability with minimum energy for intervention design.

Data: Daily closing prices for 83 listed Chinese mining-related companies (upper, middle, and lower industrial chain segments per CSRC classification) from 2020-01-02 to 2021-12-31 (486 trading days), sourced from the Choice database. Return series are used to estimate volatility spillovers and systemic risk spillovers. Volatility spillover (BEKK-GARCH): A bivariate BEKK-GARCH(1,1) is estimated pairwise. Mean equations include lagged returns; conditional covariance H_t follows H_t = C C' + A' ε_{t-1} ε_{t-1}' A + B' H_{t-1} B (presented compactly as H_t = CC' + A' H_{t-1} A + B H_{t-1} B). Shock spillover from i to j is a_{ij} and volatility spillover is b_{ij}. Total spillover SP_{i→j} = |a_{ij}| + |b_{ij}|. Systemic risk spillover (DCC-GARCH-CoVaR): Univariate GARCH provides conditional standard deviations; DCC estimates time-varying correlations ρ_{ij}^t. VaR_i^t = −Q(q) h_i^t. CoVaR_{i|j}^t = γ_i^t VaR_j^t, ΔCoVaR_{i|j}^t = γ_i^t (VaR_j^t − VaR_{j,50%}^t), with γ_i^t = ρ_{ij}^t (h_i^t / h_j^t). A sliding window is used: window length k=240 trading days (≈1 year) and step s=5 days, yielding 50 windows across 2020–2021. Industry driving and anti-risk adjustment: Edges are adjusted by industry driving effects derived from input–output backward linkages. IPC_{ij} reflects the industry driving coefficient between stocks i and j, scaled by firm market value (IPC_{ij} = x* b_{ij} x). Firms’ anti-risk ability is proxied by a Z-value index (covering scale, liquidity, profitability, structure, solvency), log-transformed with base 2.675. The adjusted risk spillover (edge weight) incorporates (1 + IPC_{ij}) × ΔCoVaR_{ij}^{t}. Risk conduction network construction: For each window, build a directed network with nodes as stocks and edges present if adjusted ΔCoVaR ≠ 0. Aggregate across 50 windows, retain edges with frequency ≥10 occurrences (≈top 80% cumulative frequency). Final graph G=(V,E,W) where E contains edges meeting the frequency threshold; edge weight W aggregates edge frequencies (per provided formula). Network control model: Represent the risk conduction network as a linear time-invariant system x'(t)=A x(t)+B u(t), y(t)=C x(t), where A is the adjacency matrix from the risk network, B selects driver (input) nodes, and u(t) are control inputs. Controllability is checked via the Kalman rank condition. Minimum-energy control is derived using the controllability Gramian W(t)=∫_0^t e^{Aτ} B B' e^{A'τ} dτ, with energy e(t)=∫_0^t ||u(t)||^2 dr = (x−x_0)' W(t)^{-1} (x−x_0). The optimal input u(t)=B' e^{−A'(t−τ)} W(t)^{-1} (x_f − x_0), t∈[0,t_f]. Control objective and scenarios: Initial state x_0 is the vector of VaRs; target x_f = 0.9 × VaR (10% reduction), consistent with observed VaR variability. Four industrial-chain cases are analyzed: whole industrial chain (WICN), and three two-layer sub-networks: upper–middle (UMN), upper–lower (ULN), and middle–lower (MLN). Driver nodes are selected among top risk-conduction nodes identified by conduction range, strength, and frequency. Control costs considered: (i) regulation cost (sum of absolute control signal energy over time), (ii) time cost (t_f to reach target), (iii) node cost (minimum number of driver nodes for global control).

- Risk conduction characteristics: Conduction range, strength, and frequency are positively associated; fewer than 20% of nodes exhibit high conduction, consistent with the Pareto principle. - Industrial-layer effects: Upper (mining) and middle (smelting) layer stocks have stronger risk conduction abilities. Midstream stocks often act as connectors in the chain, amplifying transmission. - Two-layer controllability: With key driver nodes, the ULN case achieves the highest control ratio and requires about 5 driver nodes for global control, versus ≈6 for UMN and MLN. Under complete control, control signals diminish toward 0 by t≈24; achieving a 10% risk reduction by t=8 requires very large inputs, which decline substantially by t=16 and near zero by t=24. - Three-layer controllability (WICN): <15% driver nodes can control >90% of nodes; ≈18% are needed for global control (vs ≈13% in two-layer networks). Global controllability is achieved with 11 driver nodes. Control signals approach 0 by t≈18; at t=6, inputs are on the order of tens of thousands (arbitrary units), falling below 10 by t=12. Total regulation energy is larger than in two-layer cases, and node cost is higher, though time cost is lower (t_f≈18 vs 24). - Comparative control difficulty: Whole-chain (three-layer) networks are harder to control overall (higher regulation and node costs) due to larger scale and denser correlations; among two-layer networks, ULN is the easiest, while UMN and MLN are more difficult. - Key nodes: Frequent critical nodes across scenarios include 600508.SH, 600971.SH, 600188.SH (coal mining, upstream); 600307.SH, 600231.SH (ferrous metal smelting, midstream); and others such as 600549.SH, 600961.SH, 600326.SH, 600348.SH, 002378.SZ, 600028.SH. These nodes have high conduction range/strength/frequency and are effective drivers for control. - Policy-relevant control signals: Optimal control inputs trend toward 0 over time, suggesting that strong early interventions (e.g., liquidity injections/expansionary measures) can rapidly reduce systemic risk, followed by tapering as the system approaches target risk levels.

The integrated econometrics–network control framework demonstrates that incorporating industry-chain structure and driving effects is crucial to understanding and mitigating systemic risk propagation among mining stocks. Findings confirm that risk is disproportionately transmitted by a small set of highly connected upstream and midstream nodes, aligning with a “too connected to fail” perspective. Control simulations show that targeted interventions at these key nodes can achieve significant reductions in overall risk with fewer driver nodes in simpler (two-layer) networks, while dense, whole-chain networks require more drivers and energy. The comparative analysis across UMN, ULN, MLN, and WICN clarifies where regulatory attention yields the greatest marginal benefit: upstream coal and midstream metal smelting firms. The time–energy trade-off highlights that faster risk reductions demand higher initial control energy, guiding policymakers on when to deploy stronger measures and when to taper. Overall, the approach addresses the research question by quantifying how risk propagates along the industrial chain and how to efficiently steer the system to a lower-risk state.

This paper proposes a unified framework combining BEKK-GARCH and DCC-GARCH-based ΔCoVaR with structural controllability and minimum-energy control to analyze and mitigate risk conduction in China’s mining stock industry chain. Contributions include (i) documenting that upper and midstream firms dominate risk transmission, (ii) quantifying controllability and control costs across two-layer and whole-chain networks (≈13% vs ≈18% driver nodes for global control; t_f≈24 vs ≈18), and (iii) identifying key driver stocks and case-specific strategies (focus on upstream coal in two-layer settings and midstream smelting in whole-chain control). Policy implications support a shift toward “too connected to fail” regulation and the use of timely, possibly expansionary, interventions to dampen systemic risk. Future research will incorporate real-world supervisory frameworks and policy mechanisms into the control model to enhance practical applicability.

The control framework is developed from network dynamics and structural control theory without explicitly incorporating actual regulatory concepts and policy mechanisms from the mining financial market. Integrating these supervisory elements is identified as future work.