Predicting Risk Propagation of Corporate Internet Reporting Based on Fuzzy Neural Network

The established CIR risk propagation model is shown in Figure 1. Based on the CIR indices, the process chain and value chain of information disclosure were reconstructed, and the links on the two chains were abstracted into risk propagation nodes. The paths between different links on the chains were abstracted as the edges between the nodes, forming a complex risk propagation network.

In the risk propagation network, every node is a risk carrier, and each edge is a risk propagation path. The carriers refer to all kinds of things that carry the risk flow of CIR, including tangible and intangible carriers. The tangible carriers include fund, material, and manpower of the company, while intangible carriers include policy, strategy, technology, and information.

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Figure 1. The CIR risk propagation model

The sources of CIR risks, either internal or external, exist on the overall propagation chain of the risk propagation network. It is from these risk sources that various CIR risks take shape, emerge, and start to propagate. The typical risk sources are internal opportunistic disclosure, safety of information propagation, changes in information propagation environment, and influence of mass media.

Based on the evaluation results of CIR safety, the risk adjustment threshold, risk release threshold, and risk control threshold could be determined, making risk measurement more dynamic.

3.1 Hypotheses

The propagation paths of CIR risk are formed based on the company’s process chain and value chain of the information disclosure. The propagation process bears high resemblance with the three-stage transmission of infectious disease, which consists of an incubation period, an outbreak period, and a recovery period. The CIR risk propagation can be broken down into an internal propagation and accumulation period (internal period), an external amplification and enhancement period (external period), and a stabilization and recovery period (final period). All the nodes in the risk propagation network were divided as per the period of CIR risk propagation: the number of nodes in the internal, external, and final periods at time t are denoted as TA(t), AE(t), and SR(t), respectively.

(1) It is assumed that the total number of nodes remains constant as C=TA(t)+AE(t)+SR(t). The CIR risk carriers could be divided into system carriers, suppliers and customer carriers, financial personnel carriers, fund carriers, and financial information carriers. The number of these types of carriers are denoted as C_s, C_c, C_f, C_b, and C_i, in turn. Considering that the risks propagate along the paths between different links on the process chain and value chain, the degree of correlation between carriers was expressed by the connection weight ω_ab.

(2) If two nodes have a connection weight of zero, the CIR risks will not flow between them. Let α_ab(t), β_ab(t), and γ_ab(t) be the risk adjustment threshold, risk release threshold, and risk control threshold between nodes a and b, respectively. The three thresholds can be initialized as α₀(t)=[TA_x0(t)+TA_y0(t)+TA_z0(t)]/C, β₀(t)=[AE_x0(t)+AE_y0(t)+AE_z0(t)]/C, and γ₀(t)=[SR_x0(t)+SR_y0(t)+SR_z0(t)]/C, respectively. Suppose α_ab(t) is positively correlated with the correlation gradient ρ_a(t) between the node and the process and value chains. Then, ρ_a(t)=[TA_a(t)+AE_a(t)+SR_a(t)]/[TA_c(t)+AE_c(t)+SR_c(t)], where TA_c(t), AE_c(t), and SR_c(t) are the risk adjustment threshold, risk release threshold, and risk control threshold of the company, respectively. The relationship between α_ab(t) and ρ_a(t) can be described as α_ab(t)=ω_abρ_a(t). During the internal period, node a is affected by every node in the external period connected to it. This risk propagation effect can be expressed as α_a(t)=1-∏(1-α_ab(t)).

(3) At time t, the risk propagation probability p₁(a, t) on the edge between node a in the internal period and every node in the external period connected to it depends on α₀(t), α_a(t), and the increment of risk propagation effect Δα_a+(t): p₁(a, t)=[1-α₀(t)][1-Δα_a+(t)][1-∏(1-α_ab(t))]. At time t, the state recovery probability p₂(b, t) of node b in the external period depends on β₀(t), and the increment of state recovery ability Δβ_b+(t): p₂(b, t)=1-[1－β₀(t)][1－Δβ_b+(t)].

3.2 Risk propagation model

According to the above hypotheses, the risk propagation paths are affected by the reconstruction strategy of the process chain and value chain. At time t, node a in the internal period might enter the risk state at the risk propagation probability of p₁(a, t). The loss Δα_a-(t) of influence α_a(t) of node a is positively correlated with the risk propagation probability. At time t, node b in the external period might change from the risk state to the normal state at the state recovery probability of p₂(b, t). The loss Δβ_b-(t) of state recovery ability β_b(t) of node b is negatively correlated with the state recovery probability.

According to the above analysis, the CIR risk propagation could be effectively managed, sub-effectively managed, or out of control. Based on the previous hypotheses, the CIR risk propagation can be modeled as:

$\left\{\begin{array}{l}\frac{d T A(t)}{d t}=-p_{1}(a, t) T A(t) A E(t) \\ \frac{d A E(t)}{d t}=p_{1}(a, t) T A(t) A E(t)-p_{2}(a, t) A E(t) \\ \frac{d S R(t)}{d t}=p_{2}(a, t) A E(t)\end{array}\right.$     (5)

Formula (6) must satisfy the following constraints:

$\left\{\begin{array}{c}0 \leq \int \alpha_{a}(t) d t-\Delta \alpha_{a}(i) \leq 1 \\ 0 \leq p_{1}(a, t) \leq 1 \\ 0 \leq p_{2}(a, t) \leq 1\end{array}\right.$       (6)

3.3 Model derivation

As shown in formula (5), the above set of differential equations are linked by the changes in the number of nodes TA(t), AE(t), and SR(t) in the three periods of risk propagation. Then, the risk propagation probability p₁(a, t) and state recovery probability p₂(a, t) can be respectively solved by:

$\left\{\begin{array}{l}p_{1}(a, t)=\left[1+\alpha_{0}(t)\right] \Pi\left(1-\alpha_{a b}(t)\right)+\left[1-\alpha_{0}(t)\right] \Delta \alpha_{a+}(t) \\ p_{2}(a, t)=\left[1-\beta_{0}(t)\right] \Delta \beta_{a+}(t)+\beta_{0}(t)\end{array}\right.$       (7)

Based on the relationship TA(t)=TA_x(t)+TA_y(t)+TA_z(t), the established CIR risk propagation model was decomposed in the risk propagation cycle T and subject to integration. Suppose α_a(t) reaches the ideal state in cycle T through the reconstruction of process and value chains. Then, ∫∏(1-α_ab(t))dt=0, ∫Δα_a+(t)dt→0, and ∫Δβ_b+(t)dt→0. On this basis, the evolution of TA in the internal period can be simplified as:

$\frac{\left(1+\omega_{a} \rho_{a}\right) \alpha_{0}(t)}{\omega_{a} \rho_{a}} T A_{x}(t)+\frac{1+\alpha_{0}(t)}{\omega_{a} \rho_{a}} T A_{y}(t)$

$+\left(1-\alpha_{0}(t)\right) T A_{z}(t)+2 \beta_{0}(t)-1=0$     (8)

Based on the relationship TA(t)=TA_x(t)+TA_y(t)+TA_z(t), the established CIR risk propagation model was decomposed in the risk propagation cycle T and subject to integration, revealing the evolution of AE(t) in the external period:

$\frac{1-\alpha_{0}(t)}{\omega_{a} \rho_{a}} A E_{x}(t)+\frac{\left(1+\omega_{a} \rho_{a}\right) \alpha_{0}(t)}{\omega_{a} \rho_{a}} A E_{y}(t)$

$+\left(\alpha_{0}(t)-\beta_{0}(t)\right) A E_{z}(t)-\frac{\beta_{0}(t)}{2}=0$      (9)

Based on the relationship SR(t)=SR_x(t)+SR_y(t)+SR_z(t), the established CIR risk propagation model was decomposed in the risk propagation cycle T and subject to integration, revealing the evolution of SR(t) in the final period:

$\frac{1-\alpha_{0}(t)}{\omega_{a} \rho_{a}} S R_{x}(t)+\frac{1-\alpha_{0}(t)}{\omega_{a} \rho_{a}} S R_{y}(t)$

$+\frac{\left(1+\omega_{a} \rho_{a}\right) \beta_{0}(t)}{\omega_{a} \rho_{a}} S R_{z}(t)-\beta_{0}(t)=0$      (10)

By deriving p₁(a, t) and p₂(a, t) separately, the mean risk propagation rate and mean state recovery rate of cycle T can be respectively obtained by:

$\left\{\begin{array}{l}\frac{d p_{1}(a, t)}{d t}=-\left[1-\alpha_{0}(t)\right] \frac{d\left(1-\alpha_{a}(t)\right)}{d t} \\ -\left[1+\alpha_{0}(t)\right] \frac{d \Delta \alpha_{a+}(a, t) \alpha_{i}(t)}{d t} \\ \frac{d p_{2}(a, t)}{d t}=\left[1-\beta_{0}(t)\right] \frac{d \Delta \beta_{a t}(a, t)}{d t}\end{array}\right.$      (11)

With the aid of Matlab, M functions Evo1.m, Evo2.m and Evo3.m and the M executable file Evorun.m were compiled for the results (8), (9), and (10) of CIR risk propagation in each period. The M functions were executed 60-80 times, and the results were combined and averaged. Figures 3, 4, and 5 display the results of CIR risk propagation in the internal, external, and final periods, respectively.

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Figure 3. The TA state variation in the internal period

As shown in Figure 3, during the internal period, the TA(t) value decreased gently, and the CIR risk propagation reached the equilibrium at the intersection. Before the intersection, the reconstruction focuses on the prewarning of risk propagation; after the intersection, the reconstruction focuses on the restoration from the risk state. It can be seen that the keys of the reconstruction of the process and value chains lie in the reconstruction of the risk propagation prewarning module and risk state restoration module. During the reconstruction of risk propagation prewarning module, the risk flow direction and correlation between risk carriers directly bear on the risk control effect.

As shown in Figure 4, during the external period, the AE state exhibited as a parabolic curve. The left intersection is the critical point, at which risk propagation occurred between two risk carriers; the right intersection is the critical point, at which the risk diffusion changed into risk convergence. The left intersection was slightly higher than the right intersection. This means the overall safety of the CIR increases rapidly through the reconstruction cycle.

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Figure 4. The AE state variation in the internal period

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Figure 5. The SR state variation in the internal period

As shown in Figure 5, during the final period, the intersection is the critical point, at which the risk propagation between two risk carriers gradually stabilized and returned to the normal state. Before the intersection, the CIR was highly risky; after the intersection, the CIR risks were under control, i.e., the CIR was slightly safe. At the beginning of the final period, many nodes needed to be restored in the risk propagation network, and the recovery rate was rather slow. As more and more nodes were restored, the reconstruction process was simplified, and the node recovery gradually picked up speed. This means the CIR risk propagation has relatively high fluidity.

In addition, Figures 3-5 confirm the following finding from Table 4: when the CIR risk propagation is effectively controlled, the mean risk propagation probability of all risk carriers is on the decline, while the mean state recovery probability is on the rise.

The next task is to verify the prediction effect of our BPNN on CIR risk propagation path. For this purpose, 200 sets of TA(t), AE(t), and SR(t) of CIR risk propagation model and the α_ab(t), β_ab(t), γ_ab(t), p₁(a, t), and p₂(a, t) values of each node were imported to the BPNN. Only one node output, which falls in [-1, 1], was allowed on the output layer, representing the overall evaluation result on the reconstruction and control effect of CIR risk propagation path.

The BPNN was established on Matlab, including 48 input layer nodes, 24 hidden layer nodes, and 1 output layer node. As shown in Figures 6-8, the BPNN entered the convergence state and reach the preset training accuracy of 10^-6, after 800 trainings. Besides, the BPNN predicted the CIR risk propagation path fairly accurately.

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Figure 6. The MSE of prediction results

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Figure 7. The simulated results on correlation gradient

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Figure 8. The simulation results on prediction errors