Risk Contagion of Local Government Implicit Debt Integrating Complex Network and Multi-Subject Coordination

Abstract

This article analyzes the risk contagion mechanism of local government implicit debt from the perspective of multi-subject collaboration, considering interaction effects among different influencing factors. On this basis, with the help of complex network theory and mean field theory, a risk contagion model of local government implicit debt is constructed, and then the evolution characteristics and control strategies for risk contagion of local government implicit debt are analyzed theoretically and simulated. The main findings obtained from the study are: (1) A scale-free network is not conducive to the risk contagion of local government implicit debt, while the opposite is true for a random network. (2) Information openness accuracy and information disclosure strategy both exhibit a positive “U” shaped relationship with the risk contagion of local government implicit debt. Debt management level, emotional tendency, risk preference level, credit policy robustness, accountability mechanism soundness, and perfection of laws and regulations are all negatively correlated with the risk contagion of local government implicit debt. (3) In order to effectively reduce the risk contagion intensity of local government implicit debt, local governments at all levels should continuously strengthen their debt management capabilities and information openness, and the central government should continuously improve accountability mechanisms, laws, and regulations. At the same time, financial institutions and the media should actively play the role of “stabilizers”. However, the local government implicit debt risk is an inherent risk, and its control focus should be on reducing rather than eliminating the risk.

1. Introduction

To effectively protect people’s livelihood and promote sustainable growth of the local economy, local governments at all levels obtain implicit debt funds with government credit as a guarantee, and its scale shows a trend of rapid growth [1,2]. However, with the intensification of international instability and the impact of sudden major health events, coupled with the global economic downturn, local government implicit debt is facing enormous repayment pressure, causing a local government’s implicit debt risk to continue to gather. Once a local government defaults on its implicit debt, the implicit debt risk will quickly spread through multiple channels such as information, emotion, and finance, bringing uncertainty to the entire financial market and national economic development, and may even trigger a systemic financial crisis. Therefore, in the current situation, an in-depth analysis of the risk contagion mechanism and evolution characteristics of local government implicit debt can help regulatory departments at all levels to deeply understand the risk formation and contagion rules of local government implicit debt and provide a theoretical basis and reference for optimizing the risk management level of local governments implicit debt.

Currently, research on the risk contagion of local government implicit debt mainly focuses on the impact effects exerted by four different subjects. The first is the direct influence effect of local governments at all levels. Local governments at all levels are not only the direct initiators and users of implicit debt, but also the core managers and the main risk bearers [3]. On the one hand, local governments at all levels influence the magnitude of potential implicit debt risk through their financial management capabilities [4]. On the other hand, local governments at all levels use government information openness to influence the level of contagion and attention to implicit debt risk through information channels [5]. The second subject is the indirect effect of the media. The media is not only the information publisher of local government implicit debt, but also the external supervisor of local government implicit debt [6]. The indirect impact of the media on local government implicit debt is manifested in the changes in public opinion caused by different information disclosure strategies and in the empathy between the public and the investor caused by different media emotion tendencies, which in turn affects the risk accumulation degree of local government implicit debt [5,7]. The third subject is the core disturbance effect of financial institutions. Financial institutions exert a disturbance effect on a local government’s implicit debt risk through financing channels [8]. When financial institutions maintain a stable credit policy toward local governments, it helps to reduce the default risk of local government implicit debt [9]. Meanwhile, when financial institutions belong to the investor of risk preference, they are more willing to continue providing financing to local governments [10]. These all play an important role in disrupting the risk contagion of local government implicit debt to a certain extent. The fourth subject is the macro-control effect of the central government. The central government is the main regulator of local government implicit debt and the policymaker for regulating local government implicit debt. It plays a macro adjustment effect on the overall risk contagion of local government implicit debt [11]. In other words, as the highest administrative body, the central government continuously optimizes the implicit debt behavior of local governments at the macro level by continuously improving the accountability mechanism and laws and regulations for local government implicit debt [12,13].

However, existing research on the risk contagion of local government implicit debt mainly focuses on single subjects such as local governments at all levels, the media, financial institutions, or the central government, with an emphasis on local governments at all levels. There is relatively little research on the comprehensive consideration of the synergistic effect of multiple stakeholders on the risk formation and contagion of local government implicit debt. This leads to a mismatch between existing research findings and the actual situation of risk contagion of local government implicit debt and limits the practical relevance and application effectiveness of research results. In addition, the influencing factors of different subjects also play different roles in the risk contagion of local government implicit debt. However, due to the lack of comprehensive consideration of the synergistic effects of multiple subjects in existing research, the interaction effects generated by the influencing factors of different subjects have not been fully considered. Therefore, this study analyzes the evolutionary characteristics and control strategies of risk contagion of local government implicit debt from the perspective of multi-subject collaboration and comprehensively considers the interactive effects of different influencing factors. This is not only the starting point of this study, but also the core innovation point of this study.

In addition, with the gradual deepening of the integration of network science and economic society, complex network theory provides a strong argument for explaining the complex interrelationships and impact mechanisms among economic entities. At present, complex network theory has been widely applied to financial risk contagion [14], network public opinion diffusion [15], investor behavior diffusion [16], and other aspects. Moreover, multiple subjects can form a network of local financing platform associations with information forms like common media topics, and diffuse behavior and information within the network, all of which exhibit significant complex network characteristics. These provide new ideas and effective approaches for this study on the evolutionary characteristics and control strategies of risk contagion of local government implicit debt from the perspective of multi-subject collaboration using complex network theory.

In summary, this study analyzes the risk contagion mechanism underlying local government implicit debt from the perspective of multi-subject collaboration, considering the interaction effects among different influencing factors. By constructing a risk contagion model of local government implicit debt, the evolutionary characteristics and control strategies underlying risk contagion of local government implicit debt are analyzed using theoretical and simulation analysis. The main contributions of this study include the following. (1) Different from the single-subject perspective of existing research, this study analyzes the risk contagion mechanism and characteristics underlying local government implicit debt from the perspective of multi-subject collaboration among local governments at all levels, the media, financial institutions, and the central government, which helps to enhance the objectivity and applicability of our research conclusions. (2) On the basis of the previous contribution, this study further considers the interactive effects among the influencing factors of different subjects, rather than simply analyzing the impact of a single subject’s influencing factors on the risk contagion of local government implicit debt, thereby strengthening the reliability and pertinence of our research conclusions. (3) This study not only provides a reference for regulatory authorities to deeply and comprehensively understand the risk formation and contagion mechanism underlying local government implicit debt at the theoretical level, but it also provides a basis for regulatory authorities to formulate policies that prevent and resolve local government implicit debt risk from the practical level. (4) This study addresses two important topics in the literature. First, this study is related to the ample literature on the economic effects of implicit government guarantees [17,18,19,20,21,22,23,24]. Second, this study also adds to the growing literature that focuses on the economic effects of networks [25,26,27,28,29,30]. Our main contribution is connecting these two seemingly unrelated bodies of scholarly work and showing that network structures propagate risks stemming from governments’ implicit guarantees, providing a multi-layered modeling approach to represent the actions of different players.

The structure of this article is organized as follows: The second part elaborates on the risk contagion mechanism of local government implicit debt from the perspective of multi-subject collaboration. The third part constructs a risk contagion model of local government implicit debt from the perspective of multi-subject collaboration. The fourth part theoretically analyzes the risk contagion effects of local government implicit debt under multi-subject collaboration. The fifth part simulates and analyzes the evolutionary characteristics and control strategies of risk contagion of local government implicit debt from the perspective of multi-subject collaboration. Finally, the conclusion of this article is drawn.

2. Risk Contagion Mechanism Underlying Local Government Implicit Debt from the Perspective of Multi-Subject Collaboration

2.1. Contagion Mechanism

Local government implicit debt is a debt generated when local governments promise to pay a certain amount in the future [13]. Local government implicit debt is mainly used for construction in the field of livelihood services, consumption caused by pension gaps, contingent default compensation responsibilities assumed by a government in policy financing guarantee business, and extra-budgetary debt responsibilities incurred in response to occasional events [1,31]. So, these debts have the characteristics of strong livelihood, low profitability, long periodicity, and high uncertainty and are greatly influenced by policies and economic situations [2].

In the event of an emergency or significant fluctuation in the economic situation, local government implicit debt may face difficulties in repayment, leading to local government implicit debt risk. This risk is generally transmitted through information intermediaries and the association relationship among financing platforms. That is, when a financing platform defaults on its implicit debt, once the default information is disclosed by the media or by local governments at all levels, the implicit debt risk will be transmitted to other financing platforms through information intermediaries or the association relationship among financing platforms. At the same time, financial institutions closely related to financing platforms suffer direct losses due to implicit debt risk and then feed the risk back to relevant financing platforms through credit policy adjustments and other forms. Under this influence, implicit debt risk continues to spread within and among financing platforms, with a trend of transmission from local governments at the same level to higher-level governments, or even the central government, and from government departments to financial departments, resulting in widespread economic and social impacts and leading to crises. Therefore, the process of risk contagion of local government implicit debt is mainly influenced by the interaction among local governments at all levels, the media, financial institutions, and the central government. The specific contagion mechanism is as follows (see Figure 1):

(1)

Local governments at all levels. Local governments at all levels have a direct impact on risk formation, contagion, and control of implicit debt [3]. The main factors of local governments at all levels that affect the implicit debt risk include the following. The first factor is the debt management level $\alpha$ [4]. $\alpha$ directly reflects the ability of local governments to manage implicit debt at all levels. The larger the $\alpha$, the higher the debt management level of local governments at all levels, which is more conducive to reducing the risk formation and contagion probability of implicit debt. On the contrary, the more likely it is to cause the risk formation and contagion of implicit debt. The second factor is the information openness accuracy $\eta$ [5]. reflects the openness accuracy of implicit debt information of local governments and the responsibility of local governments at all levels. However, information openness has two sides. When $\eta$ is low, the public may question the implicit debt of local governments at all levels, which can increase the potential risk of implicit debt. As $\eta$ increases, the public’s right to know guaranteed to a certain extent and local governments at all levels are also prone to establishing a good image, which helps to enhance confidence among all sectors and thus helps to reduce implicit debt risk. However, when $\eta$ is too high, a large amount of bad information is disclosed, and due to the limited cognitive ability and psychological acceptance of investors, anxiety tends to easily increase in all sectors, thereby exacerbating the risk contagion of implicit debt.

(2)

Media. The media has an indirect impact on the risk formation, contagion, and control of local government implicit debt through information feedback effects, herd effects, and panic psychology [6]. The main factors that influence local government implicit debt risk by the media include, first, the information disclosure strategy $\chi$ [6]. $\chi$ reflects the strategy chosen by the media when disclosing the implicit debt information of local governments. $\chi =0.5$ represents the media’s choice to use an objective information disclosure strategy as the dividing point for its information disclosure strategy. As $\chi$ approaches 1, the media chooses to disclose more positive information. As $\chi$ approaches 0, the media chooses to disclose more negative information. The second factor is emotional tendency [5,7]. $\phi$ represents the attitude of the media toward local government implicit debt. As $\phi$ approaches 1, the media emotion becomes more positive. As $\phi$ approaches 0, the media emotion becomes more negative.

(3)

Financial institutions. Financial institutions have become important disruptors in the risk formation, contagion, and control of local government implicit debt through financing channels [8]. The main factors affecting local government implicit debt risk by financial institutions include, first, the risk preference level $\lambda$ [9]. $\lambda$ reflects the level of tolerance of financial institutions toward a local government’s implicit debt risk. The larger the $\lambda$, the stronger the preference of financial institutions for local government implicit debt risk, the higher the level of risk tolerance, and the more likely they are to provide financing during periods when local governments experience implicit debt risk. The smaller the $\lambda$, the weaker the preference of financial institutions for implicit local government debt risk, the lower their risk tolerance, and the less willing they are to provide financing during the period when implicit debt risk occurs in local governments. The second factor is the credit policy robustness $\omega$ [10]. $\omega$ indicates whether financial institutions have stability in their credit policies toward local governments. The larger the $\omega$, the higher the stability of financial institutions’ credit policy toward local governments, and the easier it is for local governments to continuously obtain financial support. The smaller the $\omega$, the lower the stability of financial institutions’ credit policy toward local governments. Financial institutions are more likely to change their credit policy based on changes in the actual situation of local government implicit debt, which is less conducive to local governments’ sustained access to financial support.

(4)

Central government. The central government exerts a macro direct impact on the risk formation, contagion, and control of local government implicit debt through policies and regulations [11]. The main factors affecting local government implicit debt risk by the central government include, first, the accountability mechanism soundness $\rho$ [12]. $\rho$ reflects the central government’s determination and implementation of accountability for a local government’s implicit debt. The larger the $\rho$, the higher the level of accountability of the central government for a local government’s implicit debt, and the more helpful it is in suppressing a local government’s arbitrary and unreasonable implicit debt behavior. At the same time, it helps local governments better fulfill the main responsibilities and obligations of implicit debt. The smaller the $\rho$, the lower the level of accountability of the central government for a local government’s implicit debt, and the more likely local governments are to recklessly engage in unreasonable implicit debt behavior. At the same time, local governments are more likely to evade the main responsibilities and obligations of implicit debt. The second factor is the perfection of laws and regulations [13]. $\tau$ indicates whether the central government’s policies and regulations on local government implicit debt are complete. The larger the $\tau$, the more comprehensive the policies and regulations regarding local government implicit debt and the more helpful the central government is in providing guidance for local governments to make implicit debt decisions and to constrain their non-performing implicit debt behavior. The smaller the $\tau$, the fewer policies and regulations are in place to address a local government’s implicit debt. The lack of basis and constraints for local governments to adopt implicit debt behavior is more likely to encourage them to choose non-performing implicit debt.

2.2. Contagion Rules

Local governments obtain local development funds through financing platforms, but there are significant differences in the actual financial situation, planning ability, and debt management level of different local governments [13]. Therefore, this study considers three types based on the status of different financing platforms of local governments: Sensitive financing platforms, fragile financing platforms, and robust financing platforms. We use $SFP$, $FFP$, and $RFP$ to represent the three types of financing platforms, respectively. Below, we present the contagion rules for state transitions among the three types of financing platforms (see Figure 2):

(1)

After the outbreak of an emergency or significant fluctuation in the economic situation, it is assumed that the debt management ability of local governments at all levels is low, and the information openness accuracy is too high or too low. At the same time, if media information disclosure tends to be positive or negative with negative emotional tendency, financial institutions have low-risk preference and credit policies are unstable, coupled with inadequate accountability mechanisms and laws and regulations by the central government. At this point, on the one hand, sensitive financing platforms will transform into fragile financing platforms with $\psi$($0\le \psi \le 1$) probability. On the other hand, robust financing platforms will transform into sensitive financing platforms with $\theta$($0\le \theta \le 1$) probability.

(2)

We assume local governments at all levels have high debt management levels and moderate accuracy of information openness. At the same time, we assume that media information disclosure is objective and emotionally inclined, financial institutions have a high-risk preference, and credit policies are stable, coupled with sound accountability mechanisms and laws and regulations in the central government. At this point, on the one hand, some sensitive financing platforms are exempt from the impact of implicit debt risk and directly transform into robust financing platforms with $\epsilon$($0\le \epsilon \le 1$) probability. On the other hand, some fragile financing platforms have recovered from implicit debt risk and transformed into robust financing platforms with $\kappa$($0\le \kappa \le 1$) probability.

(3)

We assume that at each time period, the association network of local financing platforms enters new local financing platforms with $\xi$($0\le \xi \le 1$) probability and exit some local financing platforms with $\beta$($0\le \beta \le 1$) probability.

3. Risk Contagion Model of Local Government Implicit Debt under Multi-Subject Coordination

3.1. Model Building

To construct a risk contagion model of local government implicit debt from the perspective of multi-subject coordination, we assume $N$ as the total number of local financing platforms in the association network. $sfp$, $ffp$, and $rfp$ are the proportions of sensitive financing platforms, fragile financing platforms, and robust financing platforms, namely, $sfp=\frac{SFP}{N}$, $ffp=\frac{FFP}{N}$, $rfp=\frac{RFP}{N}$, and $sfp+ffp+rfp=1$($0\le sfp,ffp,rfp\le 1$), respectively. Moreover, we assume that at time $t$, the density of fragile financing platforms with degree $k$ is $ff{p}_k(t)$ and the probability of connecting sensitive and fragile financing platforms is $\Theta (t)$. We combine the approach of Wang et al. [32] to set the risk contagion probability of local government implicit debt from the perspective of multi-subject collaboration as:

$$\psi =\frac{1}{100}{\left[\frac{(1-e^{\frac{-1}{\sqrt{\rho }}})({\eta }^2-\eta +1){({\chi }^2-\chi +\frac{1}{2})}^{\frac{1}{2}}{\omega }^{\lambda -1}(1-e^{\frac{-1}{{\alpha }^2\tau }})}{\alpha \tau e^{\varphi }}\right]}^{\frac{1}{2}}$$

(1)

According to the risk contagion rule in Figure 2 and mean-field theory [33,34], the differential equation system of the risk contagion model of local government implicit debt from the perspective of multi-subject collaboration is represented as follows:

$$\{\begin{array}{c}\frac{dsf{p}_k(t)}{dt}=\xi -k\psi sf{p}_k(t){\Theta }^M(t)-\epsilon sf{p}_k(t)+\theta rf{p}_k(t)\\ \frac{dff{p}_k(t)}{dt}=k\psi sf{p}_k(t)\Theta (t)-\kappa ff{p}_k(t)\begin{array}{cc} & \end{array}\\ \frac{drf{p}_k(t)}{dt}=\kappa ff{p}_k(t)+\epsilon sf{p}_k(t)-\beta rf{p}_k(t)-\theta rf{p}_k(t) \end{array}$$

(2)

According to the differential system in Equation (2), if the steady-state condition is $\frac{dff{p}_k(t)}{dt}=0$, then the steady-state value $ff{p}_k(t)$ can be obtained:

$$ff{p}_k(t)=\frac{k\psi sf{p}_k(t)\Theta (t)}{\kappa }=\frac{k\xi \psi (\beta +\theta )\Theta (t)}{\beta \kappa \epsilon +k\beta \kappa \psi \Theta (t)}$$

(3)

We express the average density of fragile financing platforms as $ffp={\displaystyle {\sum }_{k}P(k)ff{p}_k(t)}$, and obtain $\Theta (t)$ according to Equation (3):

$$\Theta (t)={\displaystyle \sum _k\frac{kP(k)ff{p}_k(t)}{{\displaystyle \sum _{s}sP(s)}}=\frac{1}{<k>}{\displaystyle \sum _{k}kP(k)ff{p}_k(t)}}$$

(4)

where $<k>$ represents the network average degree of risk contagion of local government implicit debt under multi-subject collaboration.

Given that $<k>={\displaystyle {\sum }_{k}kP(k)}$ and $<k^2>={\displaystyle {\sum }_k{k}^{2}P(k)}$, it can be obtained from Equations (3) and (4) that:

$$\Theta (t)=\frac{1}{<k>}{\displaystyle \sum _{k}kP(k)\frac{k\xi \psi (\beta +\theta )\Theta (t)}{\beta \kappa \epsilon +k\beta \kappa \psi \Theta (t)}}$$

(5)

Let $\Theta =\Theta (t)$ and Equation (5) have the trivial solution $\Theta =0$. If Equation (5) has a non-trivial solution $\Theta \ne 0$, then the necessary condition is:

$$\frac{d}{d\Theta }{\left(\frac{1}{<k>}{\displaystyle \sum _{k}kP(k)\frac{k\xi \psi (\beta +\theta )\Theta (t)}{\beta \kappa \epsilon +k\beta \kappa \psi \Theta (t)}}\right)|}_{\Theta =0}\ge 1$$

(6)

That is,

$$\frac{1}{<k>}{\displaystyle \sum _{k}kP(k)\frac{k\xi \psi (\beta +\theta )}{\beta \kappa \epsilon }\ge 1}$$

(7)

Therefore, the threshold $R_0$ for the risk contagion of local government implicit debt from the perspective of multi-subject collaboration is obtained:

$$R_0=\frac{\xi \psi (\beta +\theta ){\displaystyle {\sum }_k{k}^2}P(k)}{\beta \kappa \epsilon {\displaystyle {\sum }_{k}kP(k)}}$$

(8)

By substituting Equation (1) into Equation (8), we can further obtain $R_0$ as:

$$R_0={\left[\frac{(1-e^{\frac{-1}{\sqrt{\rho }}})({\eta }^2-\eta +1){({\chi }^2-\chi +\frac{1}{2})}^{\frac{1}{2}}{\omega }^{\lambda -1}(1-e^{\frac{-1}{{\alpha }^2\tau }})}{\alpha \tau e^{\varphi }}\right]}^{\frac{1}{2}}\frac{\xi (\beta +\theta ){\displaystyle {\sum }_k{k}^2}P(k)}{100\beta \kappa \epsilon {\displaystyle {\sum }_{k}kP(k)}}$$

(9)

A value of $R_0=1$ corresponds to the threshold of whether the local government’s implicit debt risk is contagious in both the local and global association networks of local financing platforms. When $R_0<1$, the local government’s implicit debt risk gradually disappears in the global association network of local financing platforms but continues to spread with a non-zero probability in the local association network of local financing platforms. At this point, the larger the $R_0$, the greater the contagion intensity in the local association network of local financing platforms, and the more local financing platforms will be impacted. When $R_0>1$, the local government’s implicit debt risk is transmitted simultaneously in the local and global association networks of financing platforms. At this point, the larger the $R_0$, the greater the contagion intensity in the local and global association networks of financing platforms, and the more local financing platforms will be impacted.

3.2. Local Financing Platform Association Network

The subjects involved in local government implicit debt are connected through financing platforms, and an association network is formed among local financing platforms through information forms like common media topics. In the local financing platform association network, nodes represent local financing platforms and edges represent information associations among local financing platforms. We describe the algorithm for forming the local financing platform association network as follows:

(1)

At time t, there are $a_0$ fragile financing platforms and $b_0$ information correlation relationships ($a_0>0,b_0>0$).

(2)

At each time period $t_i$ ($i$ = 1, 2, 3, ……), add $a$ sensitive financing platforms to the association network, and each newly added sensitive financing platform contains $\delta$ information association relationships ($a>0,\delta >0$).

(3)

Newly added sensitive financing platforms are randomly connected to existing fragile financing platforms with probability $q$ or are selectively connected to existing fragile financing platforms with probability $(1-q)$ ($0\le q\le 1$).

(4)

During the random connection process, the probability of any existing local financing platform $i$ being selected is $\frac{1}{a_0+at}$. During the preferential attachment process, the probability of any existing local financing platform $i$ being selected is ${\Pi }_i(0\le {\Pi }_i\le 1)$:

$${\Pi }_i=\frac{k_i}{{\displaystyle \sum _j{k}_j}}$$

(10)

where $k_i$ represents the degree of existing local financing platforms $i$.

According to the above algorithm, the change rate of degree $k_i$ of local financing platform $i$ can be expressed as:

$$\begin{array}{l}\frac{\partial k_i}{\partial t}=\frac{a\delta q}{a_0+at}+(1-q)a\delta {\Pi }_i\\ =\frac{a\delta q}{a_0+at}+(1-q)a\delta \frac{k_i}{{\displaystyle \sum _j{k}_j}}\end{array}$$

(11)

Given that ${\displaystyle {\sum }_j{k}_j=2(a\delta t+b_0)}$, Equation (11) can be transformed into:

$$\frac{\partial k_i}{\partial t}=\frac{a\delta q}{a_0+at}+(1-q)a\delta \frac{k_i}{2(a\delta t+b_0)}$$

(12)

When $t\to \infty$, $at+a_0\approx at$ and $a\delta t+b_0\approx a\delta t$. In addition, based on the original conditions, $k_j(t_j)=a\delta$ can be determined. Therefore, the solution of Equation (12) can be obtained:

$$k_i=(a\theta +\frac{2\delta q}{1-q}){(\frac{t}{t_i})}^{\frac{1-q}{2}}-\frac{2\delta q}{1-q}$$

(13)

Assuming that at each identical period, new local financing platforms enter the association network, the probability density of local financing platform $i$ being selected at time $t$ is:

$$P_i=\frac{1}{at+a_0}$$

(14)

At $k_i<k$, $P(k_i(t)<k)$ can be obtained as:

$$\begin{array}{l}P(k_i(t)<k)\\ =P(t_i>t{[\frac{k(1-q)+2\delta q}{a\delta (1-q)+2\delta q}]}^{-\frac{2}{1-q}})\\ =1-P(t_i\le t{[\frac{k(1-q)+2\delta q}{a\delta (1-q)+2\delta q}]}^{-\frac{2}{1-q}})\end{array}$$

(15)

By combining Equations (14) and (15), it can be obtained that:

$$P(k_i(t)<k)=1-\frac{t}{a_0+at}{[\frac{k(1-q)+2\delta q}{a\delta (1-q)+2\delta q}]}^{-\frac{2}{1-q}}$$

(16)

And

$$\underset{t\to \infty }{\lim }P(k_i(t)<k)\approx 1-\frac{1}{a}{[\frac{k(1-q)+2\delta q}{a\delta (1-q)+2\delta q}]}^{-\frac{2}{1-q}}$$

(17)

From Equation (17), the degree distribution function of the association network can be obtained:

$$\begin{array}{l}P(k)=\frac{\partial P(k_i(t)<k)}{\partial k}\\ =\frac{2}{a[a\delta (1-q)+2\delta q]}{[\frac{k(1-q)+2\delta q}{a\delta (1-q)+2\delta q}]}^{\frac{q-3}{1-q}}\end{array}$$

(18)

Substituting Equation (18) into Equation (9), we obtain:

$$\begin{array}{l}{R}_0={\left[\frac{(1-e^{\frac{-1}{\sqrt{\rho }}})({\eta }^2-\eta +1){({\chi }^2-\chi +\frac{1}{2})}^{\frac{1}{2}}{\omega }^{\lambda -1}(1-e^{\frac{-1}{{\alpha }^2\tau }})}{\alpha \tau e^{\varphi }}\right]}^{\frac{1}{2}}\frac{\xi (\beta +\theta ){\displaystyle {\sum }_k{k}^2}P(k)}{100\beta \kappa \epsilon {\displaystyle {\sum }_{k}kP(k)}}\\ \approx {\left[\frac{(1-e^{\frac{-1}{\sqrt{\rho }}})({\eta }^2-\eta +1){({\chi }^2-\chi +\frac{1}{2})}^{\frac{1}{2}}{\omega }^{\lambda -1}(1-e^{\frac{-1}{{\alpha }^2\tau }})}{\alpha \tau e^{\varphi }}\right]}^{\frac{1}{2}}\frac{\xi (\beta +\theta ){\displaystyle {\sum }_k{k}^2}P(k){\displaystyle \underset{a\delta }{\overset{\infty }{\int }}{k}^2{[k(1-q)+2\delta q]}^{\frac{q-3}{1-q}}dk}}{100\beta \kappa \epsilon {\displaystyle \underset{a\delta }{\overset{\infty }{\int }}k{[k(1-q)+2\delta q]}^{\frac{q-3}{1-q}}dk}}\end{array}$$

(19)

4. Theoretical Analysis of Risk Contagion of Local Government Implicit Debt from the Perspective of Multi-Subject Collaboration

4.1. Equant Analysis of Risk Contagion of Local Government Implicit Debt

Theorem 1:

In the local financing platform association network, there is a unique Equant point $\tilde{\Theta }$ and $\tilde{\Theta }>0$ for the risk contagion of local government implicit debt from the perspective of multi-subject collaboration.

Proof: 

Let

$$Y_k(\Theta )=\frac{k^{2}P(k)}{<k>}{\displaystyle \sum _k\frac{\xi \psi (\beta +\theta )\Theta }{\beta \kappa \epsilon +k\beta \kappa \psi \Theta }}$$

(20)

Since $Y_k{}^{\prime }(\Theta )=\frac{k^{2}P(k)}{<k>}{\displaystyle \sum _k\frac{\xi \psi \beta \kappa \epsilon (\beta +\theta )}{{(\beta \kappa \epsilon +k\beta \kappa \psi \Theta )}^2}}>0$ and $Y_k{}^{″}(\Theta )=\frac{k^{3}P(k)}{<k>}{\displaystyle \sum _k\frac{-2\xi {\psi }^2{\beta }^2{\kappa }^2\epsilon (\beta +\theta )}{{(\beta \kappa \epsilon +k\beta \kappa \psi \Theta )}^3}}<0$, $Y_k(\Theta )$ is a monotonically increasing concave function with respect to $\Theta$. Additionally, given that $Y_k(1)=\frac{k^{2}P(k)}{<k>}{\displaystyle \sum _k\frac{\xi \psi (\beta +\theta )}{\beta \kappa \epsilon +k\beta \kappa \psi }}<\frac{kP(k)}{<k>}{\displaystyle \sum _k\frac{\xi \psi (\beta +\theta )}{\xi \psi (\beta +\theta )}}=1$ and $Y_k(0)=0$, there are at most two fixed points on $[0,1]$ for $Y_k(\Theta )=\frac{k^{2}P(k)}{<k>}{\displaystyle \sum _k\frac{\xi \psi (\beta +\theta )\Theta }{\beta \kappa \epsilon +k\beta \kappa \psi \Theta }}$. Moreover, given that ${Y_k{}^{\prime }(\Theta )|}_{\Theta =0}>1$, $Y_k(\Theta )=\frac{k^{2}P(k)}{<k>}{\displaystyle \sum _k\frac{\xi \psi (\beta +\theta )\Theta }{\beta \kappa \epsilon +k\beta \kappa \psi \Theta }}$ has a unique fixed point on $[0,1]$. Therefore, Theorem 1 is proven. □

According to Theorem 1, the equilibrium state of risk contagion of local government implicit debt is closely related to the relevant factors of the multiple subjects. Moreover, adjusting the factors corresponding to different subjects will affect the risk contagion status of local government implicit debt.

4.2. Threshold Analysis of Risk Contagion of Local Government Implicit Debt

Theorem 2:

The risk contagion threshold $R_0$ of local government implicit debt is a convex function of information openness accuracy $\eta$ and information disclosure strategy $\chi$, which first monotonically decreases and then monotonically increases. In addition, the risk contagion threshold $R_0$ of local government implicit debt is a monotonically decreasing convex function of debt management level $\alpha$, emotion tendency $\varphi$, risk preference level $\lambda$, credit policy robustness $\omega$, accountability mechanism soundness $\rho$, and perfection of laws and regulations $\tau$.

Proof: 

$$\{\begin{array}{c}\frac{\partial R_0}{\partial \eta }=2\eta -1\le 0,\hspace{0.17em}\mathrm{When} \omega >\frac{1}{2}\\ \frac{\partial R_0}{\partial \eta }=2\eta -1>0,\hspace{0.17em}\mathrm{When} \omega <\frac{1}{2}\end{array}$$

(21)

$$\frac{{\partial }^2{R}_0}{\partial {\eta }^2}=2>0$$

(22)

Therefore, it can be seen from Equations (21) and (22) that the risk contagion threshold $R_0$ of local government implicit debt is a convex function of information openness accuracy $\eta$, which monotonically decreases first and then monotonically increases.

Similarly, it can be concluded that $\{\begin{array}{c}\frac{\partial R_0}{\partial \chi }\le 0,\hspace{0.17em}\hspace{0.17em}\mathrm{When} \chi >\frac{1}{2}\\ \frac{\partial R_0}{\partial \chi }>0,\hspace{0.17em}\hspace{0.17em}\mathrm{When} \chi <\frac{1}{2}\end{array}$, $\frac{{\partial }^2{R}_0}{\partial {\chi }^2}>0$; $\frac{\partial R_0}{\partial \alpha }<0$, $\frac{{\partial }^2{R}_0}{\partial {\alpha }^2}>0$; $\frac{\partial R_0}{\partial \varphi }<0$, $\frac{{\partial }^2{R}_0}{\partial {\varphi }^2}>0$; $\frac{\partial R_0}{\partial \omega }<0$, $\frac{{\partial }^2{R}_0}{\partial {\omega }^2}>0$; $\frac{\partial R_0}{\partial \lambda }<0$, $\frac{{\partial }^2{R}_0}{\partial {\lambda }^2}>0$; $\frac{\partial R_0}{\partial \rho }<0$, $\frac{{\partial }^2{R}_0}{\partial {\rho }^2}>0$; $\frac{\partial R_0}{\partial \tau }<0$, $\frac{{\partial }^2{R}_0}{\partial {\tau }^2}>0$. It can be seen that the risk contagion threshold $R_0$ of local government implicit debt is a monotonically decreasing convex function of the debt management level $\alpha$, emotion tendency $\varphi$, risk preference level $\lambda$, credit policy robustness $\omega$, accountability mechanism soundness $\rho$, and perfection of laws and regulations $\tau$. Therefore, Theorem 2 is proven. □

According to Theorem 2, to improve the effectiveness of controlling the risk contagion of local government implicit debt, it is necessary to focus on enhancing the debt management level $\alpha$, emotion tendency $\varphi$, risk preference level $\lambda$, credit policy robustness $\omega$, accountability mechanism soundness $\rho$, and perfection of laws and regulations $\tau$. At the same time, the government and the media should respectively control information openness accuracy and information disclosure strategy within a moderate range.

5. Simulation Analysis

Based on the above-constructed risk contagion model of local government implicit debt, we next simulate and analyze the evolutionary characteristics and control strategies for risk contagion of local government implicit debt under multi-subject collaboration. This section sets benchmark values for the debt management level, information openness accuracy, information disclosure strategy, emotion tendency, risk preference level, credit policy robustness, accountability mechanism soundness, and perfection of laws and regulations based on Duan et al. [4], Guo et al. [5], Liu and Li [6], Feng et al. [9], Dong et al. [10], Cutura [12], and Zhao et al. [13]. The benchmark values for the other parameters in the model are mainly determined based on the research of Wang et al. [32], Qian et al. [35], and Chen et al. [34]. The specific benchmark value setting results for each parameter are listed in

Table 1. Benchmark values of parameters.

Parameter	Description	Benchmark Value	Value Range
$\alpha$	Debt management level	0.5	[0, 1]
$\eta$	Information openness accuracy	0.4	[0, 1]
$\chi$	Information disclosure strategy	0.6	[0, 1]
$\varphi$	Emotion tendency	0.3	[0, 1]
$\omega$	Credit policy robustness	0.7	[0, 1]
$\lambda$	Risk preference level	0.4	[0, 1]
$\rho$	Accountability mechanism soundness	0.5	[0, 1]
$\tau$	Perfection of laws and regulations	0.6	[0, 1]
$q$	The probability of random connection among newly added local financing platforms and existing local financing platforms in the network	0.2	[0, 1]
$a$	Number of new local financing platforms added to the network	20	Positive integer
$\delta$	The number of association relationships between newly added and local financing platforms in the network	20	Positive integer
$\psi$	The probability of transitioning from sensitive financing platforms into fragile financing platforms	0.3	[0, 1]
$\epsilon$	The probability of transitioning from sensitive financing platforms into robust financing platforms	0.2	[0, 1]
$\kappa$	The probability of transitioning from fragile financing platforms into robust financing platforms	0.1	[0, 1]
$\theta$	The probability of robust financing platforms re-transforming into sensitive financing platforms	0.2	[0, 1]
$\xi$	The probability of new local financing platforms entering the network	0.2	[0, 1]
$\beta$	The probability of new local financing platforms exiting the network	0.1	[0, 1]
$k$	Degree of existing local financing platforms	200	Positive integer
$N$	Total number of local financing platforms	200	Positive integer

.

5.1. Structural Evolution of the Local Financing Platform Association Network Characteristics

Figure 3 shows the risk contagion threshold and evolution characteristics of local government implicit debt under multi-subject collaboration in the different network structures of local financing platforms. In Figure 3, it can be seen that the greater the probability of random connections, the stronger the risk contagion of local government implicit debt. This indicates that as local financing platforms tend to establish associations in a random manner, a local government’s implicit debt risk is more likely to spread rapidly in both global and local networks. This further indicates that the scale-free network is not conducive to the risk contagion of local government implicit debt. However, the random network is most conducive to the risk contagion of local government implicit debt. This is contrary to the research conclusion of Wang et al. [32]. The reason for this is that this study relies on multi-subject collaboration, while Wang et al. [32] rely on single-subject collaboration. This further highlights the necessity of using multi-subject collaboration in this study. In addition, this also indicates that when controlling a local government’s implicit debt risk, the focus should be on local financing platforms that have significant potential risks and occupy a core position. Due to the huge scale of financing, the numerous stakeholders involved, and the complex related aspects of these local financing platforms, a default or bankruptcy occurs that may lead to public opinion hotspots and typical default events of local debt, causing panic among investors and financial institutions, and subsequently triggering a large-scale risk contagion of local government implicit debt. Therefore, local governments at all levels and the central government should focus on identifying core local financing platforms and conducting real-time monitoring and early warning of their potential risks.

As shown in Figure 3a,d,e–h, as the debt management level, emotion tendency, credit policy robustness, risk preference level, accountability mechanism soundness, and perfection of laws and regulations increase, the risk contagion of local government implicit debt shows a monotonically decreasing trend. This proves Theorem 2. Further analysis shows that the debt management level has the most significant impact on the risk contagion of local government implicit debt followed by credit policy robustness, perfection of laws and regulations, emotion tendency, accountability mechanism soundness, and risk preference level. This indicates that from the perspective of the single-factor impact effect of multi-subject collaboration, to reduce the risk contagion of local government implicit debt, it is necessary to focus on improving the debt management level of local governments at all levels. At the same time, efforts should be made to maintain the credit policy of financial institutions, the laws and regulations of the central government, and the accountability mechanism as sound as possible and to make the media emotion tendencies positive and the risk tolerance of financial institutions as high as possible. Furthermore, it should be noted that the relationship between risk preference level and the risk contagion of local government implicit debt is different from the research findings of Feng et al. [9]. Feng et al. [9] analyzed the risks of funds under normal use from the perspective of government guarantees, while this study investigated how to reduce risks after a crisis occurs. Both conclusions have their applicability and rationality.

As shown in Figure 3b,c, as the information openness accuracy and information disclosure strategy gradually increase, the risk contagion of local government implicit debt shows a positive “U” shaped trend of monotonic decrease first and then monotonic increase. This further confirms Theorem 2. At the same time, this indicates that the information openness accuracy and information disclosure strategy are “double-edged swords”. That is, when information openness is excessively distorted or precise, or information disclosure is excessively biased toward positive or negative values, it is highly likely that misunderstandings and suspicions will arise among investors and the public, and thus, through the “herd effect”, widespread risk contagion of local government implicit debt is caused. Therefore, local governments should try their best to maintain a relatively moderate accuracy in information disclosure while maintaining a relatively objective information disclosure strategy. However, the scale of these two tasks is difficult to grasp, which to some extent indicates difficulty in controlling the risk contagion of local government implicit debt.

5.2. Evolution Characteristics of Risk Contagion of Local Government Implicit Debt

Figure 4 shows the risk contagion threshold and evolution characteristics of local government implicit debt under multi-subject collaboration. According to Figure 4c–g,s–ab, it can be observed that when the debt management level, emotion tendency, credit policy robustness, risk preference level, accountability mechanism soundness, and perfection of laws and regulations increase simultaneously, the risk contagion of local government implicit debt shows a monotonically decreasing trend. This indicates that by comprehensively adjusting the debt management level, emotional tendency, credit policy robustness, risk preference level, accountability mechanism soundness, and perfection of laws and regulations, the risk contagion intensity of local government implicit debt can be reduced. However, to eliminate a local government’s implicit debt risk, these six factors need to be sufficiently large. The debt management level, credit policy robustness, and perfection of laws and regulations need to be almost perfect to achieve a contagion threshold of less than 1. This not only indicates that eliminating local government implicit debt risk should focus on local governments at all levels and the central government but also deeply indicates that local government implicit debt risk cannot be eradicated. This is determined by the special purpose and characteristics of local government implicit debt. Therefore, when dealing with a local government’s implicit debt risk, the focus of control should be on reducing the risk rather than eliminating it. At the same time, local governments should strengthen their own debt management level, and the central government should continuously improve accountability mechanisms, laws, and regulations. In this process, financial institutions and the media should actively play the role of “stability maintainers” and should not blindly adopt irrational behaviors such as loan withdrawals, loan restrictions, and the release of negative information.

According to Figure 4a,j,m, it can be observed that when the information openness accuracy increases simultaneously with the debt management level, credit policy robustness, and perfection of laws and regulations, the risk contagion of local government implicit debt shows a monotonically decreasing trend. Similarly, according to Figure 4b,o,r, when the information disclosure strategy increases simultaneously with the debt management level, credit policy robustness, and perfection of laws and regulations, the risk contagion of local government implicit debt also shows a monotonically decreasing trend. This reflects the impact of information openness accuracy and information disclosure strategy on the risk contagion of local government implicit debt, which is weaker than the debt management level, credit policy robustness, and perfection of laws and regulations. Moreover, synchronizing information openness accuracy or information disclosure strategy with the debt management level, credit policy robustness, and perfection of laws and regulations can also play a role in reducing the risk contagion of local government implicit debt. However, only when information openness accuracy is close to moderate or the information disclosure strategy is close to objective, and the debt management level, credit policy robustness, and perfection of laws and regulations are sufficiently large, can the contagion threshold be less than 1, which means eliminating the risk contagion of local government implicit debt. This further confirms that local government implicit debt risk cannot be eradicated.

According to Figure 4h,i,k,l,n, it can be observed that when information openness accuracy increases synchronously with information disclosure strategy, emotional tendency, risk preference level, and accountability mechanism soundness, the risk contagion of local government implicit debt shows a positive “U” shaped trend, first monotonically decreasing and then monotonically increasing. This reflects the impact of information openness accuracy on the risk contagion of local government implicit debt, which is stronger than the information disclosure strategy, emotional tendency, risk preference level, and accountability mechanism soundness. Similarly, according to Figure 4p,q, it can be observed that when the information disclosure strategy increases simultaneously with the risk preference level and accountability mechanism soundness, the risk contagion of local government implicit debt shows a positive “U” shaped trend, first monotonically decreasing and then monotonically increasing. This reflects the impact of the information disclosure strategy on the risk contagion of local government implicit debt, which is stronger than the risk preference level and accountability mechanism soundness. Moreover, synchronizing information openness accuracy with the information disclosure strategy, emotion tendency, risk preference level, and accountability mechanism soundness, or synchronizing the information disclosure strategy with the risk preference level and accountability mechanism soundness, can also reduce the risk contagion of local government implicit debt, but the risk contagion threshold of local government implicit debt has always been greater than 1. In such cases, local government implicit debt risk has been contagious in the global network of local financing platforms and cannot be eliminated. This once again confirms the conclusion obtained above.

5.3. Robustness Testing

To test the evolution robustness of risk contagion of local government implicit debt under the synergistic effect of multi-subject collaboration, we conducted sensitivity analysis on the parameters corresponding to the multiple subjects.

Table 2. Sensitivity analysis of risk contagion of local government implicit debt under the coordination of local governments at all levels and the media.

$\mathit{\alpha }$	$\mathit{\eta }$										Expectation	Variance
	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
$\chi =0.2$ $\varphi =0.3$
0.1	3.698	3.552	3.445	3.379	3.357	3.379	3.445	3.552	3.698	3.876	3.538	0.030
0.2	2.615	2.512	2.436	2.389	2.374	2.389	2.436	2.512	2.615	2.741	2.502	0.015
0.3	2.135	2.051	1.989	1.951	1.938	1.951	1.989	2.051	2.135	2.238	2.043	0.010
0.4	1.849	1.776	1.723	1.690	1.678	1.690	1.723	1.776	1.849	1.938	1.769	0.007
0.5	1.653	1.588	1.540	1.510	1.500	1.510	1.540	1.588	1.653	1.732	1.581	0.006
0.6	1.502	1.443	1.400	1.373	1.364	1.373	1.400	1.443	1.502	1.575	1.437	0.005
0.7	1.374	1.320	1.280	1.256	1.247	1.256	1.280	1.320	1.374	1.440	1.315	0.004
0.8	1.258	1.209	1.172	1.150	1.142	1.150	1.172	1.209	1.258	1.319	1.204	0.003
0.9	1.151	1.106	1.073	1.052	1.045	1.052	1.073	1.106	1.151	1.207	1.101	0.003
1	1.053	1.012	0.981	0.962	0.956	0.962	0.981	1.012	1.053	1.104	1.008	0.002
$\chi =0.5$ $\varphi =0.3$
0.1	3.424	3.290	3.190	3.129	3.108	3.129	3.190	3.290	3.424	3.589	3.276	0.025
0.2	2.421	2.326	2.256	2.213	2.198	2.213	2.256	2.326	2.421	2.538	2.317	0.013
0.3	1.977	1.899	1.842	1.807	1.795	1.807	1.842	1.899	1.977	2.072	1.892	0.008
0.4	1.712	1.645	1.595	1.564	1.554	1.564	1.595	1.645	1.712	1.795	1.638	0.006
0.5	1.530	1.470	1.426	1.398	1.389	1.398	1.426	1.470	1.530	1.604	1.464	0.005
0.6	1.391	1.336	1.296	1.271	1.263	1.271	1.296	1.336	1.391	1.458	1.331	0.004
0.7	1.272	1.222	1.186	1.163	1.155	1.163	1.186	1.222	1.272	1.334	1.218	0.004
0.8	1.165	1.119	1.085	1.065	1.058	1.065	1.085	1.119	1.165	1.221	1.115	0.003
0.9	1.066	1.024	0.993	0.974	0.968	0.974	0.993	1.024	1.066	1.117	1.020	0.002
1	0.975	0.937	0.909	0.891	0.885	0.891	0.909	0.937	0.975	1.022	0.933	0.002
$\chi =0.8$ $\varphi =0.3$
0.1	3.698	3.552	3.445	3.379	3.357	3.379	3.445	3.552	3.698	3.876	3.538	0.030
0.2	2.615	2.512	2.436	2.389	2.374	2.389	2.436	2.512	2.615	2.741	2.502	0.015
0.3	2.135	2.051	1.989	1.951	1.938	1.951	1.989	2.051	2.135	2.238	2.043	0.010
0.4	1.849	1.776	1.723	1.690	1.678	1.690	1.723	1.776	1.849	1.938	1.769	0.007
0.5	1.653	1.588	1.540	1.510	1.500	1.510	1.540	1.588	1.653	1.732	1.581	0.006
0.6	1.502	1.443	1.400	1.373	1.364	1.373	1.400	1.443	1.502	1.575	1.437	0.005
0.7	1.374	1.320	1.280	1.256	1.247	1.256	1.280	1.320	1.374	1.440	1.315	0.004
0.8	1.258	1.209	1.172	1.150	1.142	1.150	1.172	1.209	1.258	1.319	1.204	0.003
0.9	1.151	1.106	1.073	1.052	1.045	1.052	1.073	1.106	1.151	1.207	1.101	0.003
1	1.053	1.012	0.981	0.962	0.956	0.962	0.981	1.012	1.053	1.104	1.008	0.002

,

Table 3. Sensitivity analysis of risk contagion of local government implicit debt under the coordination of local governments at all levels and financial institutions.

$\mathit{\alpha }$	$\mathit{\eta }$										Expectation	Variance
	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
$\omega =0.2$ $\lambda =0.4$
0.1	5.035	4.838	4.691	4.601	4.571	4.601	4.691	4.838	5.035	5.278	4.818	0.055
0.2	3.560	3.421	3.317	3.254	3.232	3.254	3.317	3.421	3.560	3.732	3.407	0.027
0.3	2.907	2.793	2.709	2.657	2.639	2.657	2.709	2.793	2.907	3.047	2.782	0.018
0.4	2.517	2.419	2.346	2.301	2.285	2.301	2.346	2.419	2.517	2.639	2.409	0.014
0.5	2.250	2.162	2.097	2.056	2.043	2.056	2.097	2.162	2.250	2.359	2.153	0.011
0.6	2.045	1.965	1.906	1.869	1.857	1.869	1.906	1.965	2.045	2.144	1.957	0.009
0.7	1.871	1.798	1.743	1.710	1.699	1.710	1.743	1.798	1.871	1.961	1.790	0.008
0.8	1.713	1.646	1.596	1.566	1.555	1.566	1.596	1.646	1.713	1.796	1.639	0.006
0.9	1.567	1.506	1.460	1.432	1.423	1.432	1.460	1.506	1.567	1.643	1.500	0.005
1	1.434	1.378	1.336	1.310	1.302	1.310	1.336	1.378	1.434	1.503	1.372	0.004
$\omega =0.5$ $\lambda =0.4$
0.1	3.825	3.675	3.564	3.495	3.472	3.495	3.564	3.675	3.825	4.010	3.660	0.032
0.2	2.705	2.599	2.520	2.472	2.455	2.472	2.520	2.599	2.705	2.835	2.588	0.016
0.3	2.208	2.122	2.058	2.018	2.005	2.018	2.058	2.122	2.208	2.315	2.113	0.011
0.4	1.912	1.837	1.782	1.748	1.736	1.748	1.782	1.837	1.912	2.005	1.830	0.008
0.5	1.709	1.642	1.593	1.562	1.552	1.562	1.593	1.642	1.709	1.792	1.636	0.006
0.6	1.554	1.493	1.448	1.420	1.411	1.420	1.448	1.493	1.554	1.629	1.487	0.005
0.7	1.421	1.366	1.324	1.299	1.290	1.299	1.324	1.366	1.421	1.490	1.360	0.004
0.8	1.301	1.250	1.213	1.189	1.181	1.189	1.213	1.250	1.301	1.364	1.245	0.004
0.9	1.191	1.144	1.109	1.088	1.081	1.088	1.109	1.144	1.191	1.248	1.139	0.003
1	1.089	1.047	1.015	0.996	0.989	0.996	1.015	1.047	1.089	1.142	1.042	0.003
$\omega =0.8$ $\lambda =0.4$
0.1	3.322	3.192	3.095	3.036	3.016	3.036	3.095	3.192	3.322	3.482	3.179	0.024
0.2	2.349	2.257	2.189	2.147	2.132	2.147	2.189	2.257	2.349	2.462	2.248	0.012
0.3	1.918	1.843	1.787	1.753	1.741	1.753	1.787	1.843	1.918	2.010	1.835	0.008
0.4	1.661	1.596	1.548	1.518	1.508	1.518	1.548	1.596	1.661	1.741	1.589	0.006
0.5	1.485	1.426	1.383	1.357	1.348	1.357	1.383	1.426	1.485	1.556	1.421	0.005
0.6	1.350	1.297	1.257	1.233	1.225	1.233	1.257	1.297	1.350	1.415	1.291	0.004
0.7	1.234	1.186	1.150	1.128	1.121	1.128	1.150	1.186	1.234	1.294	1.181	0.003
0.8	1.130	1.086	1.053	1.033	1.026	1.033	1.053	1.086	1.130	1.185	1.081	0.003
0.9	1.034	0.994	0.964	0.945	0.939	0.945	0.964	0.994	1.034	1.084	0.990	0.002
1	0.946	0.909	0.881	0.865	0.859	0.865	0.881	0.909	0.946	0.992	0.905	0.002

,

Table 4. Sensitivity analysis of risk contagion of local government implicit debt under the coordination of local governments at all levels and central government.

$\mathit{\alpha }$	$\mathit{\eta }$										Expectation	Variance
	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
$\rho =0.2$ $\tau =0.6$
0.1	3.756	3.609	3.500	3.432	3.410	3.432	3.500	3.609	3.756	3.937	3.594	0.031
0.2	2.656	2.552	2.475	2.427	2.411	2.427	2.475	2.552	2.656	2.784	2.541	0.015
0.3	2.169	2.083	2.020	1.982	1.969	1.982	2.020	2.083	2.169	2.273	2.075	0.010
0.4	1.878	1.804	1.750	1.716	1.705	1.716	1.750	1.804	1.878	1.969	1.797	0.008
0.5	1.679	1.613	1.564	1.534	1.524	1.534	1.564	1.613	1.679	1.760	1.606	0.006
0.6	1.526	1.466	1.422	1.394	1.385	1.394	1.422	1.466	1.526	1.600	1.460	0.005
0.7	1.396	1.341	1.300	1.276	1.267	1.276	1.300	1.341	1.396	1.463	1.336	0.004
0.8	1.278	1.228	1.191	1.168	1.160	1.168	1.191	1.228	1.278	1.340	1.223	0.004
0.9	1.169	1.123	1.089	1.069	1.062	1.069	1.089	1.123	1.169	1.226	1.119	0.003
1	1.070	1.028	0.997	0.978	0.971	0.978	0.997	1.028	1.070	1.121	1.024	0.002
$\rho =0.5$ $\tau =0.6$
0.1	3.458	3.322	3.222	3.160	3.139	3.160	3.222	3.322	3.458	3.625	3.309	0.026
0.2	2.445	2.349	2.278	2.234	2.220	2.234	2.278	2.349	2.445	2.563	2.340	0.013
0.3	1.996	1.918	1.860	1.824	1.812	1.824	1.860	1.918	1.996	2.093	1.910	0.009
0.4	1.729	1.661	1.611	1.580	1.569	1.580	1.611	1.661	1.729	1.812	1.654	0.006
0.5	1.545	1.485	1.440	1.412	1.403	1.412	1.440	1.485	1.545	1.620	1.479	0.005
0.6	1.405	1.350	1.309	1.284	1.275	1.284	1.309	1.350	1.405	1.473	1.344	0.004
0.7	1.285	1.235	1.197	1.174	1.166	1.174	1.197	1.235	1.285	1.347	1.230	0.004
0.8	1.176	1.130	1.096	1.075	1.068	1.075	1.096	1.130	1.176	1.233	1.126	0.003
0.9	1.076	1.034	1.003	0.984	0.977	0.984	1.003	1.034	1.076	1.128	1.030	0.003
1	0.985	0.946	0.918	0.900	0.894	0.900	0.918	0.946	0.985	1.032	0.942	0.002
$\rho =0.8$ $\tau =0.6$
0.1	3.261	3.133	3.038	2.980	2.960	2.980	3.038	3.133	3.261	3.418	3.120	0.023
0.2	2.306	2.215	2.148	2.107	2.093	2.107	2.148	2.215	2.306	2.417	2.206	0.012
0.3	1.883	1.809	1.754	1.720	1.709	1.720	1.754	1.809	1.883	1.973	1.801	0.008
0.4	1.630	1.566	1.519	1.490	1.480	1.490	1.519	1.566	1.630	1.709	1.560	0.006
0.5	1.457	1.400	1.358	1.332	1.323	1.332	1.358	1.400	1.457	1.528	1.394	0.005
0.6	1.325	1.273	1.234	1.211	1.203	1.211	1.234	1.273	1.325	1.389	1.268	0.004
0.7	1.212	1.164	1.129	1.107	1.100	1.107	1.129	1.164	1.212	1.270	1.159	0.003
0.8	1.109	1.066	1.034	1.014	1.007	1.014	1.034	1.066	1.109	1.163	1.062	0.003
0.9	1.015	0.975	0.946	0.928	0.922	0.928	0.946	0.975	1.015	1.064	0.971	0.002
1	0.929	0.892	0.865	0.849	0.843	0.849	0.865	0.892	0.929	0.973	0.889	0.002

,

Table 5. Sensitivity analysis of risk contagion of local government implicit debt under the coordination of the media and financial institutions.

$\mathit{\chi }$	$\mathit{\varphi }$										Expectation	Variance
	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
$\omega =0.2$ $\lambda =0.4$
0.1	2.547	2.423	2.305	2.192	2.085	1.984	1.887	1.795	1.707	1.624	2.055	0.096
0.2	2.430	2.312	2.199	2.092	1.990	1.893	1.801	1.713	1.629	1.550	1.961	0.088
0.3	2.336	2.222	2.113	2.010	1.912	1.819	1.730	1.646	1.566	1.489	1.884	0.081
0.4	2.273	2.162	2.056	1.956	1.861	1.770	1.684	1.602	1.523	1.449	1.834	0.077
0.5	2.251	2.141	2.036	1.937	1.843	1.753	1.667	1.586	1.509	1.435	1.816	0.075
0.6	2.273	2.162	2.056	1.956	1.861	1.770	1.684	1.602	1.523	1.449	1.834	0.077
0.7	2.336	2.222	2.113	2.010	1.912	1.819	1.730	1.646	1.566	1.489	1.884	0.081
0.8	2.430	2.312	2.199	2.092	1.990	1.893	1.801	1.713	1.629	1.550	1.961	0.088
0.9	2.547	2.423	2.305	2.192	2.085	1.984	1.887	1.795	1.707	1.624	2.055	0.096
1	2.676	2.546	2.422	2.304	2.191	2.084	1.983	1.886	1.794	1.707	2.159	0.106
$\omega =0.5$ $\lambda =0.4$
0.1	1.935	1.840	1.751	1.665	1.584	1.507	1.433	1.363	1.297	1.234	1.561	0.056
0.2	1.846	1.756	1.671	1.589	1.512	1.438	1.368	1.301	1.238	1.177	1.490	0.051
0.3	1.774	1.688	1.605	1.527	1.453	1.382	1.314	1.250	1.189	1.131	1.431	0.047
0.4	1.727	1.642	1.562	1.486	1.414	1.345	1.279	1.217	1.157	1.101	1.393	0.044
0.5	1.710	1.626	1.547	1.472	1.400	1.332	1.267	1.205	1.146	1.090	1.379	0.043
0.6	1.727	1.642	1.562	1.486	1.414	1.345	1.279	1.217	1.157	1.101	1.393	0.044
0.7	1.774	1.688	1.605	1.527	1.453	1.382	1.314	1.250	1.189	1.131	1.431	0.047
0.8	1.846	1.756	1.671	1.589	1.512	1.438	1.368	1.301	1.238	1.177	1.490	0.051
0.9	1.935	1.840	1.751	1.665	1.584	1.507	1.433	1.363	1.297	1.234	1.561	0.056
1	2.033	1.934	1.840	1.750	1.665	1.583	1.506	1.433	1.363	1.296	1.640	0.061
$\omega =0.8$ $\lambda =0.4$
0.1	1.680	1.598	1.520	1.446	1.376	1.309	1.245	1.184	1.126	1.071	1.356	0.042
0.2	1.603	1.525	1.451	1.380	1.313	1.249	1.188	1.130	1.075	1.022	1.294	0.038
0.3	1.541	1.466	1.394	1.326	1.262	1.200	1.142	1.086	1.033	0.983	1.243	0.035
0.4	1.499	1.426	1.357	1.291	1.228	1.168	1.111	1.057	1.005	0.956	1.210	0.033
0.5	1.485	1.412	1.344	1.278	1.216	1.156	1.100	1.046	0.995	0.947	1.198	0.033
0.6	1.499	1.426	1.357	1.291	1.228	1.168	1.111	1.057	1.005	0.956	1.210	0.033
0.7	1.541	1.466	1.394	1.326	1.262	1.200	1.142	1.086	1.033	0.983	1.243	0.035
0.8	1.603	1.525	1.451	1.380	1.313	1.249	1.188	1.130	1.075	1.022	1.294	0.038
0.9	1.680	1.598	1.520	1.446	1.376	1.309	1.245	1.184	1.126	1.071	1.356	0.042
1	1.766	1.680	1.598	1.520	1.446	1.375	1.308	1.244	1.184	1.126	1.425	0.046

,

Table 6. Sensitivity analysis of risk contagion of local government implicit debt under the coordination of the media and central government.

$\mathit{\chi }$	$\mathit{\varphi }$										Expectation	Variance
	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
$\rho =0.2$ $\tau =0.6$
0.1	1.900	1.807	1.719	1.635	1.556	1.480	1.407	1.339	1.274	1.211	1.533	0.054
0.2	1.813	1.725	1.640	1.560	1.484	1.412	1.343	1.278	1.215	1.156	1.463	0.049
0.3	1.742	1.657	1.577	1.500	1.427	1.357	1.291	1.228	1.168	1.111	1.406	0.045
0.4	1.695	1.613	1.534	1.459	1.388	1.320	1.256	1.195	1.136	1.081	1.368	0.043
0.5	1.679	1.597	1.519	1.445	1.375	1.308	1.244	1.183	1.125	1.070	1.354	0.042
0.6	1.695	1.613	1.534	1.459	1.388	1.320	1.256	1.195	1.136	1.081	1.368	0.043
0.7	1.742	1.657	1.577	1.500	1.427	1.357	1.291	1.228	1.168	1.111	1.406	0.045
0.8	1.813	1.725	1.640	1.560	1.484	1.412	1.343	1.278	1.215	1.156	1.463	0.049
0.9	1.900	1.807	1.719	1.635	1.556	1.480	1.407	1.339	1.274	1.211	1.533	0.054
1	1.997	1.899	1.807	1.718	1.635	1.555	1.479	1.407	1.338	1.273	1.611	0.059
$\rho =0.5$ $\tau =0.6$
0.1	1.749	1.664	1.583	1.505	1.432	1.362	1.296	1.232	1.172	1.115	1.411	0.045
0.2	1.669	1.588	1.510	1.437	1.366	1.300	1.236	1.176	1.119	1.064	1.347	0.041
0.3	1.604	1.526	1.451	1.381	1.313	1.249	1.188	1.130	1.075	1.023	1.294	0.038
0.4	1.561	1.485	1.412	1.343	1.278	1.216	1.156	1.100	1.046	0.995	1.259	0.036
0.5	1.546	1.470	1.398	1.330	1.265	1.204	1.145	1.089	1.036	0.985	1.247	0.035
0.6	1.561	1.485	1.412	1.343	1.278	1.216	1.156	1.100	1.046	0.995	1.259	0.036
0.7	1.604	1.526	1.451	1.381	1.313	1.249	1.188	1.130	1.075	1.023	1.294	0.038
0.8	1.669	1.588	1.510	1.437	1.366	1.300	1.236	1.176	1.119	1.064	1.347	0.041
0.9	1.749	1.664	1.583	1.505	1.432	1.362	1.296	1.232	1.172	1.115	1.411	0.045
1	1.838	1.748	1.663	1.582	1.505	1.431	1.362	1.295	1.232	1.172	1.483	0.050
$\rho =0.8$ $\tau =0.6$
0.1	1.649	1.569	1.492	1.420	1.350	1.284	1.222	1.162	1.106	1.052	1.331	0.040
0.2	1.574	1.497	1.424	1.355	1.289	1.226	1.166	1.109	1.055	1.004	1.270	0.037
0.3	1.513	1.439	1.369	1.302	1.238	1.178	1.121	1.066	1.014	0.964	1.220	0.034
0.4	1.472	1.400	1.332	1.267	1.205	1.146	1.090	1.037	0.987	0.938	1.187	0.032
0.5	1.457	1.386	1.319	1.254	1.193	1.135	1.080	1.027	0.977	0.929	1.176	0.032
0.6	1.472	1.400	1.332	1.267	1.205	1.146	1.090	1.037	0.987	0.938	1.187	0.032
0.7	1.513	1.439	1.369	1.302	1.238	1.178	1.121	1.066	1.014	0.964	1.220	0.034
0.8	1.574	1.497	1.424	1.355	1.289	1.226	1.166	1.109	1.055	1.004	1.270	0.037
0.9	1.649	1.569	1.492	1.420	1.350	1.284	1.222	1.162	1.106	1.052	1.331	0.040
1	1.733	1.649	1.568	1.492	1.419	1.350	1.284	1.221	1.162	1.105	1.398	0.045

and

Table 7. Sensitivity analysis of risk contagion of local government implicit debt under the coordination of financial institutions and central government.

ω	λ										Expectation	Variance
	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
ρ = 0.2 τ = 0.6
0.1	3.885	3.462	3.086	2.750	2.451	2.185	1.947	1.735	1.547	1.378	2.443	0.708
0.2	2.844	2.624	2.421	2.234	2.061	1.902	1.755	1.619	1.494	1.378	2.033	0.243
0.3	2.370	2.231	2.101	1.978	1.863	1.754	1.651	1.555	1.464	1.378	1.834	0.111
0.4	2.082	1.989	1.900	1.815	1.733	1.656	1.581	1.511	1.443	1.378	1.709	0.056
0.5	1.883	1.819	1.757	1.697	1.639	1.583	1.529	1.477	1.427	1.378	1.619	0.029
0.6	1.735	1.691	1.648	1.607	1.566	1.527	1.488	1.451	1.414	1.378	1.550	0.014
0.7	1.618	1.590	1.562	1.534	1.507	1.480	1.454	1.428	1.403	1.378	1.496	0.007
0.8	1.524	1.507	1.490	1.474	1.457	1.441	1.425	1.410	1.394	1.378	1.450	0.002
0.9	1.445	1.438	1.430	1.423	1.415	1.408	1.400	1.393	1.386	1.378	1.412	0.001
1	1.378	1.378	1.378	1.378	1.378	1.378	1.378	1.378	1.378	1.378	1.378	0.000
ρ = 0.5 τ = 0.6
0.1	3.576	3.187	2.841	2.532	2.257	2.011	1.792	1.597	1.424	1.269	2.249	0.600
0.2	2.618	2.416	2.229	2.056	1.897	1.751	1.615	1.491	1.375	1.269	1.872	0.206
0.3	2.181	2.054	1.934	1.821	1.715	1.614	1.520	1.431	1.348	1.269	1.689	0.094
0.4	1.917	1.831	1.749	1.670	1.596	1.524	1.456	1.391	1.328	1.269	1.573	0.047
0.5	1.733	1.674	1.617	1.562	1.509	1.458	1.408	1.360	1.314	1.269	1.490	0.024
0.6	1.597	1.557	1.517	1.479	1.442	1.405	1.370	1.335	1.302	1.269	1.427	0.012
0.7	1.490	1.464	1.438	1.412	1.387	1.363	1.339	1.315	1.292	1.269	1.377	0.006
0.8	1.403	1.387	1.372	1.357	1.342	1.327	1.312	1.298	1.283	1.269	1.335	0.002
0.9	1.331	1.324	1.317	1.310	1.303	1.296	1.289	1.282	1.276	1.269	1.300	0.000
1	1.269	1.269	1.269	1.269	1.269	1.269	1.269	1.269	1.269	1.269	1.269	0.000
ρ = 0.8 τ = 0.6
0.1	3.373	3.006	2.679	2.388	2.128	1.897	1.690	1.506	1.343	1.197	2.121	0.534
0.2	2.469	2.278	2.102	1.939	1.789	1.651	1.523	1.406	1.297	1.197	1.765	0.183
0.3	2.057	1.937	1.824	1.717	1.617	1.522	1.433	1.350	1.271	1.197	1.592	0.084
0.4	1.807	1.726	1.649	1.575	1.505	1.437	1.373	1.311	1.253	1.197	1.483	0.042
0.5	1.635	1.579	1.525	1.473	1.423	1.375	1.328	1.283	1.239	1.197	1.406	0.022
0.6	1.506	1.468	1.431	1.395	1.360	1.325	1.292	1.259	1.228	1.197	1.346	0.011
0.7	1.405	1.380	1.356	1.332	1.308	1.285	1.262	1.240	1.218	1.197	1.298	0.005
0.8	1.323	1.308	1.294	1.279	1.265	1.251	1.237	1.224	1.210	1.197	1.259	0.002
0.9	1.255	1.248	1.242	1.235	1.229	1.222	1.216	1.209	1.203	1.197	1.225	0.000
1	1.197	1.197	1.197	1.197	1.197	1.197	1.197	1.197	1.197	1.197	1.197	0.000

respectively show the sensitivity analysis of risk contagion of local government implicit debt under the collaborative effect of local governments at all levels, the media, financial institutions, and central government. From Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7, under different numerical settings, the variation trend in the same parameter is basically the same, indicating strong robustness. In addition, further analysis reveals that the debt management level of local governments at all levels plays a “reinforcer” role in risk contagion. In other words, once the debt management level of local governments at all levels is insufficient, the local governments implicit debt risk will synchronously trigger larger-scale contagion in the global and local association networks under the interaction between debt management level and other factors. Therefore, the starting point and foothold for preventing and resolving local government implicit debt risk should focus on improving the debt management level of local governments at all levels, continuously optimizing management, planning, and utilization efficiencies, and financing sustainability of local government implicit debt.

In addition, according to the results of simulation and robustness, the formation and evolution of risk contagion of local government implicit debt is a complex process. This further highlights that preventing and resolving local government implicit debt is a very difficult practical problem. At the same time, the management level of debt risk varies among different local governments, which also makes it difficult for local governments to recover from implicit debt risks. Therefore, multiple entities related to local government implicit debt should collaborate and strive to overcome potential risk points that may arise, while strengthening risk management levels, improving rules and regulations, optimizing behavioral strategies, and ultimately resolving local government implicit debt.

6. Conclusions

From the perspective of multi-subject collaboration, this paper analyzes the risk contagion mechanism underlying local government implicit debt and its contagion rules. With the help of complex network theory, this study constructs a risk contagion model of local government implicit debt and analyzes the evolution characteristics and control strategies of risk contagion of local government implicit debt using theoretical deduction and numerical simulation. The main conclusions obtained from this study are explained below.

(1)

A scale-free network is not conducive to the risk contagion of local government implicit debt. However, a random network is most conducive to the risk contagion of local government implicit debt. When controlling local government implicit debt risk, special attention should be paid to local financing platforms that have significant potential risks and occupy a core position. The information openness accuracy and information disclosure strategy both exhibit a positive “U” shaped relationship with the risk contagion of local government implicit debt. In addition, the debt management level, emotional tendency, risk preference level, credit policy robustness, accountability mechanism soundness, and perfection of laws and regulations are all negatively correlated with the risk contagion of local government implicit debt.

(2)

Comprehensively adjusting the debt management level, emotion tendency, risk preference level, credit policy robustness, accountability mechanism soundness, perfection of laws and regulations, information openness accuracy, and information disclosure strategy can reduce the risk contagion intensity of local government implicit debt; however, the conditions for eradicating local government implicit debt risk are very strict. Moreover, the debt management level of local governments at all levels plays a risk “intensifier” role. From a deeper perspective, this indicates that local government implicit debt risk is an inherent risk that is difficult to eliminate. Therefore, when dealing with a local government’s implicit debt risk, the focus of control should be on reducing the risk rather than eliminating it. At the same time, local governments should strengthen their own debt management capability and information openness level, and the central government should continuously improve accountability mechanisms and laws and regulations. In this process, financial institutions and the media should actively play the role of “stability maintainers” and should not blindly adopt irrational behaviors such as loan withdrawals, loan restrictions, and the release of negative information.

This study provides theoretical and practical references for local governments to further optimize their implicit debt management capability and improve the efficiency of implicit debt planning and utilization. However, during the research process, this study found that the information correlation edges of the local financing platform association network may have different attributes, so the association network may exhibit the characteristic of a multi-layer network structure. This issue was considered in this study but will be the direction and focus in a further expansion of this study.