Ripple-Spreading Network of China’s Systemic Financial Risk Contagion: New Evidence from the Regime-Switching Model

,


Introduction
Financial security is an important part of national security; preventing and defusing fnancial risk, especially holding the bottom line of no systemic fnancial risk, is of great signifcance to ensure the smooth operation of China's economy and fnance.In recent years, many fnancial institutions in China have been engaged actively in cross-fnancial business and conduct mixed fnancial services [1], which have strengthened the linkages between fnancial institutions and increased systemic risk [2].Besides, the events of 2008 global fnancial crisis have reminded us that regulating fnancial institutions or markets in isolation and ignoring the connectedness between institutions or markets can no longer efectively prevent and control systemic fnancial risk [3].Terefore, it is essential and urgent to explore the connectedness among fnancial institutions in China and examine the spillover and contagion efects of systemic risk by considering the fnancial system as a whole.
Te global fnancial crisis in 2008 has aroused attention to the connectedness and risk contagion among fnancial institutions or markets and increased the emergence of related research methods.A commonly used measure of systemic fnancial risk is the conditional value-at-risk (CoVaR) and the delta conditional value-at-risk (ΔCoVaR) [4], which are based on a "bottom-up" perspective and refect the risk contribution of fnancial institutions to the system [5][6][7].Another commonly used measure is the marginal expected shortfall (MES) [3], which is based on an "up-bottom" perspective and focuses on the systemic risk contribution of individual fnancial institution when the system is in a crisis [8].In addition, the conditional expected shortfall (CoES) [9,10] and the SRISK risk index [11,12], etc., have also been developed to measure systemic fnancial risk.Although methods such as CoVaR, MES, CoES, and SRISK are widely used, they ignore the connectedness in the fnancial system and take less account of the risk contagion efect.While the contagion efect of risks is at the heart of the outbreak of the systemic risk or fnancial crisis, when individual or local risk emerges, diferent institutions may be afected by risk sharing, risk spreading, and risk amplifcation mechanisms, resulting in chain reactions and even the collapse of the entire fnancial system.
Network theory, another important branch of systemic risk research, provides an efective and intuitive analytical tool for systemic fnancial risk by constructing network nodes and edges, which can not only display the risk contagion paths but also identify systemically important nodes [13], and thus helps the regulators to establish a stable fnancial system and an efective regulatory system.In general, there are two types of fnancial network analysis methods.Te frst method uses detailed information on interbank asset-liability exposures to construct fnancial networks and then determines the specifc risk transmission mechanisms or simulates the impact of shocks on the network.Based on this method, scholars have established the interbank exposure network [14], interbank payment network [15,16], and asset-liability network [17,18].Although this method allows for the identifcation of specifc directions and paths of risk contagion, it relies on nonpublic data, which makes research limited.Te second method uses fnancial market data, i.e., stock prices (e.g., return, volatility, and tail risk) to build fnancial networks and then assesses risk contagion through network topology indicators.Since fnancial market data are publicly available and forward-looking, they are widely used in the study of fnancial networks and systemic risk.Related studies mainly include correlation-based networks [19][20][21], Granger causality networks [22][23][24], volatility spillover networks [25][26][27], and tail risk spillover networks [28,29].In addition, in order to take more risk information into account, some scholars have developed composite networks [30] and multilayer information spillover networks [31][32][33][34].In these fnancial networks, nodes denote fnancial institutions or markets, and edges denote relationships between nodes.By examining whether there are correlation, causality, or spillover efects between nodes, we can determine the risk contagion paths.
In summary, the research on systemic risk under fnancial network has yielded rich results.It is worth noting that, however, most of the relevant network analyses are static.Although relevant studies have used the rolling time window approach [27,30,34] to capture the dynamics of risk contagion, they are unable to model the dynamic ripple efects triggered by a contagion source, i.e., which fnancial institution is afected frst and which is afected later when a fnancial institution sufers a loss or goes bankrupt.Te ripple efect describes the gradual spreading of the efects caused by local events.In fact, the formation and evolution of many real-world complex systems depend to a large extent on the spreading of a few local events [35].Suppose a failed (lost or bankrupt) fnancial institution triggers fnancial contagion, the contagion will spread around like ripples in a calm pond.When it reaches the nearest (highest associated) fnancial institution, the institution may be infected and trigger a new ripple-spreading process, which is analogous to a ripple in a pond reaching a stake and creating a new responding ripple due to the refection efect.As the contagion continues to spread, more and more institutions may be infected.It is well known that many successful artifcial intelligence (AI) techniques are actually inspired by certain natural system or phenomena [36,37].For example, genetic algorithms (GA) are inspired by the process of natural selection and evolution, particle swarm optimization (PSO) is inspired by the learning behaviors within populations, and artifcial neural networks (ANNs) are inspired by animal brains, etc. Tese algorithms or their derived algorithms are also widely used in economic and fnancial felds [38,39].Following the common practice of learning from nature in the feld of artifcial intelligence, Hu et al. [35] developed a ripple-spreading network model (RSNM) which is inspired by the natural ripple-spreading phenomenon on the calm water surface and emphasized its application potential and fexibility in their paper.On this basis, Hu et al. [40] attempted to apply the genetic algorithm (GA) to tune the ripple-spreading related parameters and made it a great fexibility to study many real-world complex network systems.
Existing studies have shown that fnancial contagion has ripple efects [41].Looking at the latest research, several scholars have applied the ripple-spreading network to model the contagion path of fnancial risk and identify systemically important fnancial institutions or markets.For example, Su et al. [42] proposed a ripple network-based collective spillover efect approach and identifed the systemic importance of fnancial markets.Xu et al. [43] pointed out that the ripple efect is one of the most features of fnancial contagion and modeled the paths of China's systemic risk contagion under diferent contagion sources.However, the results are far from sufcient, and there are still some issues that need to be further explored.For example, few literature studies have analyzed the ripple efect of fnancial contagion in the framework of regime switching.Financial markets may experience regime shifts and nonlinear risk contagion as a result of sudden structural changes due to selling behavior of common asset holders [44], investor panic [45], and asymmetric dependence on fnancial asset returns in both upward and downward phases of the market [46].Feng et al. [47] pointed out that macroeconomic and fnancial variables often have sudden structural changes due to the exposure to external shocks such as policy changes and ignored the possible efects of regime shifts in the process of risk contagion which may lead to a signifcant bias.Terefore, it is necessary and urgent to explore the network connectedness and examine the dynamic ripple-spreading process of fnancial risk under diferent systemic risk regimes.In addition, relevant studies have examined fnancial contagion in the framework of a deterministic ripplespreading network.However, the deterministic and uncertain factors coexist in the process of fnancial contagion, and how to balance this relationship deserves further exploration.In this paper, we mainly carry out the following innovative work.
First, we improve the basic algorithm of the ripplespreading network to make it applicable to fnancial contagion and apply it to the analysis of fnancial contagion in China.Te semideterministic ripple-spreading network model (SD-RSNM) can balance the deterministic and uncertain factors in the process of fnancial contagion.By observing the instantaneous state of the ripple-spreading network, we can identify which fnancial institutions are afected frst and which are afected later.
Second, we introduce the smooth-transition vector autoregression (STVAR) model to identify the "high" and "low" state regimes of systemic risk and parameterize the ripple-spreading network on this basis.In terms of parameter specifcation, we take into account some important spatiotemporal factors afecting fnancial contagion, such as the magnitude of fnancial shock, risk amplifcation factor, risk resistance ability, and spreading speed of each fnancial institution.
Finally, we manage to identify systemically important fnancial institutions (SIFIs) based on the heterogeneous networks generated by the dynamic ripple-spreading processes under high and low systemic risk regimes.We fnd that most securities and banks and some real estate companies have the highest systemic importance in China's fnancial system.In particular, the securities sector has the strongest ripple-spreading capacity and plays an intermediary role in the system network.Te results remind fnancial regulators and government departments that systemic risk regulation should focus not only on large institutions but also on institutions with strong ripple efects.
Te remainder of the article is structured as follows: Sections 2 and 3 describe the methodology and data, respectively.In Section 4, we apply our method to Chinese fnancial institutions and present the empirical analysis results.In Section 4, we provide a brief discussion of the fndings.

Methodology
2.1.Basic Idea of the Ripple-Spreading Network.Considering the simultaneous existence of deterministic and uncertain factors in fnancial contagion, we derive the semideterministic ripple-spreading network model (SD-RSNM) based on Hu et al. [35] and Xu et al. [43] to simulate fnancial contagion in China's fnancial system.In this model, there are two classes of parameters: contagion source related and network node related.Te parameters of the contagion source include E 0 , s 0 , and d 0i .E 0 denotes the energy of the initial ripple of the contagion source, and it measures the magnitude of the systemic shock; s 0 denotes the ripple-spreading speed of the contagion source; d 0i denotes the distance between the contagion source and node i. Te parameters of the network nodes include α i , β i , s i , and d ij (i, j � 1, 2, • • • , N). α i denotes the risk amplifying factor of the node i; β i denotes the connection threshold, representing the risk resistance capability of the node i; d ij is the distance between the node i and j.Once a node is infected, it will trigger a new ripple with energy E i and speed s i .In addition, the energy parameters, i.e., E 0 and E i , decay following the same function f Decay (E i , r(i, t)) � ηE i /2πr(i, t), where η is a constant and r(i, t) is the ripple radius of the node i at time t.It is worth noting that, in the framework of SD-RSNM, the node j follows the following activation principles: if d ij ≤ r(i, t) and f Decay (E i , r(i, t)) ≥ β j , then the node j is activated by the node i, while if d ij ≤ r(i, t) and f Decay (E i , r(i, t)) < β j , then the node j is activated with probability P R (j) � 2 ω R (1− β R (j)/e source (t)) ; ω R > 0 is the probability decay coefcient.Using these parameters, the specifc simulation steps of SD-RSNM are described in Appendix A.
To visualize the basic principle of the risk ripplespreading process, we give a simple example, as shown in Figure 1, where node 0 denotes the contagion source and nodes 1-3 denote normal network nodes, i.e., fnancial institutions.Te energy of a ripple decays as it spreads; i.e., the strength of energy is refected as the thickness of the ripple.
At t � 1, the contagion source, i.e., node 0, frst triggers an initial ripple with energy E 0 and spreads out, but it has not yet reached any node.
At t � 2, the initial ripple triggered by node 0 reaches node 3. Since f Decay (E 0 , d 03 ) ≥ β 3 , node 3 is activated and a directed link from node 0 to node 3 is established.Ten, node 3 generates a response ripple with energy E 3 � α 3 f Decay (E 0 , d 03 ) that spreads out again.
At t � 3, the initial ripple triggered by node 0 reaches node 1.Since f Decay (E 0 , d 01 ) ≥ β 1 , node 1 is activated and generates the corresponding response ripple.
At t � 4, risk ripples triggered by contagion source disappear; i.e., the energy value decays to 0, while risk ripples triggered by nodes 1 and 3 continue to spread out.
At t � 5, the ripple triggered by node 1 reaches node 2; since f Decay (E 1 , d 12 ) ≥ β 2 , node 2 is activated by node 1 and generates the corresponding response ripple.
At t � 6, the ripple triggered by node 2 reaches node 1, since f Decay (E 2 , d 21 ) < β 1 , node 1 is not activated; the ripple triggered by node 1 reaches node 0, since f Decay (E 1 , d 10 ) ≥ β 0 , node 0 is activated by node 1 and generates the corresponding response ripple.At the same time, the ripple triggered by node 1 reaches node 3, although f Decay (E 2 , d 21 ) < β 1 , a dotted link is created from node 1 to 3, which is determined by chance.Ten, node 3 generates a response ripple with energy Tus, a directed fnancial network consisting of four nodes and fve edges is formed.

Parameter Specifcations.
Parameter specifcations are at the heart of the ripple-spreading network, and in this paper, we set them in the context of systemic risk regime switching.Te smooth-transition vector autoregression (STVAR) model proposed by Weise [48] is capable of examining asymmetric mechanisms in both high-and low-state regimes.Te general form of the STVAR model can be expressed as follows: where Y t is the m-dimensional endogenous variable in the period t, X t is the q-dimensional exogenous variable in the period t, p is the lag order of the STVAR model, z t is the state variable, and F(z t ) is the transition function.Given z t and F(z t ), the sample can be partitioned into two states, a lowstate variable regime (l) and a high-state variable regime (h); Π k l and Π k μ are the k-order lag term coefcient matrices of the endogenous variables in the high-and low-state regimes, respectively.Γ and Ξ are the exogenous variable coefcient matrices; μ 0 and μ 1 are the m-dimensional intercept vectors.F(z t ) is the transition function which takes the form of a logistic function.F(z t ) portrays the probability of a sample being partitioned into diferent state regimes; diag denotes the diagonal matrix; c determines the degree of smoothing of the regime transition, the larger the value of c, the faster the rate of regime transition; c is the threshold parameter for regime partitioning.
Furthermore, (1) can be equated to the following form: where Following Caggiano et al. [49], the model can be simplifed as follows: where Π * μ � Π μ − Π l , z t is the normalized state variable, i.e., z t � (z t0 − z mean )/z std , z t0 is the original observation, z mean is the mean, and z std is the standard deviation.It should be noted that, in this paper, we use systemic risk measured by CoVaR [50] as the state variable and the risk indicator, i.e., historical volatility [26] of each fnancial institution as endogenous variables in the model.
On the basis of the coefcient estimation and regime partitioning of the STVAR model, following Diebold et al. [25], we can obtain the variance contribution under high and low systemic risk regimes, i.e., how much of the future uncertainty of variable j is due to shocks in variable i: where d ij denotes the proportion of changes in i caused by the shocks of the endogenous variable j and describes the risk spillover intensity from j to i, N is the number of endogenous variables, H is the forecast period, σ −1 ii is the standard deviation of the error term for the i th equation, Σ is the covariance matrix for the error vector ε t , A h is the H-step moving average coefcient matrix, and e i is the selection vector, with one being the i th element and zeros otherwise.Since the shocks between variables are not orthogonal, the entries of each row in the variance decomposition matrix do not add up to 1. Hence, we normalize it according to the row summation approach and obtain 4 Complexity Furthermore, we can set the network parameters based on the results of the regime transformation.
Connection threshold β i : We assume that the more external shocks an institution is exposed to, the more vulnerable it is to fnancial contagion.It is similar to  d ij in the connectedness method proposed by Diebold et al. [25].Tus, based on (7), the connection threshold for institution i under high and low systemic risk regimes, i.e., β i (h) and β i (l), can be expressed as follows: Amplifying factor α i : We use the market capitalization of each fnancial institution to specify α i under high and low systemic risk regimes.For example, the average market capitalization of ICBC under a high-risk regime is 17.014 × 100 billion yuan, so we set α ICBC (h) � 17.014; the average market capitalization of ICBC under a low-risk regime is 16.276 × 100 billion yuan, so we set α ICBC (l) � 16.276.
Spreading speed s i : Following Xu et al. [43], we use the average turnover rate to specify s i under diferent systemic risk regimes.For example, the average turnover rate of ICBC under a high systemic risk regime is 0.081, so we set s ICBC (h) � 0.081; the average turnover rate of ICBC under a low systemic risk regime is 0.066, so we set s ICBC (l) � 0.066.
Market distance d ij : d ij is determined by the inverse of the volatility correlation coefcient between institutions i and j; i.e., the higher the correlation between two fnancial institutions, the shorter their distance and the easier it is for risk ripples to reach.Tus, the market distance between institutions i and j under the high-and low-risk regimes, i.e., d ij (h) and d ij (l), can be expressed as where cor ij is the correlation coefcient calculated based on history volatility data between institutions i and j.In addition, if cor ij � 0, then d ij � +∞.

Heterogeneous Network.
Given a lager enough E 0 , the ripple-spreading process will eventually form a stable fully connected network with nodes N and edges N(N − 1), which is not conducive to analyzing the systemic importance of fnancial institutions.Following Xu et al. [43], we use the variance of node degrees to identify the heterogeneous network of the ripple-spreading process under diferent systemic risk regimes.For every time instant t, let A t � (a t ij ), i, j � 1, 2, • • •, N be the adjacency matrix of the instantaneous network in the ripple-spreading process.
Te variance of node degrees at t can be expressed as follows: where D t i denotes the node degree of the node i, i.e., ji , and D t denotes the average of node degrees, i.e., D t � ( N i�1 D t i )/N; when s 2 t takes the maximum value, the network is of best heterogeneity and appropriate for analyzing systemically important fnancial institutions (SIFIs).

Data
Limited to data availability, we select 55 fnancial institutions listed before 2011 as the main research object, specifcally including 11 real estate companies, 16 banks, 14 securities, 4 insurance, and 10 diversifed fnancial institutions, as shown in Table 1.In terms of sample selection, we include some real estate companies (collectively, fnancial institutions) in our research sample due to their fnancial-like attributes.Te research sample covers the periods from January 4, 2011, to February 10, 2023.Te daily historical volatility [26] of each institution can be expressed as where H i,t , L i,t , O i,t , and C i,t are the logs of daily high, low, opening, and closing prices, respectively, and the data are obtained from the Wind database.
In the STVAR model with the systemic risk index [50] as the state variable, the results of the grid point search show that the smoothing coefcient c of the transition function is estimated to be 10 and the parameter c takes the value 1.126.Figure 2(a) shows the results of the transition function.It can be found that the transition function shows a smooth and asymptotic trend, suggesting a nonlinear relationship of asymptotic evolution of risk contagion as the state of systemic risk changes.Furthermore, the distribution of the high and low states of systemic risk over the sample period can be obtained, as shown in Figure 2(b).Te shaded areas in Figure 2(b) mark the high systemic risk regime.It can be found that the high systemic risk regime covers the period of Complexity On the basis of the above systemic risk regime identifcation, we give the behavioral parameters of the ripplespreading network under high and low systemic risk regimes, respectively, as shown in Tables 2 and 3. Te market distance, i.e., d ij (h), d ij (l), is measured by the inverse of the correlation coefcient of historical volatility between fnancial institutions.In addition, in China's fnancial system, the real estate sector occupies an important position, so we set a real estate company as the contagion source to explore the risk ripple-spreading processes.To ensure sufcient network links, the energy of initial ripple of contagion source is set as e source (h) � e source (l) � E 0 � 200π.

Ripple-Spreading Process under High Systemic Risk
Regimes.Given the model-related parameters mentioned above, we perform the risk ripple-spreading simulation processes under high systemic risk regimes according to the algorithm in Appendix A.
Figure 3 shows the dynamic ripple-spreading processes for 55 fnancial institutions, by giving some heat maps at t � 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, and 64.Te rows of the heat maps represent the ripple-spreading efect from a given fnancial institution to other fnancial institutions.For example, the frst row of Figure 3(a) represents the ripple-spreading efect from the real estate company, i.e., RE01, to other institutions.In addition, in order to better distinguish the ripple-spreading efects within and across sectors, we use dotted lines to partition the heat map by sectors, i.e., the real estate, banking, securities, insurance, and diversifed fnancial sectors.Te fve diagonal areas of each subfgure in Figure 3 indicate the risk contagion within sectors, and nondiagonal areas indicate the risk crosscontagion between sectors.For example, the top-left corner of Figure 3(a) represents the risk contagion within the real estate sector; the bottom-left corner of Figure 3(a) represents risk cross-contagion from the diversifed fnancial sector to the real estate sector.
From these heat maps, it can be found that the number of contagion links increases gradually over time.By observing the instantaneous state of the ripple-spreading network, we can identify which fnancial institutions are afected frst and which are afected later.On the whole, the contagion triggered by the real estate company, i.e., RE01, frst spreads to the real estate and banking sectors and then to the insurance and securities sectors and thus triggers the widespread contagion within the fnancial system.Specifcally, Figures 3(a) and 3(b) show that the fnancial contagion originating from the source node, i.e., RE01, frst reaches the real estate sector (RE02) and the banking sector (ABC, BOC, CCB, BCM, etc.) and then spreads to the insurance sector (CPIC, CLIC, etc.) and the securities sector (GYS).By t � 24, as shown in Figure 3(b), all banks are directly afected by the risk ripple of the contagion source.Tese fndings suggest that real estate companies have a signifcant impact on the fnancial system, especially on the banking sector.Real estate companies are linked with banks through massive amounts of debt, so when real estate risks occur, the banking sector is afected more deeply and broadly.As risk ripples continue to spread, the cross-contagion occurs among institutions beyond the source node, and the network density increases gradually, as shown in Figures 3(c)-3(l).By t � 28, as shown in Figure 3(c), the risk ripple from CCB reaches most banks such as ICBC, ABC, BOC, and BCM.Ten, the risk ripple from CCB reaches the securities sector (CMS, CJS, CITIC, etc.) and the insurance sector (CLIC, CPIC, and PAIC), as shown in Figure 3(d).Meanwhile, the risk ripples from securities institutions (CITIC and GYS) begin to spread to the fnancial system.It is worth noting that, by t � 32, all diversifed fnancial institutions are not yet afected by risk ripples and do not send any links to the system, suggesting that diversifed fnancial institutions are more distant from other fnancial institutions.
From t � 44 to t � 64, as shown in Figures 3(g)-3(l), we can fnd that the cross-contagion between fnancial institutions gradually permeates the entire fnancial system and the securities sector sends the most risk ripples to the system network.In addition, there are as many as 1,735 links at t � 52, which is signifcantly higher than that at t � 40, i.e., 603, indicating that risk ripples among fnancial institutions spread very fast.After the contagion channels are fully established in the early stage, the rapid risk contagion begins in the later stage.Terefore, it is wise for regulators to take timely measures to block the paths of risk contagion before the contagion channels are fully established.

Ripple-Spreading Process under Low Systemic Risk
Regimes.Given the model-related parameters mentioned above, we perform the risk ripple-spreading simulation Complexity processes under low systemic risk regimes according to the algorithm in Appendix A.
For a better comparative analysis with the ripplespreading network under high systemic risk regimes, Figure 4 similarly shows heat maps of the ripple-spreading process under the low systemic risk regime at t � 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, and 64.From these heat maps, we fnd that the contagion triggered by the contagion source, i.e., RE01, frst spreads to the real estate sector and then to the banking, securities, and insurance sectors and thus triggers the cross-contagion beyond the source node.Specifcally, Figures 4(a)-4(c) show that the fnancial contagion originating from the source node, i.e., RE01, frst reaches the real estate sector (RE02, RE03, RE05, etc.) and then spreads to the banking sector (ICBC, ABC, CCB, etc.), the securities sector (CJS, CITIC, GFS, etc.), and the insurance sector (CPIC, PAIC).By t � 28, as shown in Figure 4(c), most banks are directly afected by the source node, which is similar to the ripple-spreading network under high systemic risk regimes; i.e., the real estate sector has the most signifcant impact on the banking sector.Moreover, by t � 28, the number of links in the system network is 32, which is smaller than the links under the high systemic risk regime, suggesting that fnancial institutions are relatively more distant from each other under low systemic risk regimes.As the risk ripple-spreading process goes on, we fnd that the crosscontagion occurs beyond the source node, as shown in Figures 4(d)-4(f ).By t � 32, the cross-contagion occurs   8 Complexity within the securities sector, and some securities institutions such as CITIC begin to send risk ripples or links to the banking sector.By t � 36, the risk cross-contagion occurs within the banking sector.In addition, by t � 40, we can fnd that most of the links constructed by the diversifed fnancial institutions are mostly from securities institutions, while the diversifed fnancial institutions do not issue any links to the system network.Tis implies that the diversifed fnancial institutions are the main receiver of risk ripples in the Chinese fnancial system.From t � 44 to t � 64, as shown in Figures 4(g)-4(l), we fnd that the risk cross-contagion relationships become more complex and the network density increases gradually.Taken together, although risk ripples spread slower in low than in high systemic risk regimes, systemic shocks can also trigger large-scale risk contagion within the fnancial system even in low systemic risk regime as risk ripples spread.

Network Centrality Analysis of Heterogeneous Networks.
In the above analysis, we have studied the dynamic ripplespreading processes in China's fnancial system under high and low systemic risk regimes.Furthermore, we discuss the network centrality of heterogeneous ripple-spreading networks and identify the systemic importance of fnancial institutions (SIFIs).According to (10), each process can generate a stable heterogeneous network.We set the upper time limit to t � 500.Figure 5 shows the dynamic change trend of network heterogeneity.It can be seen that network heterogeneity increases sharply from 0 to a maximum value and then decreases gradually as risk ripples spread.Te network heterogeneity in high and low systemic risk regimes reaches its maximum value at t � 49 and t � 60, respectively.In this subsection, we consider four network centrality indicators, i.e., degree centrality (DC), closeness centrality (CC), eigenvector centrality (EC), and betweenness centrality (BC), based on heterogeneous networks selected above under diferent risk regimes, to identify and analyze SIFIs.Tese four network centrality indicators measure the systemic importance of fnancial institutions from diferent perspectives.In general, the higher the network centrality of a node, the higher its systemic importance in the system network.Te implications of these four centralities are presented in Appendix A. Figures 6-9 show the measurement results of four network centrality indicators for each fnancial institution under high and low systemic risk regimes.Te results are given in the form of heat maps, where darker colors indicate higher centrality values and higher systemic importance of fnancial institutions.Due to space constraints, only some of the English abbreviations of fnancial institutions are shown in heat maps, and their order from left to right is consistent with Table 1.As can be seen, whether the fnancial system is in a high-or low-risk regime, nodes with higher network centrality are mostly distributed in the securities, banking, and real estate sectors, and nodes with lower network centrality are mostly distributed in the diversifed fnancial sector.
Specifcally, Table 4 shows the top 15 SIFIs ranked based on four network centralities under high and low systemic risk regimes.Overall, the top 15 SIFIs are concentrated in the banking and securities sectors.However, there are some diferences in the ranking of fnancial institutions under high and low risk regimes.In particular, the systemic importance characteristics of some securities institutions in the high systemic risk regime are particularly signifcant.For example, securities institutions such as GYS, HZS, IS, and GFS all appear in the top 15 lists for four centralities under high systemic risk regimes and only appear in the top 15 lists for one or two centralities under low systemic risk regimes.In addition, some real estate companies, such as RE02, RE09, and RE10, appear in the top 15 lists, while diversifed      Complexity fnancial institutions do not appear in the top 15 lists.Terefore, we conclude that most securities and banks and some real estate companies have the highest systemic importance, and the diversifed fnancial institutions have the lowest importance.Furthermore, we average the network centrality indicators for each fnancial institution by sector to obtain a systemic importance ranking for each sector, and the results are shown in Table 5.It can be found that the systemic importance of the real estate sector is higher in high than in low systemic risk regimes, and supervisors should pay particular attention to the risk contagion capacity of the real estate sector when the fnancial system is under pressure.In addition, whether the fnancial system is in a high or low systemic risk regime, the securities sector has the highest "degree centrality" and "betweenness centrality," which indicates that the securities sector not only has signifcant risk ripple linkage capacity but also has signifcant risk ripple intermediation efects.Te spreading of risk ripples relies  12 Complexity heavily on information transmission from the securities sector, thus providing a new entry point for blocking risk contagion.

Conclusion
In this paper, we propose using the ripple-spreading network model to reveal the spatiotemporal evolutionary characteristics of systemic risk contagion in China's fnancial system.Compared to existing fnancial networks such as correlation networks and spillover networks, the ripplespreading network provides a new tool for modeling the dynamic process of how fnancial contagion spreading from the contagion source to the whole fnancial system.On the one hand, the dynamic ripple-spreading process can reveal which nodes are afected frst and which are afected later.
On the other hand, we can identify SIFIs based on the generated heterogeneous networks.
As for the dynamic ripple-spreading processes, we fnd that risk ripples spread much faster in high than in low systemic risk regimes.However, systemic shocks can also trigger large-scale risk contagion within the fnancial system even in low systemic risk regimes as the risk ripples continue.Excessive network connectedness among institutions can amplify fnancial shocks through contagion efects.Overall, whether the fnancial system is in a high-or low-risk regime, the risk ripples from the contagion source (i.e., a real estate company) spread frst to the real estate sector and the banking sector.On the one hand, institutions belonging to the same sector share similarities in terms of their business scope, business pattern, investment pattern, risk management, and fnancial regulation, etc., which may make it easier for the cross-contagion to occur within the sector.On the other hand, the real estate companies are linked with banks through massive amounts of debt, so when real estate risks occur, the banking sector is afected more deeply and broadly.In addition, the diversifed fnancial institutions have fewer risk interactions in the early stage, and all the links established by the diversifed institutions are mostly from securities, while it issues fewer links to the system network.Tis implies that the diversifed institutions are the main receiver of risk ripples in the Chinese fnancial system.
As for the identifcation for SIFIs, we fnd that most securities and banks and some real estate companies are the most systemically important fnancial institutions in China's fnancial system.Although we set the real estate company, i.e., RE01, as the contagion source, securities institutions exhibit the strongest risk ripple-spreading ability as the risk ripples continue, sending the most links to the system network.Especially, the systemic importance characteristics of securities institutions in high systemic risk regimes are particularly signifcant.To some extent, the results are not consistent with the previous studies, which noted that large fnancial institutions such as banks and insurances are the most SIFIs [51,52].For example, Wang et al. [51] noted that banks and insurance institutions in China contribute more to systemic risk than securities institutions.Chen et al. [52] pointed out that SIFIs are concentrated in the banking and insurance sectors.Te reason for this is that the methods used in these studies focus mainly on assessing the systemic risk contribution of fnancial institutions.Te higher the systemic risk contribution of a fnancial institution, the higher the level of systemic importance.However, the measurement of systemic risk contribution usually takes into account the impact of the size of fnancial institutions, so that large-scale fnancial institutions such as banks and insurance are identifed as SIFI.Our paper mainly focuses on the network correlation of fnancial contagion, rather than Note.G1, G2, G3, G4, and G5 represent the real estate, banking, securities, insurance, and diversifed fnancial sectors, respectively.
Complexity the systemic risk contribution.Financial institutions with the highest network correlation tend to have the highest systemic importance.At the same time, this inconsistency reminds fnancial regulators and government departments that systemic risk regulation should focus not only on large institutions but also on institutions with strong ripplespreading efects.Finally, it is worth noting that the channel mechanism of fnancial contagion is very complex, this paper only analyzes the dynamic ripple-spreading processes of risk triggered by contagion source under high and low systemic risk regimes and does not involve the exploration of specifc contagion channels or mechanisms.In the future, a more comprehensive integration of the factors afecting fnancial contagion will provide a better understanding of the contagion process of risks.

A. Simulation Steps of the Ripple-Spreading Network
Given the ripple-spreading network behavior parameters, the new dynamic ripple-spreading network model can be mathematically described as follows: Step 1. Initialize the current time instant, i.e., t � 0; initialize the current point energy of contagion source as e source (t) � E 0 , (E 0 > 0); initialize the current point energy of each network node as e nodes (i, t) � E nodes (i) � 0, i � 1, 2, • • •, N. Assume contagion source and each node have a ripple with a current radius of 0, i.e., r source (t) � 0, r nodes (i, t) � 0.
Step 2. If the stopping criteria are not satisfed, do the following: Step 2.1.Let t � t + 1, Step 2.2.Update the current radius and point energy of contagion source as r source (t) � r source (t − 1) + s 0 , e source (t) � f Decay (E 0 , r source (t)), where s 0 is the spreading speed of contagion source, i.e., the change in the radius of a ripple during one time instant, f Decay is a function defning how the point energy decays as the ripple spreads out.A typical decaying function can be defned as follows: f Decay E 0 , r source (t)  � ηE 0 2πr source (t) , (A.1) where η is a decaying coefcient and π is the mathematical constant.Clearly, η has an important infuence on the decaying speed of ripples and will therefore afect the fnal network topology.In this paper, following Xu et al. [43], we set η � 1.
Step 2.3.Check which new nodes are reached by the ripples of contagion source.Suppose d 0j is the distance between the contagion source and node j.If d 0j ≤ r source (t) and e source (t) ≥ β j , then node j is activated by contagion source, and thus, a link from the contagion source to node j is established.Node j generates a responding ripple with initial energy E nodes (j) � α j e source (t) and e nodes (j, t) � E nodes (j).In this step, we consider the uncertainty characteristics of fnancial risk contagion; i.e., if d 0j ≤ r source (t) and e source (t) < β j , then node j generates responding ripple with the following probability: P R (j) � 2 ω R 1− β R (j)/e source (t) ( ) , (A.2) where ω R > 0 is the probability decay coefcient.Obviously, the lower the ripple energy, the lower the probability of generating node behavior.
Step Finally, we can stop the simulation in Step 2 by setting an upper time limit.

B. Networks Centrality Measures
Te degree centrality of node i can be expressed as follows: where N is the number of nodes and N − 1 is the maximum out-or in-degree.If there is a directed link from node i to j, a ij � 1; otherwise, a ij � 0. Te closeness centrality for node i is calculated as follows: where d ij denotes the length of a shortest directed path from i to j.
Eigenvector centrality is a global measure of network centrality.It assigns a relative score to each node in the network, and in the contribution of a given node's score, connections to nodes with high scores are larger than connections to nodes with low scores.Te relative centrality score for node i can be defned as where i ≠ j, j ≠ k is the number of shortest directed paths linking j and k and σ jk (i) is the number of shortest directed paths linking j and k that contain node i.

Figure 1 :
Figure 1: A simple example of the semideterministic ripple-spreading network.

Figure 2 :
Figure 2: Transition function and state regime transitions.(a) Scatterplot distribution between state variable and the transition function; (b) distribution of sample values of the transition function.

Table 1 :
* Name and abbreviation of China's fnancial institutions.Complexity "the market liquidity crisis caused by the money shortage in China's banking sector in 2013," "the stock market crash in China in 2015-2016," and the outbreak of the public health event, i.e., "COVID-19" in early 2020.In particular, the aggregation characteristics of systemic risk are the most obvious during the period of "the stock market crash in China in 2015-2016."Since the end of 2014, China's stock market has seen explosive growth and high market sentiment, with investors leveraging into the market through brokerage fnancing and over-thecounter matching, resulting in an over-leveraged stock market and a serious market valuation bubble.However, by the second half of 2015, the market trend took a sharp turn for the worse, and the departure of leveraged funds accelerated the decline of the stock market, which eventually led to the outbreak of stock disasters and the high fuctuation of systemic risk.Compared with the actual situation, the model constructed in this paper can well identify the evolution of China's systemic risk in various periods, indicating that the logistic smoothswitching model adopted in this paper can efectively identify various types of crisis events, and the estimation results of the model are reasonable and reliable.Finally, it can be estimated that, in the sample range from 2011 to 2023, China's fnancial system has about 20% of the time in a state of high systemic risk.

Table 2 :
Parameters of the ripple-spreading network: high systemic risk regime.Te data in Table2are compiled by the authors based on Chapter 2.2. Note.

Table 3 :
Parameters of the ripple-spreading network: low systemic risk regime.Note.Te data in Table3are compiled by the authors based on Chapter 2.2.
Figure 6: Heat map of degree centrality for each fnancial institution.Figure 7: Heat map of closeness centrality for each fnancial institution.Figure 8: Heat map of eigenvector centrality for each fnancial institution.

Table 4 :
Te top 15 SIFIs under high and low systemic risk regimes.

Table 5 :
Systemic importance ranking of fnancial sectors under high and low systemic risk regimes.
2.4.If e nodes (i, t − 1) > 0, then update the current radius and energy of the ripple starting from node i in a similar way to the ripple from the contagion source, i.e., r nodes (i, t) � r nodes (i, t − 1)+ s i ; e nodes (i, t) � f Decay (E nodes (i), r nodes (i, t))Step 2.5.Check which new nodes are reached by the ripples of nodes.If d ij ≤ r nodes (i, t) and e nodes (i, t) ≥ β j , then node j is activated by node i, and thus, a directed link from node i to node j is established.Node j generates a responding ripple with initial energy E nodes (j) � α j e nodes (i, t) and e nodes (j, t) � E nodes (j).Likewise, we consider the uncertainty characteristics of fnancial risk contagion; i.e., if d ij ≤ r nodes (i, t) and e nodes (i, t) < β j , then node j generates responding ripple with the probability shown in equation (A.2).
is a constant.It can be written as an eigenvector equation: Ax � λx.Typically, each eigenvector will correspond to a diferent eigenvalue λ.Only the solution corresponding to the largest eigenvalue is required by the centrality measure.Te betweenness centrality for node i is calculated as follows: