# The Influence of Information Diffusion on Interbank Risk Contagion

Authors: Zhinan Li; Peilong Shen; Xiaoyuan Liu
Journal: Complexity
 (2021)

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## Abstract

In this paper, the stylized features of incomplete and asymmetric information in the interbank market leading to banks’ precautionary behaviors are introduced. Based on banks’ stylized behavioral rules, the influencing mechanism of information diffusion on interbank risk contagion is analyzed, and how the existence of information diffusion and banks’ information-obtaining ability influence the interbank risk contagion is verified through computational simulations. The results show that information diffusion can significantly accelerate the process of and magnify the extent and probability of interbank risk contagion, and improving banks’ information-obtaining ability could lower the speed and extent of the interbank risk contagion. This paper also finds and explains some special phenomena of rollover risk contagion when information diffusion exists: the saltatory risk contagion, the circular liquidity trap, and the risk discovery of information diffusion.


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## Body

## 1. Introduction

Research on financial risk contagion in the field of systemic risk has gained prevalence in recent years, with the dominant hypothesis being that of traditional finance theory. However, studies on interbank risk contagion under the premise of traditional financial assumptions can correctly predict whether the contagion occurs but are less suited for stress testing or an analysis of policy options in crises, primarily because of their lack of behavioral foundations [1].Specifically, in the interbank market, when a bank hoards liquidity because of a liquidity crisis, traditional finance theory deems that its debtor banks’ reactions are only based on their own liquidity conditions. Nevertheless, many researchers have found that banks may hoard liquidity because of precautionary motivation even without liquidity risk [2–5]. However, what are the effects of the diffusion of information regarding liquidity stress in interbank networks? To our knowledge, there has not been a clear answer until now.Therefore, the goal of this paper was to answer this question by introducing the stylized features of incomplete and asymmetric information in the interbank market, and then gaining a better understanding of the influence of information diffusion on banks’ liquidity hoarding decisions based on banks’ stylized behavioral rules and their further influence on interbank risk contagion.Specifically, because of the diffusion of incomplete information in the interbank market, healthy banks obtain different levels of information about the motivation of the bank hoarding liquidity. At the same time, information held by debtor and creditor banks is asymmetric. Therefore, after creditor banks hoard liquidity, their debtor banks will adjust their liquidity strategies (interbank lending policies) based on the information obtained even without liquidity risk, which is the stylized behavioral rules that lead to the precautionary behaviors. These behaviors lead to more information diffusion and further affect more banks.In existing studies on interbank risk contagion, the risks can be divided into three types according to different contagion paths: counterparty risk, common asset holdings risk, and rollover risk. The research on interbank risk contagion originates from the risk contagion of interbank funding networks on the path of counterparty risk. To date, the research in this field is still numerous, extensive, and in-depth. Initially, Rochet and Tirole [6] studied whether the flexibility brought by the diversification of interbank funding could make up for the rescue of the central bank from the perspective of the interbank funding relationship. Later, Allen and Gale [7], as pioneers of the study of risk contagion between banks, laid a micro foundation for studies in this field. Eisenberg and Noe [8] focused on the payment of the interbank funding network and first proved the existence and uniqueness of the settlement payment vector. They analyzed risk contagion based on the settlement payment vector and refined the contagion mechanism under the path of counterparty risk. Therefore, the vast majority of studies have adopted the settlement payment vector that Eisenberg and Noe [8] proposed or its evolution model as the settlement base of counterparty risk contagion. Subsequent studies on counterparty risk contagion can be broadly divided into three aspects: how characteristics of interbank funding networks affect counterparty risk contagion [9–14], the evolution of the interbank funding network and counterparty risk contagion [15–18], and the empirical analysis and measurement of interbank risk contagion by applying the network method to various countries and regions [19–24].Research on the contagion of common asset holdings risk starts later than that of counterparty risk, but many key studies have been conducted, and their amplification effect on risk contagion in interbank networks has been demonstrated. Dasgupta [25] was the first to find that cross-holding deposits may lead to contagious failures after liquidity shocks in incompletely correlated regions. Then, researchers at the Bank of England combined common asset holdings risk and counterparty risk and found that the “liquidity effect” and “asset market liquidity” significantly increased the extent and window of contagion [26, 27]. Caccioli et al. [28, 29], Shen and Li [30] found that interbank funding network exposure can significantly enlarge the risk contagion of overlapping portfolios. The model in this paper is an extension of Shen and Li [30] and the dynamics of asset prices are consistent with those of Caccioli et al. [29].Contagion on the path of rollover risk is closely related to counterparty risk: rollover risk is characterized by creditor banks’ liquidity-risk-inducing shocks to debtor banks’ liquidity, whereas counterparty risk is characterized by debtor banks’ default-risk-inducing shocks to creditor banks’ asset value. Therefore, many studies analyzed the relationship between assets and liabilities in the interbank funding network from the perspectives of both counterparty risk and rollover risk [31–33]. In addition, many studies focus on the contagion of both rollover risk and common asset holdings risk [34, 35]. Many scholars studied the motivation, mechanism, and impact of rollover risk and liquidity hoarding and found that they could lead to market freezes and collapses when rollover risk contagion occurs and banks consecutively hoard liquidity because of a precautionary or speculative motivation [2–5, 15, 26, 36–39]. The distinction between long-term and short-term interbank network in this paper is similar to that of Fourel et al. [33] and the dynamics of the model are based on the model by Gai et al. [4], except that we analyze only unsecured lending and add the possibility of liquidity hoarding even in the absence of a liquidity shortage. Even if the methodologies are different, this paper’s enrichment is similar to that of Acharya and Skeie [2], where banks withdraw funding for precautionary motives depending on information about the counterparties.In terms of information in risk contagion, many studies show that information asymmetry exists in the interbank market [40, 41], and the role of information diffusion in the risk contagion between banks is manifested mainly as the information interaction between different banks, leading to behavioral differentiation. Since the 1980s, research has proliferated on information diffusion and bank run behavior based on asymmetric information between financial institutions and depositors [42–44]. Many papers also focus on uncertainty on the part of the participants within the financial system. Two are most relevant for the information diffusion discussed in this paper. Acharya and Yorulmazer [45] developed a model of “information contagion” through the depositors and banks leading to collective bankruptcies. Caballero and Simsek [46] found that uncertainty about the network leads to more bank failures, a universal flight to quality, and a fire sale price for the risky asset. Different from these studies, this paper focuses on the information diffusion of liquidity hoarding behaviors in the process of rollover risk contagion.The main contribution of this paper is that it introduces the stylized features of incomplete and asymmetric information diffusion that lead to precautionary behaviors of liquidity hoarding, analyzing and modeling the influence of the information diffusion on a bank’s liquidity hoarding decision based on the stylized behavioral rules, and then verifying the influence of information diffusion and banks’ information-obtaining ability on the interbank risk contagion by computational simulations. Furthermore, some special phenomena of rollover risk contagion when information diffusion exists are found and explained: the saltatory risk contagion, the circular liquidity trap, and the risk discovery of information diffusion.This paper is organized as follows. In Section2, we analyze the mechanisms of risk contagion under the path of the three aforementioned risks (counterparty risk, common assets holdings risk, and rollover risk) and establish an interbank risk contagion model. In Section 3, we theoretically introduce the stylized features of incomplete and asymmetric information, analyze the mechanism by which information diffusion influences risk contagion, and model banks’ decision-making process with information diffusion based on the stylized behavioral rules. In Section 4, we generate the interbank network and solve the balance sheet data for each bank to lay a foundation for further simulation. In Section 5, we verify the influence of information diffusion on risk contagion and study the sensitivity of the information-obtaining ability. In Section 6, we conduct a robust analysis from three aspects. In Section 7, we analyze three special phenomena regarding rollover risk contagion under the influence of the diffusion of incomplete and asymmetric information. We present our conclusions and suggestions in Section 8.


## 2. Mechanism and Model of Risk Contagion

In this section, we theoretically analyze the risk contagion mechanism under three different risk paths and establish an interbank risk contagion model, thus laying the foundation for this paper.For financial risks, this paper studies the contagion of credit risk, liquidity risk (corresponding to funding liquidity risk and market liquidity risk), and market risk corresponding to three types of risk in the contagion process: counterparty risk (corresponding to credit risk), rollover risk (corresponding to funding liquidity risk), and common asset holdings risk (corresponding to market liquidity risk and market risk).Under the contagion path of counterparty risk, when a bank’s loss exceeds its equity, the bank faces bankruptcy from insolvency. In the liquidation stage, the bank is forced to withdraw all interbank assets from other debtor banks and convert noncash assets into cash, which is distributed to all creditor banks with the reserved cash. When the bank has inadequate cash to repay all creditor banks in full, these creditor banks suffer losses. The loss amount is the difference between the total debt and the actual repayment amount. When the loss of any creditor bank exceeds its equity, the creditor bank also goes bankrupt. Usually, interbank assets and liabilities have many counterparty banks. If more than one debtor bank fails at the same time, the losses accumulate, causing tremendous losses to a creditor bank, thereby sharply increasing its bankruptcy risk.Under the contagion path of common asset holdings risk, when a bank is experiencing liquidity distress, it immediately sells its financial asset holdings. At the same time, as a result of the limitations of market liquidity, the bank has difficulty selling these assets at their actual value in a short period—known as a fire sale. During the fire sale process, asset prices further decline, leading to other banks holding the same assets to suffer losses. Banks also go bankrupt if their losses are more than their equity.Under the contagion path of rollover risk, a bank runs into a liquidity crisis when its cash reserves cannot meet its current liquidity demand. At this time, the bank withdraws and stops rolling over its short-term interbank assets to satisfy its liquidity demand, and its debtor banks preferentially pay for the debt by cash. However, if the amount of cash is not enough for the payment, the bank also withdraws and stops rolling over its short-term interbank assets from debtor banks and even sells financial assets to meet their payment needs. Thus, on the one hand, more banks suffer from liquidity pressure; on the other hand, risk contagion may occur through the path of common asset holdings risk.Next, based on the three contagion paths of different risks, the interbank risk contagion model is established as an extension of Shen and Li [30]; referring to the model dynamics of Gai et al. [4] and the distinction between long-term and short-term interbank network of Fourel et al. [33]. An initial healthy interbank network (that is, all banks are risk-free) contains N banks. As shown in Table 1, bank assets are divided into cash, long-term interbank assets, short-term interbank assets, loans, and financial assets. Liabilities are divided into long-term interbank liabilities, short-term interbank liabilities, and deposits. The economy experiences a limited and uncertain number of times: t = 0, 1, …, T.Table 1
Bank balance sheets.
AssetsLiabilities and equityCashLong-term interbank liabilitiesLong-term interbank assetsShort-term interbank liabilitiesShort-term interbank assetsDepositsLoansEquityFinancial assetsIn terms of financial assets, suppose that the type of financial assets in the market isM and the number of financial asset m held by bank n is Sn,m. The financial asset matrix of the interbank network can be expressed as(1)Sn,m=S1,1S1,2…S1,MS2,1S2,2…S2,M…………SN,1SN,2…SN,MN∗M.The market price of financial assetm at time t is Pmt:(2)Pmt=P1tP2t…PMtM∗1.If bankj holds financial assets Sj,1,Sj,2,Sj,3,…,Sj,M at time t, then the value of the financial assets of bank j is(3)Sjt=∑m=1MSj,mPmt.In terms of long- and short-term interbank funding, interbank funding is assumed to take effect att = 0. To distinguish risk contagion along the paths of counterparty and rollover risks, these paths are represented by long-term interbank funding and short-term interbank funding, respectively. Two N∗N order matrices—LS and LL—are used to represent the short- and long-term interbank funding networks, respectively:(4)LS=0LS12…LS1NLS210…LS2N…………LSN1LSN2…0N∗N,LL=0LL12…LL1NLL210…LL2N…………LLN1LLN2…0N∗N,where LSij represents short-term interbank funding from bank i to bank j, forming the short-term interbank assets of bank i and short-term interbank liabilities of bank j. LLij represents long-term interbank funding from bank i to bank j, forming the long-term interbank assets of bank i and long-term interbank liabilities of bank j. Clearly, because a bank cannot lend to itself, the diagonal elements of the matrix must be zero. The short-term interbank assets of bank j are the sum of the elements in row j of LS: LSAj=∑i=1i≠jNLSji; the long-term interbank assets of bank j are the sum of the elements in row j of LL: LLAj=∑i=1i≠jNLLji; the short-term interbank liabilities of bank j are the sum of the elements in column j of LS:LSDj=∑i=1i≠jNLSij; and the long-term interbank liabilities of bank j are the sum of the elements in column j of LL: LLDj=∑i=1i≠jNLLij.Meanwhile, we uset = 0, 1, 2, …, T as the superscript to represent different times. At t = 0, the reserve cash of bank j is denoted by Cj0, the loans of bank j are denoted by Oj0, the deposits of bank j are denoted by Ij0, and the equity of bank j is denoted by Ej0. Thus far, the initial asset-liability relationship of bank j is(5)Sj0+Oj0+Cj0+LLAj0+LSAj0=Ij0+LLDj0+LSDj0+Ej0.At timet, if no bank fails or defaults in the entire banking system, then the total liabilities and assets of bank j are, respectively:(6)Djt=LLDjt+LSDjt+Ijt,Ajt=Sjt+Ojt+Cjt+LLAjt+LSAjt.In terms of counterparty risk contagion, if the assets of bankj are less than its liabilities at time t, it inevitably goes bankrupt and defaults on its liability. At this time, all assets are distributed among its creditor banks. Assume that deposits have a higher seniority than interbank liabilities and all banks have the same seniority with regard to long-term interbank liabilities. Therefore, if bank j can pay the full amount of deposits Ijt but cannot pay the long-term interbank liabilities, then creditor banks receive a part of the face value of their long-term interbank assets; if bank j cannot pay its deposits Ijt in full, then creditor banks receive nothing.LetXjst represent the long-term interbank liabilities that bank s can repay to bank j at time t, by definition, Xjst∈0,LLjst. Then, all assets of bank j at time t can be expressed as(7)Ajt=Sjt+Ojt+Cjt+LSAjt+∑s=1s≠jNXjst.At timet, a bank goes into bankruptcy once its assets are lower than its liabilities. If Ajt≥Ijt+LLDjt+LSDjt, then bank j has the ability to pay its liabilities in full, Xijt=LLijt; if Ajt<Ijt+LLDjt+LSDjt, then bank j goes bankrupt, leaving its long-term interbank creditor banks with payments less than the face value. If Ajt<Ijt+LSDjt (because counterparty and rollover risks are represented by long- and short-term interbank funding, respectively, short-term interbank assets are default-free in our model), then bank j is unable to repay its deposits, and long-term creditor banks lose all of their money, that is, Xijt=0. If Ijt+LSDjt<Ajt<LSDjt+LLDjt+Ijt, then bank j pays long-term interbank liabilities at only a fraction of their face value, i.e.,(8)Xijt=LLijtLLDjtSjt+Cjt+Ojt+LSAjt+∑s=1s≠jNXjst−LSDjt−Ijt.In conclusion, the amount of long-term interbank liabilities that bankj can pay to bank i at time t is(9)Xijt=LLijtLLDjtminLLDjt,Sjt+Cjt+Ojt+LSAjt+∑s=1s≠jNXjst−Ijt−LSDjt+,where ⋅+=max⋅,0.We introduce a liquidity requirement in terms of rollover risk contagion, similar to Montagna and Kok [47] and Peralta and Crisóstomo [48]:(10)Cjt≥β⋅Ijt+LSDjt,where β in the equation represents banks’ liquidity requirement ratio. According to this liquidity requirement, the cash reserve of bank j should not be less than a certain proportion of the sum of deposits and short-term interbank liabilities. If the cash reserve is less than this proportion, then the bank is in a liquidity crisis. First, the short-term interbank assets held by the bank are withdrawn to increase the cash reserve to meet the liquidity requirements. If LSAjt≥β⋅Ijt+LSDjt−Cjt, then the bank can meet the liquidity requirements by withdrawing short-term interbank assets. If LSAjt<β⋅Ijt+LSDjt−Cjt≤LSAjt+Sjt, then the bank is in a serious liquidity crisis and cannot meet the liquidity requirements even if all short-term interbank assets are withdrawn; therefore, the bank also needs to sell its financial assets in exchange for cash. If LSAjt+Sjt<β⋅Ijt+LSDjt−Cjt, then the bank cannot meet the liquidity requirements even after the sale of financial assets, leading to its default because of a lack of liquidity.In terms of common asset holdings risk contagion, the internal correlation occurs mainly through the change in financial asset prices; therefore, financial asset prices are internalized to depend on the proportion of the number of assets sold based on the previous price, which is consistent with Caccioli et al. [29]:(11)Pmt=Pmt−1⋅exp−a⋅∑1NSelln,mt−1∑1NSn,mt−1,where Selln,mt∈0,Sn,mt is the amount of financial assets m sold by bank n at time t. The a parameter is chosen such that the price drops by 10% when 10% of the asset is liquidated, i.e., a=1.0536.


## 3. Influence Mechanism of Information Diffusion on Banks’ Decision-Making

In this section, by introducing the stylized features of incomplete and asymmetric information, the influence mechanism of information diffusion on banks’ decision-making for rollover adjustments based on stylized behavioral rules are analyzed and modeled.In most existing studies on interbank risk contagion, banks are often assumed to only withdraw their short-term interbank assets when facing liquidity shortages. However, banks’ decision of whether to roll over short-term interbank assets depends not only on their liquidity condition but also on worries about future liquidity, i.e., precautionary motivation [2–5]. This paper models precautionary behavior through stylized features of incomplete and asymmetric information and banks’ stylized behavioral rules. Specifically, banks’ liquidity hoarding behaviors generate information diffusion in the market, and the level of information obtained by the rest of the banks determines the degree of their understanding of market liquidity and worries about future liquidity. Thus, the rest of the banks make decisions and adjust their rollover strategies to manage future possible liquidity shortages.Therefore, information diffusion in this paper is discussed in terms of the following phenomenon based on the stylized features of incomplete and asymmetric information and banks’ stylized behavioral rules. In the interbank short-term funding market, when a bank suffers a liquidity crisis, its liquidity hoarding behavior leads to information diffusion from the crisis bank to its debtor banks, causing debtor banks to change their rollover strategy in accordance with the information obtained. This change prompts debtor banks with no liquidity shortage to also reduce the rollover of short-term interbank assets because of their fear of future liquidity risk, further causing the deterioration of liquidity in the market. Then, information diffusion generated by the behavior of debtor banks continues to affect more banks, leading to the deterioration of the original active interbank market liquidity. Thus, more banks run into a liquidity crisis, which can result in system liquidity risk (the IMF [49] defines system liquidity risk as many financial institutions simultaneously facing difficulties when they cannot roll over short-term debt or obtain new money through universally chaotic monetary and capital markets) and bank failure.


### 3.1. Mechanism by Which Information Diffusion Influences Rollover Risk Contagion

Under the path of rollover risk contagion, risk is transmitted from creditor banks to debtor banks through the interbank short-term funding market. From the perspective of the balance sheet at the micro level, the risk contagion under the influence of information diffusion is unchanged. However, the difference is that not every bank through which information passes shows liquidity risk during the contagion process; therefore, the liquidity risk presents a saltatory contagion that is illustrated in the following example.Figure1 shows a simplified interbank short-term funding network of balance sheets with 7 banks (A, B, C, D, E, F, and G), and the connecting arrow represents the initial flow of funds. When A and B are caught in a liquidity crisis, the short-term interbank lending that they lend to C and D is withdrawn. Assume that under the path of rollover risk contagion, C and D have no liquidity risk because of their abundant cash, and the risk does not continue to contagion. However, when information diffusion exists, if C and D face liquidity hoarding by A and B, they try their best to obtain information about the motivations of A and B. Given the incompleteness and asymmetry of information, C and D cannot realize thoroughly whether A and B hoard liquidity because of a liquidity crisis or normal adjustments. Thus, C and D reduce the rollover of their short-term assets because of concerns over future liquidity uncertainty. C and D withdraw short-term assets from E and G, respectively, whereas F is in a liquidity crisis from the simultaneous withdrawal from C and D. Under the influence of information diffusion, liquidity risk is transmitted from A and B to F through C and D, whereas C and D do not experience liquidity risk. Apparently, contagion is transmitted from A and B to F in a saltatory manner.Figure 1
Simplified interbank short-term funding network of balance sheet. The figure shows the simplified interbank short-term funding network of a balance sheet for 7 banks (A, B, C, D, E, F, G), and the connecting arrow represents the initial flow of funds from creditor banks’ short-term interbank assets to debtor banks’ short-term interbank liabilities.


### 3.2. Banks’ Decision-Making Model

In the interbank market, a bank hoards liquidity for two possible motivations: if liquidity risk exists, the bank has to meet the liquidity requirements; if no liquidity risk exists, the bank adjusts the rollover strategy because of information diffusion as the creditor bank withdraws its short-term assets. Liquidity hoarding behavior under these two types of motivations cause information to be diffused from creditor banks to debtor banks and, surely, a bank’s ability to obtain information determines whether it can clearly judge the motivation of creditor banks, resulting in debtor banks adjusting their rollover strategies to different degrees. Next, we establish a bank decision-making model to quantify the banks’ stylized behavioral rules based on the motivation of creditor banks and the information-obtaining ability of debtor banks.As Table2 shows, when the debtor bank has a stronger ability to obtain information, it can more accurately determine the motivation of the creditor bank’s liquidity hoarding. Thus, debtor banks will be pessimistic about future liquidity conditions and largely reduce rollover when the creditor bank hoards liquidity because of liquidity risk. In addition, debtor banks will surely not need to worry about future liquidity conditions and reduce rollover when creditor banks face no liquidity risk. When a debtor bank has a weaker ability to obtain information, it is difficult to know whether its creditor bank is hoarding liquidity because of risk; therefore, its liquidity rollover moderately decreases because of concerns over future unknown conditions.Table 2
Influence factors of the degree of rollover reduction.
Debtor banks: degree of rollover reductionDebtor banks: ability to obtain informationStrongerWeakerMotivation of creditor banks’ liquidity hoarding: whether there is liquidity riskYesLargerModerateNoSmallerModerateTo quantify the degree of rollover reduction based on these stylized behavioral rules, suppose that after banki reduces the rollover of short-term interbank assets in period t − 1, bank i’s debtor bank j subsequently reduces the rollover of short-term interbank assets from bank j’s debtor bank s in period t. The degree of bank j’s rollover reduction from bank s (i.e., the withdrawal ratio of short-term interbank assets) is represented by(12)Yjst=LSjstCjtIjt+LSDjt<β,LSjst⋅ρjtCjtIjt+LSDjt≥β,s=1,2,3,…,N,s≠i≠j,where Cjt/Ijt+LSDjt<β and Cjt/Ijt+LSDjt≥β represents whether bank j is in a liquidity crisis. According to this formula, when bank j faces liquidity risk, it withdraws all short-term interbank assets; when bank j faces no liquidity risk, it partially reduces the degree of short-term interbank asset rollovers according to the information obtained from the creditor bank’s liquidity hoarding behavior. ρjt represents the withdrawal ratio determined by(13)ρjt=1+σj2CitIit+LSDit<β,1−σj2CitIit+LSDit≥β,σj∈0,100%,where σj∈0,100% represents bank j’s ability to obtain information. Bank j can obtain complete information when σj=100% and cannot obtain any information when σj=0. A larger σj means that bank j has a greater ability to obtain information. It should be noted that ρjt only plays a role when bank j is not in a liquidity crisis (i.e., when Cjt/Ijt+LSDjt≥β in equation (12) holds), and the calculation of ρjt in equation (13) is based on whether bank i (withdrawing short-term interbank assets from bank j in the last period) falls into a liquidity crisis. Corresponding to the analysis in Table 2, when bank i faces liquidity risk, the larger bank j’s ability to obtain information means that the withdrawal ratio is closer to 1. When bank i has no liquidity risk, the larger bank j’s ability to obtain information means that the withdrawal ratio is closer to 0. At the same time, the effect of an improvement in the bank’s information-obtaining ability for decision-making is assumed to exhibit the law of diminishing margins. That is, the change in liquidity withdrawal gradually decreases as the information-obtaining ability improves. In addition, banks with larger asset scales usually have higher internal management levels and industry status, leading to better information-obtaining ability. Therefore, the bank’s information-obtaining ability is further assumed to be in direct proportion to its asset scale.


## 3.1. Mechanism by Which Information Diffusion Influences Rollover Risk Contagion

Under the path of rollover risk contagion, risk is transmitted from creditor banks to debtor banks through the interbank short-term funding market. From the perspective of the balance sheet at the micro level, the risk contagion under the influence of information diffusion is unchanged. However, the difference is that not every bank through which information passes shows liquidity risk during the contagion process; therefore, the liquidity risk presents a saltatory contagion that is illustrated in the following example.Figure1 shows a simplified interbank short-term funding network of balance sheets with 7 banks (A, B, C, D, E, F, and G), and the connecting arrow represents the initial flow of funds. When A and B are caught in a liquidity crisis, the short-term interbank lending that they lend to C and D is withdrawn. Assume that under the path of rollover risk contagion, C and D have no liquidity risk because of their abundant cash, and the risk does not continue to contagion. However, when information diffusion exists, if C and D face liquidity hoarding by A and B, they try their best to obtain information about the motivations of A and B. Given the incompleteness and asymmetry of information, C and D cannot realize thoroughly whether A and B hoard liquidity because of a liquidity crisis or normal adjustments. Thus, C and D reduce the rollover of their short-term assets because of concerns over future liquidity uncertainty. C and D withdraw short-term assets from E and G, respectively, whereas F is in a liquidity crisis from the simultaneous withdrawal from C and D. Under the influence of information diffusion, liquidity risk is transmitted from A and B to F through C and D, whereas C and D do not experience liquidity risk. Apparently, contagion is transmitted from A and B to F in a saltatory manner.Figure 1
Simplified interbank short-term funding network of balance sheet. The figure shows the simplified interbank short-term funding network of a balance sheet for 7 banks (A, B, C, D, E, F, G), and the connecting arrow represents the initial flow of funds from creditor banks’ short-term interbank assets to debtor banks’ short-term interbank liabilities.


## 3.2. Banks’ Decision-Making Model

In the interbank market, a bank hoards liquidity for two possible motivations: if liquidity risk exists, the bank has to meet the liquidity requirements; if no liquidity risk exists, the bank adjusts the rollover strategy because of information diffusion as the creditor bank withdraws its short-term assets. Liquidity hoarding behavior under these two types of motivations cause information to be diffused from creditor banks to debtor banks and, surely, a bank’s ability to obtain information determines whether it can clearly judge the motivation of creditor banks, resulting in debtor banks adjusting their rollover strategies to different degrees. Next, we establish a bank decision-making model to quantify the banks’ stylized behavioral rules based on the motivation of creditor banks and the information-obtaining ability of debtor banks.As Table2 shows, when the debtor bank has a stronger ability to obtain information, it can more accurately determine the motivation of the creditor bank’s liquidity hoarding. Thus, debtor banks will be pessimistic about future liquidity conditions and largely reduce rollover when the creditor bank hoards liquidity because of liquidity risk. In addition, debtor banks will surely not need to worry about future liquidity conditions and reduce rollover when creditor banks face no liquidity risk. When a debtor bank has a weaker ability to obtain information, it is difficult to know whether its creditor bank is hoarding liquidity because of risk; therefore, its liquidity rollover moderately decreases because of concerns over future unknown conditions.Table 2
Influence factors of the degree of rollover reduction.
Debtor banks: degree of rollover reductionDebtor banks: ability to obtain informationStrongerWeakerMotivation of creditor banks’ liquidity hoarding: whether there is liquidity riskYesLargerModerateNoSmallerModerateTo quantify the degree of rollover reduction based on these stylized behavioral rules, suppose that after banki reduces the rollover of short-term interbank assets in period t − 1, bank i’s debtor bank j subsequently reduces the rollover of short-term interbank assets from bank j’s debtor bank s in period t. The degree of bank j’s rollover reduction from bank s (i.e., the withdrawal ratio of short-term interbank assets) is represented by(12)Yjst=LSjstCjtIjt+LSDjt<β,LSjst⋅ρjtCjtIjt+LSDjt≥β,s=1,2,3,…,N,s≠i≠j,where Cjt/Ijt+LSDjt<β and Cjt/Ijt+LSDjt≥β represents whether bank j is in a liquidity crisis. According to this formula, when bank j faces liquidity risk, it withdraws all short-term interbank assets; when bank j faces no liquidity risk, it partially reduces the degree of short-term interbank asset rollovers according to the information obtained from the creditor bank’s liquidity hoarding behavior. ρjt represents the withdrawal ratio determined by(13)ρjt=1+σj2CitIit+LSDit<β,1−σj2CitIit+LSDit≥β,σj∈0,100%,where σj∈0,100% represents bank j’s ability to obtain information. Bank j can obtain complete information when σj=100% and cannot obtain any information when σj=0. A larger σj means that bank j has a greater ability to obtain information. It should be noted that ρjt only plays a role when bank j is not in a liquidity crisis (i.e., when Cjt/Ijt+LSDjt≥β in equation (12) holds), and the calculation of ρjt in equation (13) is based on whether bank i (withdrawing short-term interbank assets from bank j in the last period) falls into a liquidity crisis. Corresponding to the analysis in Table 2, when bank i faces liquidity risk, the larger bank j’s ability to obtain information means that the withdrawal ratio is closer to 1. When bank i has no liquidity risk, the larger bank j’s ability to obtain information means that the withdrawal ratio is closer to 0. At the same time, the effect of an improvement in the bank’s information-obtaining ability for decision-making is assumed to exhibit the law of diminishing margins. That is, the change in liquidity withdrawal gradually decreases as the information-obtaining ability improves. In addition, banks with larger asset scales usually have higher internal management levels and industry status, leading to better information-obtaining ability. Therefore, the bank’s information-obtaining ability is further assumed to be in direct proportion to its asset scale.


## 4. Generating an Interbank Network

Before verifying the theory in this paper using a computational simulation, in this section, the interbank network is simulated and balance sheet data for each bank are solved.The real interbank network has typical scale-free characteristics [50–53]. The interbank funding network in this paper is established by referring to the Barratt-Barthelemy-Vespignani-based (BBV-based) directed weighted network model proposed by Wang et al. [54]. Two scale-free networks, each including 25 banks (given limited computing power, we have to balance the calculated amount (robustness) and number of nodes (banks)). To maintain 5000 simulation rounds, the number of nodes (banks) is set at 25. More nodes do not change the results of this paper but lead to exponentially increasing computation time. In addition, too many nodes result in a lack of clarity in the explanation of the exact contagion process in Section 7 and are generated to represent long- and short-term interbank funding networks. During the generating process, the node and line weights change when a new node bank joins. Thus, the connection preference is affected when new nodes are added, making the interbank network more similar to the relationship in the real interbank market. The network model’s generation steps and rules are as follows:First, the initial network is set as a fully connected network consisting ofn0 nodes; that is, bidirectional edges connected any two nodes, and the initial weight of each edge is f0.Second, the growth and development characteristics of the banking network lead to the assumption that the following two types of growth occur in each period: (1) adding a new nodes in the network with a probability of q, given the addition of the new node s, generated n new edges connecting s with other existing old nodes (n ≤ n0). Among these nodes, the number of outgoing edges obeys binomial distribution B (m, p), and the number of incoming edges obeys binomial distribution B (m, 1 − p) and p ∈ [0, 1]; and (2) adding n new edges in the existing old nodes of the network with a probability of 1 − q, and n ≤ n0.Third, according to the connection preference characteristics of the new node, in the first type of case in the second step, the probability that the outgoing edge of the new nodes selects node i to connect among many existing old nodes is(14)Πs⟶i=Giin∑j=0NGjin,where Giin represents the in-weight of node i. The probability that the incoming edge of the new node s selects node i to connect among many existing nodes is(15)Πi⟶s=Giout∑j=0NGjout,where Giout represents the out-weight of node i. In the second type of case in the second step, the selection of the out node and the in node of the new edge also follows these preference characteristics.Fourth, the addition of a new node affects the existing node relationship in the network. Set the initial weight of each newly added edge asf0. When the new node s generates an outgoing edge, that is, the edge points from the new node s to the old node i—a new incoming edge for old node i—then, the evolution rules of the node and edge weights of node i are as follows:(16)Giin⟶Giin+f0,ΔFij=δiFijGiout,Fij⟶Fij+ΔFij,Giout⟶Giout+δi,Gjin⟶Gjin+ΔFij,where Fij represents the weight of the edge from node i to node j, and ΔFij represents the change in Fij. Node j is the existing connected node of node i, and δi is the additional amount of increase in the outgoing edge weight caused by the increase in the incoming edge weight of node i.When the new node s generates an incoming edge, that is, the edge points from the old nodei to the new node s, a new outgoing edge for the old node i, then the evolution rules of the node and edge weights of node i are as follows:(17)Giout⟶Giout+f0,ΔFji=δiFjiGiin,Fji⟶Fji+ΔFji,Giin⟶Giin+δi,Gjout⟶Gjout+ΔFji,where node j is the existing connected node of node i, and δi is the additional amount of increase in the incoming edge weight caused by the increase in the outgoing edge weight of node i.Adding a new edge only causes the weight change in two existing nodes. Therefore, if the newly added edge points from the old nodei to the old node j, the weight evolution rule is as follows:(18)Giout⟶Giout+f0,Gjin⟶Gjin+f0,Fji⟶Fji+f0.Fifth, new nodes and corresponding edges are constantly added and the original edge and node weights are updated until the total number of nodes reachesN. Then, the BBV network generation and evolution process end, and the final matrix is recorded.Using the evolution rules of the aforementioned weighted directed BBV network model, the long- and short-term interbank lending networks (LS and LL) of the banking industry are generated, respectively. Here, the total number of bank nodes isN = 25. At the same time, to ensure that the balance sheets of bank nodes are relatively reasonable, the initial weight assignment and passive increment are δi = f0 = 10, the number of initial nodes and newly added edges is n = n0 = 3, and the evolution probability parameters are p=0.5 and q = 0.5. After the two networks are generated, nodes are numbered 0–24 according to the sum of the interbank assets and liabilities from large to small to ensure the consistency of the relative scale of interbank assets and liabilities (this adjustment ensures that banks with large interbank liabilities also have large interbank assets to prevent any one bank’s imbalance of assets and liabilities). On this basis, the initial balance sheets of 25 banks are further determined by setting the numerical relationship between assets and liabilities.First, according to the loan-to-deposit ratio of China’s commercial banking industry of 70.55% (China Financial Stability Report (2018)http://www.pbc.gov.cn/goutongjiaoliu/113456/113469/3656006/index.html), the numerical relationship between the bank’s loans and deposits in the model is set as Oj=0.7⋅Ij.Second, Peralta and Crisóstomo [48] show that the liquidity requirement is set as β=5%, and equity capital accounts for 6% of total assets.Third, we further make a simplifying assumption to complete solvable equations: as depository institutions, commercial banks derive income mainly from loans and financial assets, and the funding mainly comes from deposits and interbank borrowings:(19)Sj0+Oj0=Ij0+LLDj0+LSDj0.Therefore, the relationship between a single bank’s assets and liabilities in the initial time can be set as(20)Oj0=0.7⋅Ij0,Cjt=5%⋅Ijt+LSDjt,EjSj0+Oj0+Cj0+LLAj0+LSAj0=6%,Sj0+Oj0=Ij0+LLDj0+LSDj0,Sj0+Oj0+Cj0+LLAj0+LSAj0=Ij0+LLDj0+LSDj0+Ej0.Among them,LLDj0,LSDj0,LLAj0,LSAj0 can be obtained by summing the row or column elements of LL and LS in the interbank funding matrix. Therefore, the five unknown values of Sj0,Oj0,Ij0,Cj0,Ej0 can be solved using these equations. Finally, the balance sheet data of 25 banks at t = 0 can be obtained.In most existing studies on interbank risk contagion, balance sheet data are randomly generated by assuming that assets and liabilities obey a certain distribution, and equity is used to balance the articulation [4, 26, 27, 48, 55]. However, these methods cannot guarantee scale-free network characteristics and are missing the formation process of interbank networks. Therefore, this paper first uses the network evolution rule of a directed weighted network model based on the BBV to generate long- and short-term interbank networks with a scale-free nature. Then, each bank’s balance sheet data can be obtained by solving articulation equations, including liquidity requirements, leverage ratios, loan-to-deposit ratios, and other assumptions.


## 5. Basic Computational Simulation

In this section, some basic theories are verified through computational simulation using Python 3.6. We first compare and analyze the process and result of risk contagion without and with information diffusion when only rollover risk exists and all three risks exist. We further study the influence of information-obtaining ability on risk contagion using a parameter sensitivity analysis ofσ.


### 5.1. Influence of Information Diffusion on Rollover Risk Contagion

The analysis in Section3 shows that information diffusion mainly influences the contagion of rollover risk. Therefore, the influence of information diffusion on risk contagion is first simulated when only rollover risk exists. Medium-scale bank 12 is set as the initial shocked bank. The initial liquidity shock was set as the bank’s cash decreases to 0. The mean values of 5000 repeated simulations without (panel a) or with (panel b) information diffusion are recorded, as shown in Figure 2.Figure 2
Influence of information diffusion on contagion of rollover risk. (a) Without information diffusion. (b) With information diffusion. The figure depicts the process of interbank risk contagion with or without information diffusion under the liquidity shock when only rollover risk exists. The horizontal axis represents time, the vertical axis represents the number of banks in a liquidity crisis, the orange line represents the cumulative number of banks at each time, and the blue line represents the incremental number of banks at each time.
(a)(b)From the perspective of the result, approximately 9 banks finally fall into a liquidity crisis when no information diffusion exists, and approximately 17.5 banks finally fall into a liquidity crisis when information diffusion exists. From the perspective of the process, approximately 2 and 1.5 banks experience a liquidity crisis during the peak of the liquidity crisis att = 4 and t = 5, respectively, when no information diffusion exists, and approximately 4 and 5 banks experience a liquidity crisis during the peak of the liquidity crisis at t = 3 and t = 4, respectively, when information diffusion exists. From the perspective of probability, out of the 5000 repeated simulations, a total of 2199 instances of risk contagion occur (The occurrence of risk contagion means more than 1 bank (the initial shocked bank) falls into a liquidity crisis after the initial shock) when information diffusion does not exist, and 4933 instances of risk contagion occur when information diffusion exists. These findings indicate that when only rollover risk exists, the existence of information diffusion significantly magnifies the extent and probability of interbank risk contagion and accelerates the interbank risk contagion process.


### 5.2. Influence of Information Diffusion on Three Risks Contagion

Then, the influence of information diffusion on risk contagion is simulated when counterparty risk, rollover risk, and common asset holdings risk all exist. The mean values of 5000 repeated simulations without (panel a) or with (panel b) information diffusion are recorded, as shown in Figure3.Figure 3
Influence of information diffusion on contagion of all three risks. (a) Without information diffusion. (b) With information diffusion. The figure depicts the process of interbank risk contagion with or without information diffusion under a liquidity shock when rollover risk, counterparty risk, and common asset holdings risk exist. The horizontal axis represents time, and the vertical axis represents the number of banks in liquidity crisis, the number of defaulted banks, or the average price of financial assets. The orange line represents the cumulative number of banks at each time, and the blue line represents the incremental number of banks at each time.
(a)(b)From the perspective of the result, when information diffusion does not exist, approximately 2 banks default, the average price of the financial assets declines by only 12%, and approximately 8 banks are in a liquidity crisis. By contrast, the existence of information diffusion increases the number of banks in a liquidity crisis to approximately 17.5, leading to a near 30% decline in the average price of the financial assets and an increase in the number of defaulted banks to approximately 5.5. From the perspective of the process, the difference in risk contagion with and without information diffusion is similar to that in Section5.1. From the perspective of probability, in the 5000 repeated simulations, a total of 3423 instances of risk contagion occur when information diffusion does not exist, and 4998 instances of risk contagion occur when information diffusion exists. These findings indicate that when counterparty risk, rollover risk, and common asset holdings risk all exist, the existence of information diffusion also significantly magnifies the extent and probability of interbank risk contagion and accelerates the process of interbank risk contagion.Because of the addition of counterparty and common asset holdings risk, the superposition effect of the three risks could be further analyzed. Att = 0, the initial shocked bank inevitably encounters a serious liquidity crisis. On the one hand, it stops rolling over its short-term interbank assets, resulting in liquidity risk being transmitted to other banks through the path of rollover risk. On the other hand, it liquidates financial assets to obtain cash, resulting in a fire sale and declining asset prices. Declining asset prices further leads more banks to go bankrupt due to the loss of financial asset devaluation. Further, given the path of counterparty risk, the defaulted bank leads to a default of long-term debt, causing the creditor bank to suffer losses and leading to further risk contagion. Constant bankruptcy, liquidity hoarding, fire sales of financial assets, and declining asset prices form a vicious circle and superposition effect, resulting in constant risk contagion. The system does not stabilize until no new banks go bankrupt or run into a liquidity crisis. This finding confirms the superposition effect of risk contagion through the path of counterparty risk, rollover risk, and common asset holdings risk in existing studies.


### 5.3. Influence of Information-Obtaining Ability on Risk Contagion

Based on the analysis in Section3, increasing banks’ information-obtaining ability enables them to obtain more information and realize clearer situations of its creditor bank hoarding liquidity, which changes their decisions on liquidity hoarding and further influences interbank risk contagion. In this section, the effectiveness of the information-obtaining ability on interbank risk contagion is verified through a sensitivity analysis of the σ parameter. Specifically, how is the banking industry’s risk contagion affected under a liquidity shock when the information-obtaining ability σ parameter of all banks changes in the range of 0%–100% at a 10% step? To ensure the robustness of the results, 5000 repeated simulations are performed under any σ value, and the average value is taken as the result for recording. The results in Figure 4 show that an increase in σ has the following effects on the interbank risk contagion.Figure 4
Influence of information-obtaining ability on risk contagion. The figure describes the influence of information-obtaining ability on the process of interbank risk contagion under a liquidity shock when three risks exist. The axis “sigma” represents the value of the information-obtaining ability parameterσ, the axis “time” represents time, and the vertical axis “proportion” represents the proportion of banks in a liquidity crisis, the proportion of defaulted banks, and the average price of financial assets, respectively.From the perspective of the result, asσ increases, the proportion of banks facing a liquidity crisis declined from 70% to 15%, the proportion of defaulted banks declined from 26% to 5%, and financial asset prices decreased from 35% to 10%. These results indicate that the improvement in the bank’s information-obtaining ability significantly reduces the extent of the interbank risk contagion.From the perspective of the process, an increase inσ results in a flattening of the outbreak of the interbank risk contagion, and the proportion of crisis banks and price declines at the peak period declined significantly. These results show that an improvement in banks’ information-obtaining ability during the contagion process significantly reduces the severity of the peak of the interbank risk contagion.From the perspective of the threshold, an increase inσ from 0% to 70% only lowers the outbreak of the liquidity crisis and has little effect on the final proportion of banks facing a liquidity crisis. Meanwhile, the bank bankruptcy ratio and the declining ratio of the average price of financial assets also decrease smoothly. However, when σ exceeds 70%, it shows a very significant impact, which causes the proportion of crisis banks at the peak of the liquidity crisis to decline significantly, and the final proportion of banks in liquidity crisis declines significantly. Meanwhile, the bank bankruptcy ratio and the ratio of the average price of financial assets also decrease rapidly.Therefore, as banks’ information-obtaining ability improves with increasingσ, they are less affected by incomplete and asymmetric information, which significantly reduces the extent of the interbank contagion risk. This finding is consistent with our theoretical analysis: when the bank’s information-obtaining ability is improved, it reduces unnecessary liquidity hoarding and further reduces the extent and spread of the interbank risk contagion.


## 5.1. Influence of Information Diffusion on Rollover Risk Contagion

The analysis in Section3 shows that information diffusion mainly influences the contagion of rollover risk. Therefore, the influence of information diffusion on risk contagion is first simulated when only rollover risk exists. Medium-scale bank 12 is set as the initial shocked bank. The initial liquidity shock was set as the bank’s cash decreases to 0. The mean values of 5000 repeated simulations without (panel a) or with (panel b) information diffusion are recorded, as shown in Figure 2.Figure 2
Influence of information diffusion on contagion of rollover risk. (a) Without information diffusion. (b) With information diffusion. The figure depicts the process of interbank risk contagion with or without information diffusion under the liquidity shock when only rollover risk exists. The horizontal axis represents time, the vertical axis represents the number of banks in a liquidity crisis, the orange line represents the cumulative number of banks at each time, and the blue line represents the incremental number of banks at each time.
(a)(b)From the perspective of the result, approximately 9 banks finally fall into a liquidity crisis when no information diffusion exists, and approximately 17.5 banks finally fall into a liquidity crisis when information diffusion exists. From the perspective of the process, approximately 2 and 1.5 banks experience a liquidity crisis during the peak of the liquidity crisis att = 4 and t = 5, respectively, when no information diffusion exists, and approximately 4 and 5 banks experience a liquidity crisis during the peak of the liquidity crisis at t = 3 and t = 4, respectively, when information diffusion exists. From the perspective of probability, out of the 5000 repeated simulations, a total of 2199 instances of risk contagion occur (The occurrence of risk contagion means more than 1 bank (the initial shocked bank) falls into a liquidity crisis after the initial shock) when information diffusion does not exist, and 4933 instances of risk contagion occur when information diffusion exists. These findings indicate that when only rollover risk exists, the existence of information diffusion significantly magnifies the extent and probability of interbank risk contagion and accelerates the interbank risk contagion process.


## 5.2. Influence of Information Diffusion on Three Risks Contagion

Then, the influence of information diffusion on risk contagion is simulated when counterparty risk, rollover risk, and common asset holdings risk all exist. The mean values of 5000 repeated simulations without (panel a) or with (panel b) information diffusion are recorded, as shown in Figure3.Figure 3
Influence of information diffusion on contagion of all three risks. (a) Without information diffusion. (b) With information diffusion. The figure depicts the process of interbank risk contagion with or without information diffusion under a liquidity shock when rollover risk, counterparty risk, and common asset holdings risk exist. The horizontal axis represents time, and the vertical axis represents the number of banks in liquidity crisis, the number of defaulted banks, or the average price of financial assets. The orange line represents the cumulative number of banks at each time, and the blue line represents the incremental number of banks at each time.
(a)(b)From the perspective of the result, when information diffusion does not exist, approximately 2 banks default, the average price of the financial assets declines by only 12%, and approximately 8 banks are in a liquidity crisis. By contrast, the existence of information diffusion increases the number of banks in a liquidity crisis to approximately 17.5, leading to a near 30% decline in the average price of the financial assets and an increase in the number of defaulted banks to approximately 5.5. From the perspective of the process, the difference in risk contagion with and without information diffusion is similar to that in Section5.1. From the perspective of probability, in the 5000 repeated simulations, a total of 3423 instances of risk contagion occur when information diffusion does not exist, and 4998 instances of risk contagion occur when information diffusion exists. These findings indicate that when counterparty risk, rollover risk, and common asset holdings risk all exist, the existence of information diffusion also significantly magnifies the extent and probability of interbank risk contagion and accelerates the process of interbank risk contagion.Because of the addition of counterparty and common asset holdings risk, the superposition effect of the three risks could be further analyzed. Att = 0, the initial shocked bank inevitably encounters a serious liquidity crisis. On the one hand, it stops rolling over its short-term interbank assets, resulting in liquidity risk being transmitted to other banks through the path of rollover risk. On the other hand, it liquidates financial assets to obtain cash, resulting in a fire sale and declining asset prices. Declining asset prices further leads more banks to go bankrupt due to the loss of financial asset devaluation. Further, given the path of counterparty risk, the defaulted bank leads to a default of long-term debt, causing the creditor bank to suffer losses and leading to further risk contagion. Constant bankruptcy, liquidity hoarding, fire sales of financial assets, and declining asset prices form a vicious circle and superposition effect, resulting in constant risk contagion. The system does not stabilize until no new banks go bankrupt or run into a liquidity crisis. This finding confirms the superposition effect of risk contagion through the path of counterparty risk, rollover risk, and common asset holdings risk in existing studies.


## 5.3. Influence of Information-Obtaining Ability on Risk Contagion

Based on the analysis in Section3, increasing banks’ information-obtaining ability enables them to obtain more information and realize clearer situations of its creditor bank hoarding liquidity, which changes their decisions on liquidity hoarding and further influences interbank risk contagion. In this section, the effectiveness of the information-obtaining ability on interbank risk contagion is verified through a sensitivity analysis of the σ parameter. Specifically, how is the banking industry’s risk contagion affected under a liquidity shock when the information-obtaining ability σ parameter of all banks changes in the range of 0%–100% at a 10% step? To ensure the robustness of the results, 5000 repeated simulations are performed under any σ value, and the average value is taken as the result for recording. The results in Figure 4 show that an increase in σ has the following effects on the interbank risk contagion.Figure 4
Influence of information-obtaining ability on risk contagion. The figure describes the influence of information-obtaining ability on the process of interbank risk contagion under a liquidity shock when three risks exist. The axis “sigma” represents the value of the information-obtaining ability parameterσ, the axis “time” represents time, and the vertical axis “proportion” represents the proportion of banks in a liquidity crisis, the proportion of defaulted banks, and the average price of financial assets, respectively.From the perspective of the result, asσ increases, the proportion of banks facing a liquidity crisis declined from 70% to 15%, the proportion of defaulted banks declined from 26% to 5%, and financial asset prices decreased from 35% to 10%. These results indicate that the improvement in the bank’s information-obtaining ability significantly reduces the extent of the interbank risk contagion.From the perspective of the process, an increase inσ results in a flattening of the outbreak of the interbank risk contagion, and the proportion of crisis banks and price declines at the peak period declined significantly. These results show that an improvement in banks’ information-obtaining ability during the contagion process significantly reduces the severity of the peak of the interbank risk contagion.From the perspective of the threshold, an increase inσ from 0% to 70% only lowers the outbreak of the liquidity crisis and has little effect on the final proportion of banks facing a liquidity crisis. Meanwhile, the bank bankruptcy ratio and the declining ratio of the average price of financial assets also decrease smoothly. However, when σ exceeds 70%, it shows a very significant impact, which causes the proportion of crisis banks at the peak of the liquidity crisis to decline significantly, and the final proportion of banks in liquidity crisis declines significantly. Meanwhile, the bank bankruptcy ratio and the ratio of the average price of financial assets also decrease rapidly.Therefore, as banks’ information-obtaining ability improves with increasingσ, they are less affected by incomplete and asymmetric information, which significantly reduces the extent of the interbank contagion risk. This finding is consistent with our theoretical analysis: when the bank’s information-obtaining ability is improved, it reduces unnecessary liquidity hoarding and further reduces the extent and spread of the interbank risk contagion.


## 6. Robustness Test

In Section5, the influence of information diffusion and information-obtaining ability on interbank risk contagion is verified. To ensure that our results are more reliable, three types of robustness tests are carried out in this section: changing the initial shocked bank, changing the initial shock type, and changing the network structure. Meanwhile, because the influence of information-obtaining ability is the sensitivity analysis of the parameter σ when information diffusion exists, the robustness test in this section only needs to focus on the influence of information diffusion on interbank risk contagion.


### 6.1. Change in Initial Shocked Bank

Based on the simulation in Section5.2, the initial shocked bank is changed from medium-scale bank 12 to larger-scale bank 4, and 5000 repeated simulations are carried out under the condition of liquidity shock and the three risks.The results are shown in Figure5. Approximately 8 banks fail because of risk contagion when information diffusion does not exist, the average price of financial assets declines by only 40%, and approximately 15 banks experience a liquidity crisis. In contrast, the existence of information diffusion increases the number of banks in a liquidity crisis to nearly 19, leading to a more than 50% decline in the average price of financial assets. Moreover, the number of defaulted banks increases to approximately 11. These findings indicate that the influence of information diffusion on interbank risk contagion is still significant after changing the initial shocked bank.Figure 5
Influence of information diffusion after bank 4 is shocked. (a) Without information diffusion. (b) With information diffusion. The figure describes the process of interbank risk contagion with or without information diffusion after bank 4 initially suffered a liquidity shock when three risks exist. The horizontal axis represents time, and the vertical axis represents the number of banks in a liquidity crisis, the number of defaulted banks, or the average price of financial assets. The orange line represents the cumulative number of banks at each time, and the blue line represents the incremental number of banks at each time.
(a)(b)Furthermore, a comparison of Figures3 and 5 show that changing the initial shocked bank from bank 12 to bank 4 increases the absolute result of interbank risk contagion in each panel. That is, the number of banks that eventually fall into a liquidity crisis increased from 8 or 17.5 to 15 or 19, the number of bankrupt banks increased from 2 or 5.5 to 8 or 11, and the average price of financial assets declined from 12% or 30% to 40% or 50%, respectively. However, the relative result of interbank risk contagion with or without information diffusion is constant regardless of the initial shocked bank. Therefore, changing the initial shocked bank affects the ultimate level of interbank risk contagion because of the shocked bank’s scale in the network; however, this change had no effect on a comparison of the results of whether information diffusion exists.


### 6.2. Change in Initial Shock Type

The initial shock type is changed from liquidity shock to bankruptcy shock (the initial shocked bank’s loans decrease to 0, leading to bankruptcy att = 0), and 5000 repeated simulations are carried out under the condition of liquidity shock and three risks.The results are shown in Figure6. Approximately 3 banks fail when information diffusion does not exist, the average price of financial assets declines by 12%, and approximately 7.5 banks face a liquidity crisis. In contrast, the existence of information diffusion increases the number of banks facing a liquidity crisis to 17, leading to an obvious 30% decline in the average price of financial assets, and an increase in the number of defaulted banks to approximately 7. After changing the initial shock type, the influence of information diffusion on interbank risk contagion is still significant.Figure 6
Influence of information diffusion after the bankruptcy shock. (a) Without information diffusion. (b) With information diffusion. The figure depicts the process of interbank risk contagion with or without information diffusion after a bankruptcy shock when three risks exist. The horizontal axis represents time, and the vertical axis represents the number of banks in a liquidity crisis, the number of defaulted banks, or the average price of financial assets. The orange line represents the cumulative number of banks at each time, and the blue line represents the incremental number of banks at each time.
(a)(b)


### 6.3. Change in Network Structure

The systemic risk properties of financial networks are extremely sensitive to their network features [56]. Thus, this paper discusses the influence of information diffusion on interbank risk contagion in different network structures. Referring to network structures in Ramadiah et al. [56], the free-scale BBV network is changed to three different networks, including a core-peripheral structure, a modular assortative structure, and a modular disassortative structure (as shown in Figure 7). The influence of information diffusion on interbank risk contagion in these network structures is verified, and the results are shown in Figure 8. To compare the robustness of different structures, the results of the BBV network structure are also shown in Figure 8.Figure 7
Three block network structures. (a) Core-peripheral. (b) Modular assortative. (c) Modular disassortative. The figure depicts three types of block network structures. In the top row, it shows the shape of the three special block network structures: core-peripheral, modular assortative, and modular disassortative. In the bottom row, it reports the heatmaps of the connection probability matrices (with red and blue representing denser and sparser regions of the network, respectively).
(a)(b)(c)Figure 8
Influence of information diffusion in different networks. (a) Without information diffusion. (b) With information diffusion. The figure describes the process of interbank risk contagion with or without information diffusion in different networks under a liquidity shock when three risks exist. The horizontal axis represents time. The vertical axis represents the incremental (cumulative) number of banks in a liquidity crisis, the incremental (cumulative) number of defaulted banks, or the average price of financial assets. The blue, orange, green and red lines represent the BBV, core-peripheral, modular assortative and modular disassortative networks, respectively.
(a)(b)For each network structure, 5000 repeated simulations are carried out under the condition of liquidity shock and three risks exist. As to the core-peripheral structure, the existence of information diffusion increases the number of banks facing a liquidity crisis from 2 to 20, the number of defaulted banks increases from 0 to 6, and the decline in financial assets prices changes from 4% to 34%. As for the modular assortative structure, the existence of information diffusion increases the number of banks facing a liquidity crisis from 10 to 17, the number of defaulted banks increases from 1 to 9, and the decline in financial asset prices changes from 6% to 42%. As for the modular disassortative structure, the existence of information diffusion increases the number of banks facing a liquidity crisis from 3 to 12, the number of defaulted banks increases from 1 to 6, and the decline in financial assets price changes from 4% to 28%. From the perspective of the process, given the results of this robustness test section, the existence of information diffusion also clearly makes the risk of an outbreak faster, more extensive, more frequent, and more concentrated. This research shows that the conclusion of the influence of information diffusion on interbank risk contagion in a BBV network still holds true in a core-peripheral structure, a modular assortative structure, and a modular disassortative structure.In addition, by comparing the robustness (a lower number of banks in liquidity crisis, a lower number of defaulted banks, and a lower decrease in the average price of financial assets all indicate a more robust network) of different network structures based on the results, we can find that the core-peripheral structure and modular disassortative structure are more robust than the BBV structure and modular assortative structure when information diffusion does not exist, but the robustness of the core-peripheral structure declines sharply after adding the information diffusion. It indicates that information diffusion could affect the robustness of interbank network structures to varying degrees.


## 6.1. Change in Initial Shocked Bank

Based on the simulation in Section5.2, the initial shocked bank is changed from medium-scale bank 12 to larger-scale bank 4, and 5000 repeated simulations are carried out under the condition of liquidity shock and the three risks.The results are shown in Figure5. Approximately 8 banks fail because of risk contagion when information diffusion does not exist, the average price of financial assets declines by only 40%, and approximately 15 banks experience a liquidity crisis. In contrast, the existence of information diffusion increases the number of banks in a liquidity crisis to nearly 19, leading to a more than 50% decline in the average price of financial assets. Moreover, the number of defaulted banks increases to approximately 11. These findings indicate that the influence of information diffusion on interbank risk contagion is still significant after changing the initial shocked bank.Figure 5
Influence of information diffusion after bank 4 is shocked. (a) Without information diffusion. (b) With information diffusion. The figure describes the process of interbank risk contagion with or without information diffusion after bank 4 initially suffered a liquidity shock when three risks exist. The horizontal axis represents time, and the vertical axis represents the number of banks in a liquidity crisis, the number of defaulted banks, or the average price of financial assets. The orange line represents the cumulative number of banks at each time, and the blue line represents the incremental number of banks at each time.
(a)(b)Furthermore, a comparison of Figures3 and 5 show that changing the initial shocked bank from bank 12 to bank 4 increases the absolute result of interbank risk contagion in each panel. That is, the number of banks that eventually fall into a liquidity crisis increased from 8 or 17.5 to 15 or 19, the number of bankrupt banks increased from 2 or 5.5 to 8 or 11, and the average price of financial assets declined from 12% or 30% to 40% or 50%, respectively. However, the relative result of interbank risk contagion with or without information diffusion is constant regardless of the initial shocked bank. Therefore, changing the initial shocked bank affects the ultimate level of interbank risk contagion because of the shocked bank’s scale in the network; however, this change had no effect on a comparison of the results of whether information diffusion exists.


## 6.2. Change in Initial Shock Type

The initial shock type is changed from liquidity shock to bankruptcy shock (the initial shocked bank’s loans decrease to 0, leading to bankruptcy att = 0), and 5000 repeated simulations are carried out under the condition of liquidity shock and three risks.The results are shown in Figure6. Approximately 3 banks fail when information diffusion does not exist, the average price of financial assets declines by 12%, and approximately 7.5 banks face a liquidity crisis. In contrast, the existence of information diffusion increases the number of banks facing a liquidity crisis to 17, leading to an obvious 30% decline in the average price of financial assets, and an increase in the number of defaulted banks to approximately 7. After changing the initial shock type, the influence of information diffusion on interbank risk contagion is still significant.Figure 6
Influence of information diffusion after the bankruptcy shock. (a) Without information diffusion. (b) With information diffusion. The figure depicts the process of interbank risk contagion with or without information diffusion after a bankruptcy shock when three risks exist. The horizontal axis represents time, and the vertical axis represents the number of banks in a liquidity crisis, the number of defaulted banks, or the average price of financial assets. The orange line represents the cumulative number of banks at each time, and the blue line represents the incremental number of banks at each time.
(a)(b)


## 6.3. Change in Network Structure

The systemic risk properties of financial networks are extremely sensitive to their network features [56]. Thus, this paper discusses the influence of information diffusion on interbank risk contagion in different network structures. Referring to network structures in Ramadiah et al. [56], the free-scale BBV network is changed to three different networks, including a core-peripheral structure, a modular assortative structure, and a modular disassortative structure (as shown in Figure 7). The influence of information diffusion on interbank risk contagion in these network structures is verified, and the results are shown in Figure 8. To compare the robustness of different structures, the results of the BBV network structure are also shown in Figure 8.Figure 7
Three block network structures. (a) Core-peripheral. (b) Modular assortative. (c) Modular disassortative. The figure depicts three types of block network structures. In the top row, it shows the shape of the three special block network structures: core-peripheral, modular assortative, and modular disassortative. In the bottom row, it reports the heatmaps of the connection probability matrices (with red and blue representing denser and sparser regions of the network, respectively).
(a)(b)(c)Figure 8
Influence of information diffusion in different networks. (a) Without information diffusion. (b) With information diffusion. The figure describes the process of interbank risk contagion with or without information diffusion in different networks under a liquidity shock when three risks exist. The horizontal axis represents time. The vertical axis represents the incremental (cumulative) number of banks in a liquidity crisis, the incremental (cumulative) number of defaulted banks, or the average price of financial assets. The blue, orange, green and red lines represent the BBV, core-peripheral, modular assortative and modular disassortative networks, respectively.
(a)(b)For each network structure, 5000 repeated simulations are carried out under the condition of liquidity shock and three risks exist. As to the core-peripheral structure, the existence of information diffusion increases the number of banks facing a liquidity crisis from 2 to 20, the number of defaulted banks increases from 0 to 6, and the decline in financial assets prices changes from 4% to 34%. As for the modular assortative structure, the existence of information diffusion increases the number of banks facing a liquidity crisis from 10 to 17, the number of defaulted banks increases from 1 to 9, and the decline in financial asset prices changes from 6% to 42%. As for the modular disassortative structure, the existence of information diffusion increases the number of banks facing a liquidity crisis from 3 to 12, the number of defaulted banks increases from 1 to 6, and the decline in financial assets price changes from 4% to 28%. From the perspective of the process, given the results of this robustness test section, the existence of information diffusion also clearly makes the risk of an outbreak faster, more extensive, more frequent, and more concentrated. This research shows that the conclusion of the influence of information diffusion on interbank risk contagion in a BBV network still holds true in a core-peripheral structure, a modular assortative structure, and a modular disassortative structure.In addition, by comparing the robustness (a lower number of banks in liquidity crisis, a lower number of defaulted banks, and a lower decrease in the average price of financial assets all indicate a more robust network) of different network structures based on the results, we can find that the core-peripheral structure and modular disassortative structure are more robust than the BBV structure and modular assortative structure when information diffusion does not exist, but the robustness of the core-peripheral structure declines sharply after adding the information diffusion. It indicates that information diffusion could affect the robustness of interbank network structures to varying degrees.


## 7. Further Discussion

The influence of information diffusion on the contagion of the three risks fully illustrates the importance of adding information diffusion to the study of interbank risk contagion. Next, some special phenomena of rollover risk contagion under the influence of information diffusion from a subtler perspective are shown and discussed. Note that, different from the previous general analysis, these phenomena are special cases of rollover risk contagion because of the influence of information diffusion, meaning that they may occur under certain conditions in reality.


### 7.1. Saltatory Risk Contagion

To seek evidence of saltatory risk contagion in the theoretical analysis, we extract the contagion process of a single simulation and the corresponding short-term interbank funding network, as shown in Table3 and Figure 9. Figure 9(a) represents the short-term interbank funding network structure of the banking system, nodes 0–24 represent the twenty-five banks, and the connecting arrow represents the initial flow of funds. Figure 9(b) shows the contagion process of rollover risk after bank 24 (the reason for setting node 24 as the initial shocked bank node is as follows: as node 24 has the lowest in-degree and out-degree, it has few connections with other banks, and the initial rollover risk contagion route is simple. Observing the direction of the connection in Figure 9(a) to help readers understand the risk contagion process is easy) suffers from an initial liquidity shock corresponding to the bank with a liquidity crisis at each time in Table 3.Table 3
Times and states of banks.
Time (t)01234567Bank in liquidity crisis2411, 13, 20, 218, 9, 1410, 15, 18, 23171Figure 9
Network structure and contagion process of the saltatory risk contagion. (a) Short-term interbank funding network. (b) Contagion process of rollover risk. The figure describes the interbank network structure and contagion process of rollover risk when saltatory risk contagion occurs. Panel (a) is the short-term interbank funding network structure of the banking system, nodes 0–24 represent the twenty-five banks, and the connecting arrow represents the initial flow of funds. Panel (b) shows the process of rollover risk contagion after bank 24 suffers a liquidity shock. The horizontal axis represents time, the vertical axis represents the number of banks in a liquidity crisis, the orange line represents the cumulative number of banks at each time, and the blue line represents the incremental number of banks at each time.
(a)(b)As shown in Figure9 and Table 3, the first shocked bank 24 stops the rollover of short-term interbank assets after it falls into a liquidity crisis at t = 0. Because only one outgoing edge exists from bank 24 to bank 1, bank 1 suffers from a liquidity shock at t = 1 but does not fall into a liquidity crisis given its large scale and abundance of cash. However, given the influence of information diffusion, bank 1 reduces its rollover of short-term interbank assets at t = 2. The short-term interbank funding network from bank 1 to other banks has many edges, among which banks 11, 13, 20, and 21 experience a liquidity crisis at t = 2. Clearly, from t = 0 to t = 2, liquidity risk is transmitted from bank 24 to banks 11, 13, 20, and 21 through bank 1, and bank 1 does not face a liquidity crisis during this period.Saltatory risk contagion is a phenomenon that is difficult for us to understand in reality. After a bank’s liquidity crisis, its debtor banks still maintain healthy. However, other banks, which do not have a direct financing connection with the initial liquidity crisis bank but with its debtor banks, subsequently fall into a liquidity crisis. At present, researchers study rollover risk contagion without banks’ behavioral foundations, making it difficult to find the theoretical basis of this saltatory risk contagion. The influence mechanism of information diffusion in interbank markets with stylized features of incomplete and asymmetric information and stylized behavioral rules of banks provide a reasonable explanation.


### 7.2. Circular Liquidity Trap

Because of the influence mechanism of information diffusion, the liquidity risk is transmitted from creditor banks to debtor banks along the direction of the short-term interbank funding network. Even if a bank is not in a liquidity crisis, it may still reduce the rollover of short-term interbank assets because of information diffusion, causing more debtor banks to suffer from liquidity shocks. Then, when the short-term interbank funding network has ring structures, a terrible phenomenon occurs—defined as a “circular liquidity trap.”To illustrate the formation of this phenomenon and its influence, we select part of a short-term interbank funding network, as shown in Figure10(a). Banks 0, 1, and 6 form a ring network, and many external banks exist: 2, 4, 5, 11, 13, 14, 15, 18, and 23. External creditor bank 2 of bank 6 in the ring network is set as the initial shocked bank. The contagion process is shown in Figure 10(b), and Table 4 shows the corresponding bank stopping rollover because of the liquidity crisis and the bank reducing rollover because of the influence of information diffusion at each time.Figure 10
Network structure and contagion process of the circular liquidity trap. (a) Short-term interbank funding network. (b) Contagion process of rollover risk. The figure describes the interbank network structure and contagion process of rollover risk when the circular liquidity trap occurs. Panel (a) is part of a short-term interbank financing network that we chose, and the connecting arrow represents the initial flow of funds. Panel (b) describes the process of rollover risk contagion after bank 2 suffers a liquidity shock. The horizontal axis represents time, the vertical axis represents the number of banks in a liquidity crisis, the orange line represents the cumulative number of banks at each time, and the blue line represents the incremental number of banks at each time.
(a)(b)Table 4
Times and states of the banks.
Time01234567Bank stopping rollover because of a liquidity crisis223116, 4, 150, 141, 58, 13Bank reducing rollover because of the influence of information diffusion601We find that bank 2 falls into a liquidity crisis after the initial shock att = 0; thus, it stops the rollover of short-term interbank assets. Although bank 2’s debtor bank 6 does not face a liquidity crisis, bank 6 also reduces the rollover of short-term interbank assets at t = 1 under the influence of information diffusion. Then, bank 0 reduces the rollover of short-term interbank assets at t = 2, leading to bank 1 again reducing the rollover of short-term interbank assets at t = 3.After a cycle of rollover risk contagion, bank 6 once again suffers a liquidity shock from bank 1 att = 4 and falls into a liquidity crisis. Therefore, bank 6 stops the rollover of residual short-term interbank assets, leading bank 0 to face a liquidity crisis at t = 5 and bank 1 to fall into a liquidity crisis at t = 6. So far, three banks in the ring network are in a liquidity crisis because of circular liquidity withdrawal behavior under the influence of information diffusion. Meanwhile, nine banks outside the ring network are in a liquidity crisis because of the spillover effect of the bank liquidity withdrawal of their debtor banks 0, 1, and 6 in the ring network.In reality, circulating information diffusion and liquidity hoarding can lead to accumulating liquidity risk in the ring network until all banks fall into a liquidity crisis and the ring network of short-term interbank markets enters a liquidity freeze state. During this period, external debtor banks of the ring network suffer from extreme liquidity pressure of round after round of liquidity withdrawals (For example, each time that bank 1 reduces the rollover of short-term interbank assets because of the influence of information diffusion, bank 13 experiences liquidity, and the continuous accumulation of this type of shock eventually leads bank 13 to experience a liquidity crisis), easily resulting in systemic liquidity risk.


### 7.3. Risk Discovery

In Section5, the presented results are the mean value of 5000 repeated simulations, which show that information diffusion has a significant expansion effect on interbank risk contagion. However, among the simulation cases, we find that an interesting phenomenon occurs with a small probability: the existence of information diffusion reduces the number of banks experiencing a liquidity crisis. Extracting this special case of the contagion process, Figure 11(a) shows the complete short-term interbank funding network. The risk contagion process of this special case without or with information diffusion is shown in Figures 11(c) and 11(d). Table 5 shows the corresponding banks in a liquidity crisis at each time without or with information diffusion. The existence of information diffusion eventually makes bank 4 successfully avoid a liquidity crisis. Therefore, to better understand the network relationship characteristics of bank 4, Figure 11(b) separately depicts the network structure around this node.Figure 11
Network structure and contagion process of the risk discovery. (a) Whole network structure. (b) Network structure around bank 4. (c) Contagion process without information diffusion. (d) Contagion process with information diffusion. The figure describes the interbank network structure and contagion process of rollover risk when the risk is discovered. Panel (a) shows the whole short-term interbank funding network, panel (b) separately depicts the network structure around bank 4, and the connecting arrow represents the initial flow of funds. Panels (c) and (d) show the risk contagion process without or with information diffusion. The horizontal axis represents time, the vertical axis represents the number of banks in a liquidity crisis, the orange line represents the cumulative number of banks at each time, and the blue line represents the incremental number of banks at each time.
(a)(b)(c)(d)Table 5
Times and states of the banks.
Time0123456Bank in liquidity crisisFigure11(c)128, 166, 9, 113, 14, 21, 240, 2, 5, 7, 10, 201,4, 13, 15, 17, 19, 2218, 23Figure11(d)128, 162, 3, 5, 6, 9, 10, 11, 13, 14, 17, 19, 21, 241, 7, 15, 18, 20, 22, 230Comparing the cases in which information diffusion exists and does not exist shows that the influence of information diffusion mainly starts att = 2, bringing forward the peak of the liquidity crises from t = 4, 5 to t = 2. As shown in Figure 11(a), the first liquidity-shocked bank 12 points to banks 8, 16, and 0 in the short-term interbank funding network. Bank 12’s liquidity hoarding behavior leads to the liquidity crisis of banks 8 and 16 at t = 1. When information diffusion does not exist, bank 0 does not adjust its rollover strategy, and the rollover risk is transmitted from banks 8 and 16 to their debtor banks 6, 9, and 11 at t = 2. When information diffusion exists, bank 0’s liquidity hoarding behavior from information diffusion at t = 1 transmits risk to more banks at t = 2. Therefore, thirteen banks experience a liquidity crisis at t = 2, achieving the peak of the liquidity risk outbreak at this time.Why does information diffusion allow bank 4 to avoid a liquidity crisis? The reasons are as follows. According to the theoretical analysis in Section3, banks withdraw all short-term interbank assets in a liquidity crisis and only withdraw part of the short-term interbank assets because of information diffusion. As shown in Figure 11(b), the short-term interbank liabilities of bank 4 come from banks 0, 1, 6, and 19. When information diffusion exists, banks 6 and 19 experience a liquidity crisis at t = 2, bank 1 faces a liquidity crisis at t = 3, and bank 0 falls into a liquidity crisis at t = 4. Therefore, these four creditor banks of bank 4 withdraw their short-term interbank assets at different times, providing plenty of time for bank 4 to adjust its liquidity strategy. At the same time, because of the high clustering coefficient of banks 0 and 1 in the network, they start adjusting the rollover strategy because of information diffusion at the initial times of risk contagion (i.e., t = 0, 1). Bank 4 also adjusts the rollover strategy in advance (withdraw some of its short-term interbank assets) because of information diffusion from banks 0 and 1 at t = 1, 2, providing an opportunity for bank 4 to discover possible future liquidity risks and change its liquidity strategy in advance to prevent the real arrival of a future liquidity crisis.Therefore, bank 4 successfully avoids the liquidity crisis through the “risk discovery” of information diffusion because its creditor banks constantly withdraw their short-term interbank assets (hoarding liquidity because of information diffusion) at different times from bank 4, which makes bank 4 constantly withdraw its short-term interbank assets to increase its liquidity even without liquidity shortage. The occurrence of risk discovery of information diffusion, a special coincidence, is based on the special structural characteristics of the short-term interbank funding relationship in the network: enough diversification of bank 4’s interbank funding and low uniformity of interbank funding networks, which allows banks 4 to constantly adjust its liquidity but avoid its creditor banks’ simultaneous liquidity hoarding.


## 7.1. Saltatory Risk Contagion

To seek evidence of saltatory risk contagion in the theoretical analysis, we extract the contagion process of a single simulation and the corresponding short-term interbank funding network, as shown in Table3 and Figure 9. Figure 9(a) represents the short-term interbank funding network structure of the banking system, nodes 0–24 represent the twenty-five banks, and the connecting arrow represents the initial flow of funds. Figure 9(b) shows the contagion process of rollover risk after bank 24 (the reason for setting node 24 as the initial shocked bank node is as follows: as node 24 has the lowest in-degree and out-degree, it has few connections with other banks, and the initial rollover risk contagion route is simple. Observing the direction of the connection in Figure 9(a) to help readers understand the risk contagion process is easy) suffers from an initial liquidity shock corresponding to the bank with a liquidity crisis at each time in Table 3.Table 3
Times and states of banks.
Time (t)01234567Bank in liquidity crisis2411, 13, 20, 218, 9, 1410, 15, 18, 23171Figure 9
Network structure and contagion process of the saltatory risk contagion. (a) Short-term interbank funding network. (b) Contagion process of rollover risk. The figure describes the interbank network structure and contagion process of rollover risk when saltatory risk contagion occurs. Panel (a) is the short-term interbank funding network structure of the banking system, nodes 0–24 represent the twenty-five banks, and the connecting arrow represents the initial flow of funds. Panel (b) shows the process of rollover risk contagion after bank 24 suffers a liquidity shock. The horizontal axis represents time, the vertical axis represents the number of banks in a liquidity crisis, the orange line represents the cumulative number of banks at each time, and the blue line represents the incremental number of banks at each time.
(a)(b)As shown in Figure9 and Table 3, the first shocked bank 24 stops the rollover of short-term interbank assets after it falls into a liquidity crisis at t = 0. Because only one outgoing edge exists from bank 24 to bank 1, bank 1 suffers from a liquidity shock at t = 1 but does not fall into a liquidity crisis given its large scale and abundance of cash. However, given the influence of information diffusion, bank 1 reduces its rollover of short-term interbank assets at t = 2. The short-term interbank funding network from bank 1 to other banks has many edges, among which banks 11, 13, 20, and 21 experience a liquidity crisis at t = 2. Clearly, from t = 0 to t = 2, liquidity risk is transmitted from bank 24 to banks 11, 13, 20, and 21 through bank 1, and bank 1 does not face a liquidity crisis during this period.Saltatory risk contagion is a phenomenon that is difficult for us to understand in reality. After a bank’s liquidity crisis, its debtor banks still maintain healthy. However, other banks, which do not have a direct financing connection with the initial liquidity crisis bank but with its debtor banks, subsequently fall into a liquidity crisis. At present, researchers study rollover risk contagion without banks’ behavioral foundations, making it difficult to find the theoretical basis of this saltatory risk contagion. The influence mechanism of information diffusion in interbank markets with stylized features of incomplete and asymmetric information and stylized behavioral rules of banks provide a reasonable explanation.


## 7.2. Circular Liquidity Trap

Because of the influence mechanism of information diffusion, the liquidity risk is transmitted from creditor banks to debtor banks along the direction of the short-term interbank funding network. Even if a bank is not in a liquidity crisis, it may still reduce the rollover of short-term interbank assets because of information diffusion, causing more debtor banks to suffer from liquidity shocks. Then, when the short-term interbank funding network has ring structures, a terrible phenomenon occurs—defined as a “circular liquidity trap.”To illustrate the formation of this phenomenon and its influence, we select part of a short-term interbank funding network, as shown in Figure10(a). Banks 0, 1, and 6 form a ring network, and many external banks exist: 2, 4, 5, 11, 13, 14, 15, 18, and 23. External creditor bank 2 of bank 6 in the ring network is set as the initial shocked bank. The contagion process is shown in Figure 10(b), and Table 4 shows the corresponding bank stopping rollover because of the liquidity crisis and the bank reducing rollover because of the influence of information diffusion at each time.Figure 10
Network structure and contagion process of the circular liquidity trap. (a) Short-term interbank funding network. (b) Contagion process of rollover risk. The figure describes the interbank network structure and contagion process of rollover risk when the circular liquidity trap occurs. Panel (a) is part of a short-term interbank financing network that we chose, and the connecting arrow represents the initial flow of funds. Panel (b) describes the process of rollover risk contagion after bank 2 suffers a liquidity shock. The horizontal axis represents time, the vertical axis represents the number of banks in a liquidity crisis, the orange line represents the cumulative number of banks at each time, and the blue line represents the incremental number of banks at each time.
(a)(b)Table 4
Times and states of the banks.
Time01234567Bank stopping rollover because of a liquidity crisis223116, 4, 150, 141, 58, 13Bank reducing rollover because of the influence of information diffusion601We find that bank 2 falls into a liquidity crisis after the initial shock att = 0; thus, it stops the rollover of short-term interbank assets. Although bank 2’s debtor bank 6 does not face a liquidity crisis, bank 6 also reduces the rollover of short-term interbank assets at t = 1 under the influence of information diffusion. Then, bank 0 reduces the rollover of short-term interbank assets at t = 2, leading to bank 1 again reducing the rollover of short-term interbank assets at t = 3.After a cycle of rollover risk contagion, bank 6 once again suffers a liquidity shock from bank 1 att = 4 and falls into a liquidity crisis. Therefore, bank 6 stops the rollover of residual short-term interbank assets, leading bank 0 to face a liquidity crisis at t = 5 and bank 1 to fall into a liquidity crisis at t = 6. So far, three banks in the ring network are in a liquidity crisis because of circular liquidity withdrawal behavior under the influence of information diffusion. Meanwhile, nine banks outside the ring network are in a liquidity crisis because of the spillover effect of the bank liquidity withdrawal of their debtor banks 0, 1, and 6 in the ring network.In reality, circulating information diffusion and liquidity hoarding can lead to accumulating liquidity risk in the ring network until all banks fall into a liquidity crisis and the ring network of short-term interbank markets enters a liquidity freeze state. During this period, external debtor banks of the ring network suffer from extreme liquidity pressure of round after round of liquidity withdrawals (For example, each time that bank 1 reduces the rollover of short-term interbank assets because of the influence of information diffusion, bank 13 experiences liquidity, and the continuous accumulation of this type of shock eventually leads bank 13 to experience a liquidity crisis), easily resulting in systemic liquidity risk.


## 7.3. Risk Discovery

In Section5, the presented results are the mean value of 5000 repeated simulations, which show that information diffusion has a significant expansion effect on interbank risk contagion. However, among the simulation cases, we find that an interesting phenomenon occurs with a small probability: the existence of information diffusion reduces the number of banks experiencing a liquidity crisis. Extracting this special case of the contagion process, Figure 11(a) shows the complete short-term interbank funding network. The risk contagion process of this special case without or with information diffusion is shown in Figures 11(c) and 11(d). Table 5 shows the corresponding banks in a liquidity crisis at each time without or with information diffusion. The existence of information diffusion eventually makes bank 4 successfully avoid a liquidity crisis. Therefore, to better understand the network relationship characteristics of bank 4, Figure 11(b) separately depicts the network structure around this node.Figure 11
Network structure and contagion process of the risk discovery. (a) Whole network structure. (b) Network structure around bank 4. (c) Contagion process without information diffusion. (d) Contagion process with information diffusion. The figure describes the interbank network structure and contagion process of rollover risk when the risk is discovered. Panel (a) shows the whole short-term interbank funding network, panel (b) separately depicts the network structure around bank 4, and the connecting arrow represents the initial flow of funds. Panels (c) and (d) show the risk contagion process without or with information diffusion. The horizontal axis represents time, the vertical axis represents the number of banks in a liquidity crisis, the orange line represents the cumulative number of banks at each time, and the blue line represents the incremental number of banks at each time.
(a)(b)(c)(d)Table 5
Times and states of the banks.
Time0123456Bank in liquidity crisisFigure11(c)128, 166, 9, 113, 14, 21, 240, 2, 5, 7, 10, 201,4, 13, 15, 17, 19, 2218, 23Figure11(d)128, 162, 3, 5, 6, 9, 10, 11, 13, 14, 17, 19, 21, 241, 7, 15, 18, 20, 22, 230Comparing the cases in which information diffusion exists and does not exist shows that the influence of information diffusion mainly starts att = 2, bringing forward the peak of the liquidity crises from t = 4, 5 to t = 2. As shown in Figure 11(a), the first liquidity-shocked bank 12 points to banks 8, 16, and 0 in the short-term interbank funding network. Bank 12’s liquidity hoarding behavior leads to the liquidity crisis of banks 8 and 16 at t = 1. When information diffusion does not exist, bank 0 does not adjust its rollover strategy, and the rollover risk is transmitted from banks 8 and 16 to their debtor banks 6, 9, and 11 at t = 2. When information diffusion exists, bank 0’s liquidity hoarding behavior from information diffusion at t = 1 transmits risk to more banks at t = 2. Therefore, thirteen banks experience a liquidity crisis at t = 2, achieving the peak of the liquidity risk outbreak at this time.Why does information diffusion allow bank 4 to avoid a liquidity crisis? The reasons are as follows. According to the theoretical analysis in Section3, banks withdraw all short-term interbank assets in a liquidity crisis and only withdraw part of the short-term interbank assets because of information diffusion. As shown in Figure 11(b), the short-term interbank liabilities of bank 4 come from banks 0, 1, 6, and 19. When information diffusion exists, banks 6 and 19 experience a liquidity crisis at t = 2, bank 1 faces a liquidity crisis at t = 3, and bank 0 falls into a liquidity crisis at t = 4. Therefore, these four creditor banks of bank 4 withdraw their short-term interbank assets at different times, providing plenty of time for bank 4 to adjust its liquidity strategy. At the same time, because of the high clustering coefficient of banks 0 and 1 in the network, they start adjusting the rollover strategy because of information diffusion at the initial times of risk contagion (i.e., t = 0, 1). Bank 4 also adjusts the rollover strategy in advance (withdraw some of its short-term interbank assets) because of information diffusion from banks 0 and 1 at t = 1, 2, providing an opportunity for bank 4 to discover possible future liquidity risks and change its liquidity strategy in advance to prevent the real arrival of a future liquidity crisis.Therefore, bank 4 successfully avoids the liquidity crisis through the “risk discovery” of information diffusion because its creditor banks constantly withdraw their short-term interbank assets (hoarding liquidity because of information diffusion) at different times from bank 4, which makes bank 4 constantly withdraw its short-term interbank assets to increase its liquidity even without liquidity shortage. The occurrence of risk discovery of information diffusion, a special coincidence, is based on the special structural characteristics of the short-term interbank funding relationship in the network: enough diversification of bank 4’s interbank funding and low uniformity of interbank funding networks, which allows banks 4 to constantly adjust its liquidity but avoid its creditor banks’ simultaneous liquidity hoarding.


## 8. Conclusion and Suggestions

This paper introduces the stylized features of incomplete and asymmetric information to establish the decision-making model of banks’ liquidity hoarding based on stylized behavioral rules. It further analyzes the influence of information diffusion and information-obtaining ability on interbank risk contagion and verifies the theory through computational simulation. The results show that information diffusion has a significant impact on risk contagion, resulting in a higher probability and extent of contagion, an accelerated contagion process, and enhanced risky banks at the peak of risk contagion. This finding fully illustrates the importance of considering information diffusion in the study of interbank risk contagion and puts forward new ideas and breakthroughs for existing theories. The robustness tests show that the conclusion is still valid after replacing the initial shocked bank, initial shock type, and network structure. In addition, we find that improving banks’ information-obtaining ability (theσ parameter) reduces banks’ unnecessary liquidity hoarding, which significantly reduces the interbank risk contagion after σ > 70%.In further research, this paper discusses three special phenomena of rollover risk contagion influenced by information diffusion: the saltatory risk contagion, the circular liquidity trap, and the risk discovery of information diffusion. Through computational simulations, the formation process and results of the three phenomena are depicted, and their reasons and influences are theoretically analyzed to provide a reference for the establishment of regulatory risk policies for real interbank systems.The existence of the saltatory risk contagion and the circular liquidity trap reveals the hidden saltatory and explosive characteristics of rollover risk contagion under the influence of information diffusion. Accidental risk phenomena seemingly have an inevitable internal path, creating significant challenges for the supervision of systemic risk in banking systems. To this end, we make the following suggestions. First, regulators should attach great importance to the disclosure of bank information and continuously require banks to disclose their liquidity and other risk indicators to enhance the completeness and symmetry of information diffusion. Second, in view of the already existing incomplete and asymmetric information in the market, systemic risk supervision should not only focus on the exposed crisis but also pay attention to every path node in saltatory risk contagion to discover the real cause and path of risk formation and outbreak. Third, regulators should be highly alert to the formation of interbank funding ring networks to prevent banks in the ring network from falling into the circular liquidity trap because of information diffusion.Finally, this paper determines the risk discovery of information diffusion that could reduce interbank risk contagion. It also provides new ideas for the supervision of interbank funding. On the one hand, a single bank should diversify its interbank funding from more banks, providing more chances to discover the liquidity risk through information diffusion. On the other hand, the diversification of interbank funding should avoid the formation of complete networks, preventing simultaneous liquidity hoardings of all banks. These two aspects of supervision can provide enough time and possibility for banks to adjust their liquidity levels by using the risk discovery function of information diffusion.However, because of the limitations to real interbank funding data, this paper cannot analyze real banking risk status. However, we propose a theory and verify it using a simulation method. Regulators can use real interbank data to conduct stress tests based on the theory and methods proposed in this paper. In addition, how to make the risk discovery of information diffusion plays a greater role through specific allocation strategies of interbank funding, and macroprudential supervision of interbank funding networks is also worthy of further study.



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*Source: 8887518-2021-03-26.xml*