Aref Mahdavi Ardekani
email: aref.mahdavi@univ-paris1.fr

Liquidity, Interbank Network Topology and Bank Capital

Keywords: JEL Classification: G21, G28, L14 Interbank network topology, Bank regulatory capital, Liquidity risk, Basel III JEL Classification: G21, G28, L14 Interbank network topology, Bank regulatory capital, Liquidity risk, Basel III

By applying the interbank network simulation, this paper examines whether the causal relationship between capital and liquidity is influenced by bank positions in the interbank network. While existing literature highlights the causal relationship that moves from liquidity to capital, the question of how interbank network characteristics affect this relationship remains unclear. Using a sample of commercial banks from 28 European countries, this paper suggests that banks' interconnectedness within interbank loan and deposit networks affects their decisions to set higher or lower regulatory capital ratios when facing higher illiquidity. This study provides support for the need to implement minimum liquidity ratios to complement capital ratios, as stressed by the Basel Committee on Banking Regulation and Supervision. This paper also highlights the need for regulatory authorities to consider the network characteristics of banks.

Introduction

Following the recent financial crises, much effort has been made by financial regulators to monitor the capital and liquidity of banks in order to enhance the stability of financial markets.

Concurrently, the Basel Committee on Banking Regulation and Supervision introduced its unbonded capital and liquidity constraints under the Basel III guideline (BCBS, 2010). While the effectiveness of Basel III has been extensively debated in the literature (e.g., [START_REF] Allahrakha | Do higher capital standards always reduce bank risk? The impact of the Basel leverage ratio on the U.S. triparty repo market[END_REF][START_REF] Le | Capital requirements and banks performance under Basel-III: A comparative analysis of Australian and British banks[END_REF][START_REF] Merkl | Banks' regulatory buffers, liquidity networks and monetary policy transmission[END_REF][START_REF] Petersen | A Basel perspective on bank leverage[END_REF], recent studies have also pointed to the existence of interrelationships between bank capital and liquidity [START_REF] Berger | Bank liquidity creation[END_REF][START_REF] Bhattacharya | Contemporary banking theory[END_REF][START_REF] Diamond | Liquidity Risk , Liquidity Creation , and Financial Fragility : A Theory of Banking[END_REF][START_REF] Gorton | Liquidity provision, bank capital, and the macroeconomy[END_REF][START_REF] Von Thadden | Bank capital adequacy regulation under the new Basel accord[END_REF]. However, existing studies neglect the role of banks' sophisticated Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée) interconnectedness in the interbank market in determining the causal relationship between capital and liquidity.

The most prominent function of banks is liquidity creation by funding long-term illiquid assets with short-term liquid liabilities. This leads banks to hold illiquid assets and provide liquidity to stimulate the whole economy. In addition, however, it makes banks vulnerable to the risk of unexpected withdrawals of short-term liabilities which have been invested in illiquid assets, and consequently raises the probability of bank failure. The interbank market plays a critical role in facilitating liquidity transformation through the channelling of short-term liquid funds between banks with surpluses and those with shortages. It links banks and financial institutions based on their bilateral liquidity needs. Although the mechanism of liquidity redistribution in the interbank market could alleviate such liquidity shocks among banks, access to this market is not equally granted to all banks. Moreover, banks have traditionally been reluctant to cooperate in unsecured interbank markets during turmoil.

Capital and liquidity regulations aim to strengthen bank solvency and liquidity positions.

The new capital ratios require banks to maintain a certain amount of Tier 1 and Tier 2 capital against all banks' balance sheets and off-balance sheets exposures to 'constrain the build-up of excessive leverage' in financial institutions. Furthermore, the liquidity constraints require banks to hold a certain amount of liquid assets over a one-month horizon (Liquidity Coverage Ratio, LCR) and to maintain sufficient stable funds over a one-year horizon (Net Stable Funding Ratio, NSFR) to protect them from liquidity shocks. The Basel III capital and liquidity requirements are, however, independent of the banks' network topology or the quality of their interconnectedness in the interbank network. [START_REF] Ardekani | Do banks change their liquidity ratios based on network characteristics[END_REF] document that the ways banks set their liquidity ratios are dependent on their interbank network characteristics. [START_REF] Distinguin | Bank regulatory capital and liquidity: Evidence from US and European publicly traded banks[END_REF], [START_REF] Fu | Bank capital and liquidity creation in asia pacific[END_REF] and [START_REF] Horváth | Bank Capital and Liquidity Creation: Granger-Causality Evidence[END_REF] all additionally highlight the causal relationship that moves from liquidity to capital.

Accordingly, the relationship between a bank's capital and liquidity might also be influenced by their position in the interbank network.

Literature on the causal relationship between capital and liquidity has often proposed contradictory findings. [START_REF] Distinguin | Bank regulatory capital and liquidity: Evidence from US and European publicly traded banks[END_REF] suggest complementarity between liquidity and capital for large banks and substitutionary effects for small banks. By working on a sample of US Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

and European banks, they find that large (small) banks reduce (increase) their regulatory capital when they face higher illiquidity. Through investigating the causal relationship between liquidity and regulatory capital of banks in Asia-Pacific economies, [START_REF] Fu | Bank capital and liquidity creation in asia pacific[END_REF] find a negative interrelationship between capital and liquidity regardless of bank size. [START_REF] Horváth | Bank Capital and Liquidity Creation: Granger-Causality Evidence[END_REF] propose substitutionary impacts of liquidity on the capital of Czech banks. They demonstrate that banks enhance their capital ratios when they face higher illiquidity. Because empirical literature does not suggest any clear-cut findings regarding the causal relationship that moves from liquidity to capital, the reason behind these findings remains an open issue. Those findings could possibly be explained by considering the banks' positions in the interbank network. [START_REF] Matz | Liquidity Risk Measurement and Management Liquidity Risk Measurement and Management[END_REF] argue that higher liquidity creation results in a bank's higher exposure to liquidity risk. Since increasing capital might improve banks' ability to raise more funds from the market or reduce depositor runs, they strengthen their solvency standards when they face higher illiquidity. However, if they are well positioned in the interbank network, they have wider access to the wholesale liquid funds which can reduce their demand for higher capital. This paper examines how ease of access to the interbank market could influence the interrelationship between banks' capital and liquidity. Specifically, I hypothesise that weakly interconnected banks might target a higher capital ratio to improve their solvency when they face higher illiquidity. Because they have weaker access to interbank funds, they might substitute capital for liquidity to offset their liquidity constraints and to promote their fundraising capacity.

Moreover, I suggest that a stronger network position may lead banks to substitute less capital for liquidity, presumably because they believe wider access to wholesale liquidity funds could alleviate the liquidity constraints and soften depositors' requirements for solvency standards.

Banks may therefore set different capital ratios depending on their position in the interbank market when they face higher illiquidity. Additionally, banks target different ratios during normal times and distress periods, which could also be differently shaped depending on their network positions.

This study is also related to literature that analyses the topology of bank networks to draw an accurate picture of the significant role played by the shape of interbank network connectedness in bank liquidity risk, systemic risk and the contagion of financial shocks to the economy [START_REF] Ardekani | Do banks change their liquidity ratios based on network characteristics[END_REF][START_REF] Borges | Systemic risk in the Angolan interbank payment system-a network approach[END_REF][START_REF] Capponi | Systemic risk mitigation in financial networks[END_REF][START_REF] Glasserman | How likely is contagion in financial networks[END_REF][START_REF] Huang | A financial network perspective of financial institutions' systemic risk contributions[END_REF][START_REF] Paltalidis | Transmission Channels of Systemic Risk and Contagion in the European Financial Network[END_REF][START_REF] Souza | Insolvency and contagion in the Brazilian interbank market[END_REF]. [START_REF] Iori | A network analysis of the Italian overnight money market[END_REF], [START_REF] Kuzubaş | Network centrality measures and systemic risk: An application to the Turkish financial crisis[END_REF] Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

and [START_REF] Soramäki | The topology of interbank payment flows[END_REF] all investigate the efficiency of banking networks by analysing the topological characteristics of payment systems in different countries. [START_REF] Martinez-Jaramillo | An empirical study of the Mexican banking system's network and its implications for systemic risk[END_REF] and [START_REF] Rørdam | The topology of Danish interbank money flows[END_REF] assess the network topology of interbank exposure and payment systems in different countries and compare their characteristics. [START_REF] Gabrieli | A network view on interbank market freezes[END_REF] suggest that strong network positions would result in higher bank access to the interbank liquidity market. [START_REF] Chinazzi | Post-mortem examination of the international financial network[END_REF] and [START_REF] Soramäki | The topology of interbank payment flows[END_REF] highlight the interaction between banks in the interbank market and their willingness to connect with banks of different sizes. This study uses network topology statistics to measure the strength of banks' interconnectedness in the interbank network and to quantify how banks can more or less easily gain access to wholesale liquid funds. This research distinguishes systemwide from local network positions in the interbank market. While local network topology quantifies a bank's immediate access to interbank funds, systemwide network topology measures the crucial role that each bank plays in the whole interbank network.

Although existing literature has addressed the causal relationship between bank capital and liquidity, these studies have neglected the role of interconnectedness among banks in the interbank network. This paper contributes to existing literature by investigating how the interbank network characteristics influence the relationship between a bank's capital and liquidity. My results suggest that, whereas weak interbank interconnectedness strengthens the substitutionary effect of liquidity on capital, broader access to the interbank market works as liquidity insurance and weakens this relationship during normal times. During times of economic distress, my findings suggest that strongly local interconnected banks do not substitute capital for liquidity. This is presumably because they have broader access to the interbank funds, and thus are less pressured by depositor runs. My results do, however, exhibit that strongly system-wide interconnected banks set lower capital ratios when they face higher illiquidity.

The rest of this article is laid out as follows. Section 2 describes the data, variables and methodology of the study, while section 3 presents the results of the study. Robustness checks are reported in section 4. Section 5 provides a conclusion.

Sample, Variables and Methodology

Sample

My sample consists of 506 commercial banks established in 28 European countries 1 . I construct the interbank networks by including all available commercial, investment and real estate banks in the Bankscope database in each country 2 . Therefore, network statistics in this study capture connections of banks in my sample (only commercial) with all possible banks in their respective countries.

The selected sample period is from 2001 to 2013. Accounting data (annual financial statements) for individual banks are obtained from Bankscope Fitch IBCA. Bankscope reported balance sheets and income statements for 1,238 commercial banks for the countries considered in this study. After eliminating banks for which Bankscope does not report information on variables of interest, the final sample of banks consists of 506 banks.

[Insert Table 1]

Definition of Variables

This section presents dependent variables, different independent variables reflecting interbank network characteristics and control variables that are introduced in my estimations.

Descriptive statistics and definitions of these variables are provided in Table 2. The extreme bank year observations for my dependent and bank-level control variables winsorised (1% lowest and highest values).

[Insert Table 2]

Total Capital Ratio (TCR)

The Basel Committee on Banking Regulation and Supervision has introduced a new capital ratio that requires banks to maintain a certain amount of Tier 1 and Tier 2 capital against all the 1 Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden and the United Kingdom. 2 This study omits savings and mutual banks from the sample, subject to their specificities in terms of interbank network relationships [START_REF] Bis | Marrying the Macro-and Micro-Prudential Dimensions of Financial Stability[END_REF]Boss & Elsinger, 2004;[START_REF] Worms | The reaction of bank lending to monetary policy measures in Germany[END_REF]. Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée) bank's balance sheet and off-balance sheet exposures in order to 'constrain the build-up of excessive leverage' in financial institutions. The total capital ratio is defined according to Basel III guidelines as Tier 1 plus Tier 2 capital divided by risk-weighted assets (RWA):

!"# = !"#$ &'!"#$ ( )*+
Tier 1 capital represents a bank's core capital, which includes a shareholder's equity and noncumulative preferred shares. Tier 2 capital is defined as complementary capital and consists of hybrid instruments and subordinated debts3 .

Inverse of Structural Liquidity Indicator (I. NSFR)

In addition to the Total Capital Ratio (TCR), the Basel Committee introduced an international framework to assess banks' liquidity. Its longer horizon liquidity requirements, the 'net stable funding ratio' (NSFR) (BCBS, 2010), requires banks to reduce liquidity mismatch by financing their illiquid assets with less risky and more stable funds. It is a structural tool for efficient liquidity measurement, as it scrutinises two sides of the balance sheet and classifies assets and liabilities as illiquid, semi-liquid and liquid, assigning weights to each component. To measure the illiquidity, and consistent with [START_REF] Distinguin | Bank regulatory capital and liquidity: Evidence from US and European publicly traded banks[END_REF], this study uses the inverse of liquidity regulatory ratio. This ratio is defined as 4 : !. #$%& = "#$%&'#( )*+%,-+. /-)01# .%,(/ 23)&1)01# )*+%,-+. /-)01# .%,(/ The available amount of stable funds is the total amount of bank capital; the share of time deposits and stable demand deposits (maturity of less than one year) that would be expected to stay within the bank; and liabilities with a maturity equal or greater than one year. The required amount of stable funding is defined as the amount of assets that are used as collateral for borrowing during a period of liquidity stress, or those that cannot be easily monetised. Basel III requires banks to set their NSFR ratio above 1 (100%). Estimating NSFR according to BCBS (2010) guidelines computed. Additionally, the LCR measurements are based on liquidity shocks over one month (a short horizon). The network statistics variables in my work are computed annually. Hence, their time horizon is in compliance with the NSFR.

(2)

(1) Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée) is difficult due to the unavailability of data on a detailed breakdown of the balance sheet; therefore, I approximate it with Bankscope data using the weights defined by [START_REF] Vazquez | Bank funding structures and risk: Evidence from the global financial crisis[END_REF] 5 .

Interbank Network

This paper aims to examine whether the causal relationship between bank capital and liquidity is influenced by interbank network characteristics. To achieve this, network variables are extracted from interbank lending-borrowing relationships. Construction of the interbank network requires an estimation of the bilateral transactions between each pair of banks in the interbank market. A crucial shortcoming when studying the interbank exposure is the unavailability of bilateral transaction data, as financial authorities do not require banks to report such information in most European countries. The only available data is the aggregate interbank loans and deposits in each bank. The most relevant and applied method for constructing the interbank network based on the aggregate interbank transactions is the minimum density (MD) algorithm, proposed by [START_REF] Anand | Filling in the blanks: network structure and interbank contagion[END_REF].

A notable point of applying this method is its economic rationality: producing and maintaining extra network linkages in the interbank market is costly and should be minimised.

Moreover, it uses known features of the interbank network, as explained in the literature. For instance, [START_REF] Anand | Filling in the blanks: network structure and interbank contagion[END_REF] highlight the extreme cost of connections with all possible banks in the network. They also explain that the interbank loan and deposit market has hierarchical attributes. [START_REF] Cocco | Lending relationships in the interbank market[END_REF] suggest that banks connect to counterparties with minimum correlated liquidity shocks, which leads to sparse networks. [START_REF] Chiu | Relationships in the interbank market[END_REF] and [START_REF] Cocco | Lending relationships in the interbank market[END_REF] argue that a long-term lending relationship is the base requirement for banks' connections in the interbank network. Building on such work, and by considering the core-periphery characteristics of the interbank network, [START_REF] Craig | Interbank tiering and money center banks[END_REF] show its tiering properties. [START_REF] Ardekani | Do banks change their liquidity ratios based on network characteristics[END_REF] construct their interbank exposure network based on an MD algorithm, and examine the relationship between interbank network characteristics and liquidity ratios of European banks. I follow [START_REF] Ardekani | Do banks change their liquidity ratios based on network characteristics[END_REF] and apply the MD algorithm of Anand 5 Table B1 presents its detailed weighted components. [START_REF] Vazquez | Bank funding structures and risk: Evidence from the global financial crisis[END_REF] explain the departure from Basel III weights. For instance, 100% weight is assigned to total amount of loans, as splitting loans subject to their type or maturity is not possible. An average weight of 35% is assigned to other earning assets, as they are supposed to be more liquid. The interbank market allows an exchange of short-term funds among banks based on their liquidity preferences and forms complex networks of debtors and creditors. Therefore, it facilitates liquidity transmission in the financial system. To construct my interbank network, I consider each bank as a node Ni and the lending-borrowing relationship between each bank as a directed link Lij that connects node i to node j, where Lij ≠ Lji. Lij represents a loan from bank i to bank j and Lji denotes deposits from bank j to bank i. In addition to direct linkages between banks, path length of Ni to Nj where denoted by PLij measures a number of edges between two banks and shows the network distance between two different banks. While there might be several paths from Ni to Nj, dij quantifies the shortest distance, that is the minimum number of edges between node i and j.

Centrality is a concept developed and used in the social network to identify the importance, influence and power of each entity in the corresponding network. However, growing literature on financial economics (e.g., Boss et al., 2004;[START_REF] Langfield | Mapping the UK interbank system[END_REF][START_REF] Minoiu | A network analysis of global banking: 1978-2010[END_REF][START_REF] Rørdam | The topology of Danish interbank money flows[END_REF]) have adopted it to optimally assess the risk, vulnerability and performance of financial networks. This study employs centrality measurements to assess the accessibility of the interbank market funds to each bank by identifying the interconnectedness among banks in the interbank network.

Local Network Variables

Local network variables measure the immediate connectedness of banks with their counterparts in the interbank network, and are comprised of In-Degree and Out-Degree. In-Degree quantifies the number of immediate incoming links (direct lenders), and Out-Degree measures the number of immediate outgoing links (direct borrowers).

& , "-= ' ( ," ,
Where ( ," takes a value of one if there is an interbank loan from bankj to banki, and zero otherwise.

& , ./0 = ' ( ", ,
Where ( ", takes a value of one if there is interbank deposit from banki to bankj, and zero otherwise.

I define network dummy variables to distinguish between strong and weak local interconnectedness. HIn-Degree and HOut-Degree dummy variables take a value of one if their values are greater than or equal to the median value.

Banks with strong local network positions in the interbank markets are more specialised for diversifying their interbank borrowing or lending, which helps them to maintain their linkages even in times of economic distress. According to the substitutionary effect of liquidity on capital, banks set higher capital ratios to strengthen their solvency and increase their fundraising capabilities when facing higher illiquidity. I assume that the fundraising capability of illiquid banks is less affected by their solvency if they have higher lending and borrowing diversification in the interbank market. Likewise, I expect that a strong local position in the interbank network weakens this substitutionary effect when banks face higher illiquidity.

System-Wide Network Variables

System-wide network variables measure a bank's access to the interbank funds of the whole network. Those variables quantify the network characteristics of each bank compared with all existing banks in the interbank market, and also provide a wider image of a bank's interconnectedness compared with local measurements.

Betweenness Centrality quantifies the systemic position of each bank in the whole network. Where / ,"1 is defined as the number of geodesic paths between bank j and bank k that pass-through bank i. Eventually, PageRank measures the importance of each bank's counterparties in determining the significant role of a bank in the interbank network.

)*+,**--*.. " = ' / ,"1 / ,1 ,21 (4)
01/*#1-2(4) = (1 -8) 9 + 8 ' 01/*#1-2(;) !((;) ,∈45(")
Where for bank i, TL is the total number of links that depart from its out degree, and d is a factor that [START_REF] Winograd | The PageRank Citation Ranking: Bringing Order to the Web[END_REF] recommended setting at 0.85. I introduce system-wide network dummy variables to differentiate between strong and weak access of banks to the interbank funds in the whole network. HBetweenness and HPageRank dummy variables take a value of one if their values are greater than or equal to the median value.

A bank's strong system-wide position in the interbank network increases their access to the interbank funds, but is also a potential source of contagion and systemic risk. On that note, the impact of system-wide network variables on the relationship between capital and liquidity depends on the economic situation (crisis vs. normal). Overall, I expect that banks with strong system-wide interconnectedness substitute less capital for liquidity. I also assume that wide accessibility to interbank funds increases banks' fundraising capabilities and reduces pressure to further strengthen their solvency7 .

Validity of the Simulated Network

Anand et al. ( 2018) compare a variety of methods for reconstructing networks for 25 distinct financial markets, and suggest that the best method to preserve the structure of links is Minimum Density. I further prove this simulation technique's validity by analysing whether the simulated interbank network has characteristics in common with the interbank network in the real world.

Small world and scale-free properties are the common features of the real-word interbank network, as suggested by the empirical literature. A network has the scale-free properties when its Degree distribution follows the power law. By analysing the network topology of the interbank market in several different countries, Boss and Elsinger (2004), [START_REF] Iori | A network analysis of the Italian overnight money market[END_REF], [START_REF] Lenzu | Systemic risk on different interbank network topologies[END_REF], Martinez-Jaramillo et al. ( 2014), [START_REF] Rönnqvist | Bank networks from text: interrelations, centrality and determinants[END_REF] and [START_REF] Xu | A dynamic network model for interbank market[END_REF] highlight that interbank Degree distribution follows the power law. In this study, I investigate whether my simulated network has the common feature of the real-world interbank network by testing whether its Degree distribution fits the power law. The probability distribution function of networks that obey power law is:

P(x) ∝ x -α
Where α is a scaling parameter with typical value that usually lies in the range 2<α<3 (sporadic exceptions are allowed).

The first step to test whether data distribution fits the power law is determination of the lower-bound value of the data Xmin, and to subsequently test the distribution for the value greater than Xmin. To estimate Xmin and consequently the scaling parameter α, I follow [START_REF] Clauset | On the frequency of severe terrorist events[END_REF]. Finally, to show whether the Degree distribution of my simulated network fits power law, I estimate goodness-of-fit between power law and my data. I reject the hypothesis that Degree follows the power law distribution if the p-value is less than or equal to 0.1.

The Degree log-log plots are illustrated in Figures 2 and3. Section A shows the plots for the whole sample (28 European countries), and section B illustrates the plot for six selected European countries. Table 3 presents the results for the power-law fits and the corresponding pvalues, and shows that the Degree distribution of my simulated interbank network obeys power law.

[Insert Table 3]

Control Variables

My study also includes a set of control variables known to affect the capital and liquidity of banks. In my capital equation, I control for bank size by introducing the logarithm of total assets (Size); loan loss provision to net interest income (LLP_NIR) as a proxy of the riskiness of bank assets; and the Z-Score, which is an indicator of a bank's distance to bankruptcy in my regressions.

<.=>?* = In the liquidity equation, I control for bank network characteristics by using the network dummy (HNetwork). I also consider the Net Interest Margin (NIM) as a proxy of bank profitability, the ratio of bank total assets to country total assets as a proxy for market power (MKT_POW) and the Central Bank policy rate (CB_Policy) as a proxy for monetary policy. Eventually, I include crises dummy.

The correlation matrix is presented in Table 4. The correlation coefficients between independent variables are low except between bank size, as measured by the natural logarithm of total assets (Size) and some network measures. To check whether such correlation affects my results, I perform a robustness test. I replace Bank Size with a size dummy variable which is not correlated with my network measures. All the specifications yield qualitatively similar results (see 4.3). Thus, all the results presented below are those obtained with Bank Size and network variables introduced simultaneously8 .

[Insert Table 4]

Methodology

In this paper, I question whether bank network topology could influence the interrelationship between bank liquidity and capital. Specifically, I use individual bank network indicators based on their loans to other banks and deposits from other banks and test how their interactions with liquidity affect banks' capital ratios. However, the causal relationship between capital and liquidity as documented in the literature poses potential serial correlation and endogeneity issues. To tackle these issue, I follow [START_REF] Distinguin | Bank regulatory capital and liquidity: Evidence from US and European publicly traded banks[END_REF] and [START_REF] Sclip | Large EU banks' capital and liquidity: Relationship and impact on credit default swap spreads[END_REF] in using the GMM simultaneous equations model. GMM is more efficient than 2SLS (two-stage least squares) regression because it considers the heteroskedasticity of errors. Additionally, it is robust to the error distribution [START_REF] Distinguin | Bank regulatory capital and liquidity: Evidence from US and European publicly traded banks[END_REF][START_REF] Hall | Advanced Texts in Econometrics: Generalized Method of Moments[END_REF].

!"# ",0 = @ A + @ & A. 9CD# ",0 + @ ( E9*+,(F) ",,,0 + @ ; E9*+,(F) ",,,0 * A. 9CD# ",0 + @ B ) ",,,05& + @ C " ,,0 + @ D "?4.*. 0 + H ",0

A. 9CD# ",0 = @ A + @ & !"# ",0 + @ ( E9*+,(F) ",,,0 + @ ; ) ",,,05& + @ B " ,,0 + @ C "?4.*. 0 + H ",0

In the capital equation, I regress capital ratio on the interaction between the inverse of Basel-III liquidity ratio (I.NSFR) and interbank network topology, and also a set of controls identified in the previous literature. Where !"# ",0 is a Total Capital Ratio, @ A is a constant, H ",0 is an error term.

A. 9CD# ",
In the liquidity equation, I regress the illiquidity ratio on the TCR and a set of controls identified in previous literature. White estimator results in standard errors that are robust within panel correlation and to crosssectional heteroscedasticity. Finally, I include time and cross-section fixed effects in the regressions.

Results

I first investigate the link between illiquidity and capital of European banks with different interbank network positions. I then look at how this interrelationship is influenced by network topology during both normal times and crisis times.

Impact of Network Topology on Causal Relationships between Bank Liquidity and Capital

To determine whether the relationship between liquidity and capital is affected by bank topology in the interbank network, I first investigate this interrelationship without considering the interbank network interactions (column 1 of Table 5). Higher illiquidity pushes banks to raise their capital ratios. Banks that face higher illiquidity are more inclined to strengthen their solvency standards and consequently improve their fundraising abilities. Likewise, concerning liquidity equation and consistent with the 'Risk Absorption' theory, a higher regulatory capital ratio is associated with higher illiquidity. [START_REF] Bhattacharya | Contemporary banking theory[END_REF] and von Thadden (2004) show that higher capital allows banks to absorb greater risk, which results in higher liquidity creation.

These results highlight that banks consider capital as a substitute for liquidity when facing higher illiquidity. Columns 2 through 5 of Table 5 present results of the effect of bank network topology on the liquidity-capital relationship.

Banks with a small number of immediate interbank borrowers (Out-Degree) or lenders (In-Degree) substitute capital for liquidity. Because they have less access to interbank funds, they strengthen their solvency to improve their fundraising abilities. There is no significant evidence for the impact of illiquidity on capital for banks with strong local network positions, which may indicate that these banks do not substitute capital for liquidity. The broader local access to the interbank market and stronger ability of interbank lending-borrowing diversification works as liquidity insurance and weakens (or eradicates) the substitutionary effect. Indeed, they might easily raise funds without compromising their solvency.

Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

Concerning system-wide network measures and similar to my findings for local network variables, banks that face higher illiquidity strengthen their solvency when they are characterised by weak PageRank. Because of weak linkages to highly connected counterparties, they might be less confident in their liquidity funding capabilities in the interbank network. Therefore, they strengthen their solvency to improve their external fundraising capabilities. However, illiquidity does not significantly influence the capital of banks with weak Betweenness. Surprisingly, banks with strong intermediation role in the interbank market (Betweenness) or banks that are strongly interconnected with centrally positioned peers (PageRank) set lower capital ratios when facing higher illiquidity. Increasing global accessibility to wholesale liquid funds might strengthen bank fundraising abilities, which could explain the negative sign of the relationship.

Concerning control variables, the most relevant factor that explains regulatory capital is Bank Size. A negative and significant effect of Bank Size on TCR demonstrates that larger banks set a lower capital ratio. In regards to liquidity determinants, NIM, CB_Policy and Crises_Dummy are the most relevant factors. Results show that more profitable banks set lower liquidity ratios.

Additionally, raising central bank policy rates or emergent financial crises correspond with banks' decision to set lower liquidity ratios.

In summary, I find that a strong system-wide position leads banks to set a lower capital ratio when they face higher illiquidity. By contrast, banks with weak local network positions and banks that are weakly interconnected with centrally positioned peers strengthen their solvency standards.

[Insert Table 5] Eurosystem. Under such circumstances, the role played by network interconnectedness is expected to change dramatically.

Impact of Network Topology on Causal Relationships between

Moreover, the financial crisis had a severe impact on bank fundraising ability. It raised bank funding costs and liquidity risks. [START_REF] Schanz | A joint calibration of bank capital and liquidity ratios[END_REF] argues that depositors are more risk-averse during crisis times. They force banks to strengthen their solvency and offer a higher deposit rate to induce them to roll their funds over. These depositor requirements (higher bank solvency and higher deposit rate) could possibly raise the substitutionary impact of liquidity on capital. In this section, I examine how the ease of a bank's accessibility to the interbank market might impact the substitutionary effects of liquidity on capital during both normal and crisis times.

The results presented in Table 6 show that banks with strong local interconnectedness do not substitute capital for liquidity during normal times or crisis times. By contrast, banks with weak local positions set higher capital ratios when they face illiquidity during both periods.

During normal times, banks with low system-wide access to the interbank market substitute capital for liquidity. The broader system-wide position, however, works as liquidity insurance and weakens these substitutionary effects. Crises weaken the substitutionary effects of liquidity on capital for banks with a weak system-wide position. Although a weak system-wide position indicates less access to the interbank liquid funds, it could also be interpreted as fewer counterparty and systemic risks during crisis times. However, banks with strong system-wide access to the interbank market start targeting lower capital ratios when they face higher illiquidity. Because of the too-interconnectedness-to-fail status, bailout expectations are higher for such interconnected intermediaries during crisis times. These expectations boost the confidence their interbank peers have in these highly interconnected banks, leading them to take advantage of their system-wide network positions during crises which enables them to benefit from lower fluctuations in interbank borrowing rates. Consequently, bank managers might consider interbank funds as stable, and substitute them for capital when they face higher illiquidity. Therefore, by targeting a lower capital ratio, they might achieve higher profits.

[Insert

Robustness Checks

To check the robustness of my results, I conduct several sensitivity analyses as described in the following sections.

Alternative Capital Ratio

To check the robustness of my results, I conduct regression by replacing TCR with the Tier-1 capital ratio defined as Tier-1 capital divided by risk-weighted assets (RWA). As presented in Table C1, the results are consistent with my baseline model.

Alternative Measure of Strong Network Interconnectedness

I also determine whether the interrelationship between liquidity and capital is influenced by extremely strong interconnectedness in the interbank network. For this purpose, I replace my network dummy with IE9*+,(F), which takes a value of one if the value of the network variable is greater than or equal to the value of the 90 th percentile. As shown in Table C2, the results remained unchanged.

Size Dummy Variable

With the exception of HPageRank, the logarithm of bank total assets (Bank Size) and network dummy variables are correlated (ranging from 47% to 56%). To ensure that this correlation do not affect my results, I perform a robustness test by replacing Bank Size with a dummy variable (Size_Dummy) that takes the value of one for small banks (banks with total assets less than one billion Euro) and zero otherwise. As shown in Table C3, the main results remain unchanged.

Additional Controls

To check the robustness of the simultaneous equations and correspondent to the literature, I also add controls for Bank Size and GDPperCap on the liquidity equation. The results remain unchanged9 .

Estimation of NSFR with Different Weights

To check the robustness of the NSFR ratio estimation, I apply minimum, maximum and extreme case weights (0.5, 0.85 or 1) for demand and savings deposits, as documented by the Basel accords. The conclusion remained unchanged 9 .

Conclusion

Existing literature has thus far neglected to address how the causal relationship between bank liquidity and capital is influenced by its network characteristics. In this paper, I augment traditional capital-liquidity relationship models with network statistics to assess whether and how this relationship depends on a bank's local and system-wide network characteristics in the interbank market. Using a GMM simultaneous equations approach applied to a dataset of listed and unlisted banks from 28 European countries, my study shows that bank capital ratio is not only explained by the macro environment and the individual bank characteristics outlined in the literature, but also by the relationship between liquidity and interbank network topology. My findings suggest that while banks with weak local or system-wide interbank positions improve their solvency when they face higher illiquidity, those with strong local interconnectedness do not substitute capital for liquidity. Likewise, strongly system-wide interconnected banks set a lower capital ratio when they face higher illiquidity.

In addition, my findings highlight that, in times of financial crisis, banks with a strong system-wide position in the interbank network weaken their solvency when they face higher illiquidity; this is presumably because of higher bailout expectations. As the higher system-wide interconnectedness leads banks to set lower liquidity ratios during crisis times [START_REF] Ardekani | Do banks change their liquidity ratios based on network characteristics[END_REF], these banks are more exposed to the insolvency and liquidity risks than others during turmoil. My findings support the need to implement minimum liquidity ratios in complement with capital ratios, as stressed by the Basel Committee on Banking Regulation and Supervision.

However, my findings also cast doubt on the current uniform liquidity and capital requirements for banks with different interbank topology. Adding liquidity ratios to capital ratios could be more relevant for institutions with strong system-wide positions than for weakly interconnected ones.

Presumably, strong system-wide interconnected banks might underestimate liquidity and insolvency risk because of their too-interconnected-to-fail status.

Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

In-Degree

Total numbers of interbank lenders to bank

HIn-Degree

Dummy variable that takes a value of one if bank's In-Degree is greater than or equal to the mean value of Country's In-Degree 0,369 0,483 0,000 0,000 1,000

Out-Degree

Total numbers of interbank borrowers from bank HOut-Degree

Dummy variable that takes a value of one if bank's Out-Degree is greater than or equal to the mean value of Country's Out-Degree 0,355 0,479 0,000 0,000 1,000

Betweenness

The ratio of links between bank j and bank k that passed through bank i compared to the total number of links between bank j and bank k. This table presents regression results using GMM simultaneous equations model for an unbalanced panel of European commercial banks over the 2001-2013 period by introducing the interaction the network dummy and I.NSFR. employ two steps GMM estimator with robust standard error:

HBetweenness

!"# !,# = & $ + & % (. *+,# !,# + & & -*./0(2) !,',# + & ( -*./0(2) !,',# * (. *+,# !,# + & ) 5 !,',#*% + & + " ',# + & , "678.8 # + 9 !,# (. *+,# !,# = & $ + & % !"# !,# + & & -*./0(2) !,',# + & ( 5 !,',#*% + & ) " ',# + & + "678.8 # + 9 !,#
TCR is capital ratio and I.NSFR is my illiquidity measurement, Network statistics dummies (HNetw) include HIn-degree, HOut-degree, HBetweenness and HPageRank. Bi,t-1 is a vector of bank-level control. Cj is a vector of country-level control variables. Crises is a dummy variable Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

for financial crises (2007)(2008)(2009)(2010)(2011)(2012). I include time and cross-section fixed effects in the regressions and I use the Huber-White estimator. This table presents regression results for crisis times and normal times using GMM simultaneous equations model for an unbalanced panel of European commercial banks over the 2001-2013 period by introducing the interaction the network dummy, crises dummy and I.NSFR. I employ two steps GMM estimator with robust standard error:

!"# !,# = & $ + & % (. *+,# !,# + & & -*./0(2) !,',# + & ( -*./0(2) !,',# * (. *+,# !,# + & ) "678.8(2) !,',# * (. *+,# !,# + & + -*./0(2) !,',# * (. *+,# !,# * "678.8(2) !,',# + & , 5 !,',#*% + & -" ',# + & . "678.8 # + 9 !,# (. *+,# !,# = & $ + & % !"# !,# + & & -*./0(2) !,',# + & ( 5 !,',#*% + & ) " ',# + & + "678.8 # + 9 !,#
TCR is capital ratio and I.NSFR is my illiquidity measurement, Network statistics dummies (HNetw) include HIn-degree, HOut-degree, HBetweenness and HPageRank. Bi,t-1 is a vector of bank-level control. Cj is a vector of country-level control variables. Crises is a dummy variable for financial crises (2007)(2008)(2009)(2010)(2011)(2012) I define a set of probabilities Q that captures disassortative characteristics of an interbank network:

M #$ ∝ max P )3_ ) # ):_ ) $ , ):_ ) $ )3_ ) # Q
Consistent with the probability Q, there is an increase in the lending probability of i to j if i is a small lender to a large borrower or i is a large lender to a small borrower j. The selection process continues by highest value loading a matrix of exposure following the like-selection priority determination:

IEij = Min {IL_Di, ID_Dj}
Eventually, the interbank network is produced by this maximisation function:

RBS 1 T()*)U()*) + V ǁ M)
!"

P(IE):

The probability distribution for all plausible network formations.

R:

The relative entropy function.

q: A scaling parameter (reflecting the finding solutions' weight with similar features to the prior matrix Q. TCR is capital ratio and I.NSFR is illiquidity measurement, Network statistics dummies (HNetw) include HIn-degree, HOut-degree, HBetweenness and HPageRank. Bi,t-1 is a vector of bank-level control. Cj is a vector of country-level control variables. Crises is a dummy variable for financial crises (2007)(2008)(2009)(2010)(2011)(2012)

  Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée) (3) et al. (2015) to simulate bilateral exposure network; I then compute my network centrality measurements 6 .

Figure 1

 1 Figure 1 illustrates the configuration of the simulated interbank network in France, a selected European country among my sample in 2013. Each node represents a bank in the interbank network, and the linkages represent the interbank lending-borrowing relationships. Highlighted banks are those with the most significant network interconnectedness among others. In order to capture the interconnectedness of banks in the interbank network, I categorise my network variables into two subgroups: local and system-wide network variables.

  Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

  Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)Where ROAmma3 is the three-year rolling window average return on assets defined as the ratio of net income to average total assets, (Equity/TA)mma3 represents the three-year rolling window average of equity to total assets and ROAsdma3 represents the three-year rolling window standard deviation of the Return on Assets. I also include the Return on Equity (ROE). My country-level control variable comprises the Natural logarithm of GDP per capita (GDPperCa). GDPperCa is a country's gross domestic product per capita. Given that European banks have recently weathered both the global financial crisis of 2007-2009 and the European sovereign debt crisis of 2010-2012, I construct a dummy variable to capture the effects of both crises. This dummy variable takes the value of one in the aforementioned crisis years, and zero otherwise.

  ) ",05& is a vector of bank-level control variables including Net Interest Margin (NIM) and Market Power (MKT_POW). " ,,0 is a vector of country-level control variable that comprises the Central bank Policy rate (CB_Policy). Standard errors are clustered at the bank level. To deal with possible endogeneity, and consistent with[START_REF] Distinguin | Bank regulatory capital and liquidity: Evidence from US and European publicly traded banks[END_REF] and[START_REF] Sclip | Large EU banks' capital and liquidity: Relationship and impact on credit default swap spreads[END_REF], I replace all bank-level controls with their one-year lagged value in both equations. Subsequent to testing for cross-section and time random versus fixed effects, I apply random effect estimations using the Huber-White estimator in both equations. Using the Huber-Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

  impact of interbank network topology on the interrelationship between bank liquidity and capital ratio within crisis periods by looking at both the global financial crisis of 2007-2009 and the European sovereign debt crisis of 2010-2012. Both crises significantly influenced the interconnectedness of European banks, as banks were reluctant to deal with each other on unsecured interbank markets during these periods and preferred to interact through the Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

Figure- 1 :Figure- 2 :

 12 Figure-1: Interbank network configurations of selected country (France-2013):

Figure- 3 :

 3 Figure-3: Degree log-log plot for six selected European countries. It consists of three large size banking industry (France, Germany and Italy)and three small size banking industry (Czech Republic, Poland and Slovenia). This figure shows the cumulative distribution functions P(x) and their maximum likelihood power-law fits for Degree distribution of four selected European countries.

  . I include time and cross-section fixed effects in the regressions and I use the Huber-White estimator. Hansen Test is used. VIF test reports multicollinearity checks among all variables. All dependent and bank-level control variables are winsorized at 1% -99%. Standard errors are shown in parentheses. *, **, *** indicate significance at the 10%, 5%, and 1% level, respectively. Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

  2001-2013 period by introducing the interaction the network dummy and I.NSFR. I employ two steps GMM estimator with robust standard error: !"# !,# = & $ + & % (. *+,# !,# + & & -*./0(2) !,',# + & ( -*./0(2) !,',# * (. *+,# !,# + & ) 5 !,',#*% + & + " ',# + & , "678.8 # + 9 !,# (. *+,# !,# = & $ + & % !"# !,# + & & -*./0(2) !,',# + & ( 5 !,',#*% + & ) " ',# + & + "678.8 # + 9 !,#

  . I include time and cross-section fixed effects in the regressions and I use the Huber-White estimator. Hansen Test is used. VIF test reports multicollinearity checks among all variables. All dependent and bank-level control variables are winsorized at 1% -99%. Standard errors are shown in parentheses. *, **, *** indicate significance at the 10%, 5%, and 1% level, respectively. Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

Table 6 ]

 6 Documents

de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

Table 1 :

 1 Distribution of banks and of the final sample

	Country Name	No. Obs
	AUSTRIA	136
	BELGIUM	75
	BULGARIA	89
	CROATIA	65
	CYPRUS	50
	CZECH REPUBLIC	58
	DENMARK	400
	ESTONIA	36
	FINLAND	48
	FRANCE	164
	GERMANY	183
	GREECE	75
	HUNGARY	60
	IRELAND	62
	ITALY	608
	LATVIA	38
	LITHUANIA	41
	LUXEMBOURG	86
	MALTA	15
	NETHERLANDS	131
	POLAND	88
	PORTUGAL	70
	ROMANIA	81
	SLOVAKIA	22
	SLOVENIA	101
	SPAIN	107
	SWEDEN	101
	UNITED KINGDOM	236
	Total	3226

Italy Slovenia

Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

Table 2 :

 2 Descriptive Statistics and definitions of my variables

	Variables	Definition	Mean	Sd	Min	Median Max
		Capital measure
	TCR	Total Capital Ratio	15,022	6,065	9,000	13,400	40,230
		Liquidity measure
	I.NSFR	Inverse of Net stable funding ratio	2,619	4,561	0,155	1,198	43,728
		Network Variables

  Dummy variable that takes a value of one if bank's Betweenness is greater than or equal to the mean value of Country's BetweennessPageRankRatio that indicates to what extent the importance of counterparties could determine the importance of each bank

			0,428	0,495	0,000	0,000	1,000
	HPageRank	Dummy variable that takes a value of one if bank's PageRank is	0,704	0,456	0,000	1,000	1,000
		greater than or equal to the mean value of Country's PageRank
		Controls
	Bank Size	Logarithm of total assets	15,393	1,974	10,639	15,423	18,201
	Z-Score	Indicator of bank distance to bankruptcy	60,235	70,053	3,284	34,356	311,580
	ROE	Return on Equity	7,392	10,547	-19,134	7,925	28,440
	LLP_NIR	Loans loss provisions to net interest revenue	23,691	27,148	-13,684	15,632	100,562
	MKT_POW	Market power measured by bank total assets divided by country	0,070	0,123	0,000	0,010	0,874
		total assets
	NIM	Net Interest Margin	2,620	1,528	0,132	2,388	7,026
	GDPperCa	Natural logarithm of GDP per capita	27,136	1,598	22,235	27,969	30,790
	Policy_Rate	ECB policy rates	1,911	1,316	0,000	2,000	7,750
	Crises	Dummy variable for crisis times. Takes a value of one for the	0,542	0,498	0,000	1,000	1,000
		persiods of 2007-2009 (Global Financial Crisis) & 2010-
		2012(European Sovereign Debt Crisis)

Table 3 :

 3 Power-law fits This table provides Xmin as a lower-bound for Degree, the parameters for power-law fits, the goodness-of-Fits and its corresponding p-value and α as the scaling parameter. Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

		Xmin	α	Goodness of Fit	p-value	Log-Likelihood
	France	18	3,5	0,1246	0,619	-25,8138
	Germany	3	2,18	0,0694	0,421	-128,743
	Italy	3	2,27	0,08	0,469	-62,3853
	Czech	2	2,49	0,089	0,505	-30,0925
	Slovenia	2	2,83	0,0871	0,673	-14,6528
	Poland	2	2,93	0,0383	0,896	-33,3666
	All Sample	4	2,23	0,0311	0,469	-565,1287

Table 4 :

 4 Correlation matrix of explanatory and control variables

		TCR	I.NSFR	HInDegree HOutDegree	HBetweenness HPageRank	Size	Size_Dummy Z-Score	LLP_NIR ROE	GDPperCap	crises	NIM	MKT_POW	CB_Policy
	TCR	1.0000
	I.NSFR	0.0861	1.0000
	HInDegree	-0.1812	-0.0136	1.0000
	HOutDegree	-0.1351	-0.0686	0.4873	1.0000
	HBetweenness	-0.1584	-0.0821	0.6449	0.6034	1.0000
	HPageRank	-0.1212	-0.1218	0.3671	0.2338	0.4617	1.0000
	Bank Size	-0.3136	-0.0839	0.5629	0.5606	0.4710	0.1822	1.0000
	Size_Dummy	0.2848	0.0688	-0.3788	-0.3581	-0.3133	-0.0739	-0.7428	1.0000
	Z-Score	-0.0261	0.0866	-0.0432	-0.0506	-0.0847	-0.1065	0.0111	-0.0500	1.0000
	LLP_NIR	-0.0205	0.0571	0.0293	-0.0106	0.0281	0.0153	0.0234	0.0010	-0.2021	1.0000
	ROE	-0.0886	-0.0766	0.0805	0.1099	0.1054	0.0994	0.0739	-0.0830	0.0717	-0.5108	1.0000
	GDPperCap	-0.0459	0.1230	-0.0482	-0.0754	-0.1500	-0.3176	0.1192	-0.0772	0.1560	-0.0865	-0.0375	1.0000
	crises	0.0327	0.0463	-0.0121	-0.0202	-0.0104	-0.0549	0.0573	-0.0502	-0.1156	0.1300	-0.1211	0.0159	1.0000
	NIM	0.0391	-0.1308	-0.1869	-0.1808	-0.1205	0.0348	-0.4527	0.3651	-0.0890	0.0278	0.1593	-0.0969	-0.0701	1.0000
	MKT_POW	-0.1278	-0.0874	0.4612	0.4928	0.4716	0.3004	0.4738	-0.2737	-0.0836	0.0039	0.1566	-0.2595	-0.0557	-0.0055	1.0000
	CB_Policy	-0.1325	-0.1136	0.0247	0.0368	0.0289	0.0644	-0.0307	0.0225	0.0903	-0.3178	0.3403	-0.0352	-0.2407	0.0875	0.0570	1.0000

Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

Table 5 :

 5 Baseline GMM model of interaction between network topology and liquidity on bank capital ratio

		1	2	3	4	5
	VARIABLES		HIn-Degree	HOut-Degree	HBetweenness	HPageRank
	Capital Equation
	I.NSFR (1)	2.552***	2.838***	2.600***	0.606	2.513**
		(0.963)	(0.621)	(0.553)	(0.771)	(1.088)
	HNetw		6.363***	7.305***	6.146***	4.778***
			(1.703)	(1.702)	(1.821)	(1.802)
	HNetw*I.NSFR (2)		-2.883***	-2.599***	-0.847	-2.627**
			(0.628)	(0.553)	(0.771)	(1.102)
	Bank Size	-1.100***	-0.591**	-0.915***	-1.190***	-0.578***
		(0.369)	(0.261)	(0.278)	(0.198)	(0.211)
	Z-score	-0.000376	-0.00577*	-0.00269	-0.000721	-0.00421
		(0.00240)	(0.00305)	(0.00254)	(0.00281)	(0.00317)
	LLP_NIR	-0.0244	-0.0288**	-0.0142	-0.0137	-0.0145
		(0.0215)	(0.0132)	(0.0106)	(0.0113)	(0.0130)
	ROAE	-0.0728	-0.0614*	-0.0380	-0.0590**	-0.0245
		(0.0603)	(0.0328)	(0.0271)	(0.0278)	(0.0267)
	GDPperCa	-2.139	-1.111**	-0.861*	0.556	-1.497
		(1.379)	(0.529)	(0.516)	(0.718)	(1.220)
	Crises	0.170	0.285	0.455	0.854*	-0.380
		(0.645)	(0.404)	(0.405)	(0.449)	(0.754)
	Constant	84.51**	48.76***	45.86***	15.10	59.69*
		(40.39)	(16.14)	(15.95)	(19.50)	(32.74)
	Liquidity Equation
	TCR	0.528*	0.284**	0.206**	0.223	0.313***
		(0.282)	(0.113)	(0.0981)	(0.287)	(0.108)
	HNetw		0.515*	-0.467*	-0.933	-0.0517
			(0.305)	(0.240)	(2.916)	(0.300)
	NIM	-0.400**	-0.196***	-0.245***	-0.336***	-0.327***
		(0.164)	(0.0724)	(0.0719)	(0.102)	(0.105)
	MKT_POW	0.692	-0.330	1.084	1.816	0.496
		(1.548)	(1.186)	(1.124)	(4.717)	(1.276)
	CB_Policyrate	-0.0170	-0.167**	-0.197***	-0.144	-0.0973
		(0.190)	(0.0709)	(0.0659)	(0.170)	(0.0941)
	Crises	-0.406**	-0.353**	-0.334**	-0.333	-0.365**
		(0.207)	(0.165)	(0.161)	(0.205)	(0.174)
	Constant	-4.038	-0.912	0.707	0.823	-0.913
		(4.193)	(1.720)	(1.479)	(5.438)	(1.545)
	Wald-test
	1+2		-.044	.001	-.240***	-.113**
	Observations	3,226	3,226	3,226	3,226	3,226
	No. Banks	506	506	506	506	506
	VIF Test	1.42	1.5	1.51	1.47	1.48
	Hansen's J	0.653703	4.28696	4.19729	2.91254	2.57347
	Hansen's J p-value	0.7212	0.3686	0.3800	0.2331	0.4622

  Hansen Test is used. VIF test reports multicollinearity checks among all variables. All dependent and bank-level control variables are winsorized at 1% -99%. Standard errors are shown in parentheses. *, **, *** indicate significance at the 10%, 5%, and 1% level, respectively.

Table 6 :

 6 GMM model of interaction between network topology and liquidity on bank capital ratio during normal times and crisis times

		1	2	3	4	5
	VARIABLES		HIn-Degree	HOut-Degree	HBetweenness HPageRank
	Capital Equation
	I.NSFR (1)	4.253*** 6.759***	5.397***	6.755***	4.853***
		(1.630)	(2.563)	(1.639)	(1.974)	(1.285)
	HNetw		9.248	9.043**	7.592***	4.976***
			(6.713)	(4.374)	(2.498)	(1.105)
	HNetw*I.NSFR (2)		-7.364	-5.729*	-6.211***	-4.601***
			(5.049)	(3.196)	(1.831)	(1.208)
	Crises*I.NSFR (3)	-4.209**	-6.296***	-5.076***	-6.411***
		(1.638)	(2.349)		(1.897)
	HNetw*I.NSFR *Crises (4)		6.369	4.879*	5.186***	3.470***
			(4.408)	(2.785)	(1.510)	(0.984)
	Liquidity Equation
	TCR	0.533*	0.188	0.229	0.164	0.357***
		(0.284)	(0.309)	(0.263)	(0.254)	(0.0972)
	Wald-test
	1+2	.044	-.605	-.331	.544***	.252**
	1+3		.462*		.344***	.234***
	1+2+3+4		-.531	-.527	-.680*	-.895***
	Observations	3,226	3,226	3,226	3,226	3,226
	No. Banks	506	506	506	506	506
	VIF Test	1.71	2.21	2.05	2.05	2.08
	Hansen's J	1.17504	2.66928	1.97063	2.94736	3.09101
	Hansen's J p-value	0.5557	2.66928	0.1604	0.3998	0.5427
	Bank-Level Controls	Yes	Yes	Yes	Yes	Yes
	Country-level Controls	Yes	Yes	Yes	Yes	Yes

Table C2 :

 C2 Baseline GMM model of interaction between network topology and liquidity on TCR capital ratio: Network dummy is defined based on 90-percentile value.This table presents regression results using GMM simultaneous equations model for an unbalanced panel of European commercial banks over the

		1	2	3	4
	VARIABLES	HIn-Degree	HOut-Degree	HBetweenness HPageRank
	Capital Equation
	I.NSFR (1)	2.408***	2.240***	0.122	4.092
		(0.510)	(0.457)	(0.795)	(2.996)
	HNetw	6.294***	6.808***	3.955**	6.086
		(1.535)	(1.570)	(1.764)	(3.941)
	HNetw*I.NSFR (2)	-2.450***	-2.296***	-0.426	-4.314
		(0.522)	(0.466)	(0.804)	(3.098)
	Liquidity Equation
	TCR	0.242**	0.201**	0.101	0.330***
					(0.113)
	Wald-test
	1+2	-.0428	-.0562	-.304***	-.221*
	Observations	3,226	3,226	3,226	3,226
	No. Banks	506	506	506	506
	Hansen's J	5.24486	5.62496	2.22564	2.62603
	Hansen's J p-value	0.2631	0.2290	0.3286	0.4529
	Bank-level Controls	Yes	Yes	Yes	Yes
	Country-level Controls	Yes	Yes	Yes	Yes

Table C3 :

 C3 Baseline GMM model of interaction between network topology and liquidity on TCR capital ratio -replace logarithm of total assets by size dummyThis table presents regression results using GMM simultaneous equations model for an unbalanced panel of European commercial banks over the 2001-2013 period by introducing the interaction the network dummy and I.NSFR. I employ two steps GMM estimator with robust standard error:!"# !,# = & $ + & % (. *+,# !,# + & & -*./0(2) !,',# + & ( -*./0(2) !,',# * (. *+,# !,# + & ) 5 !,',#*% + & + " ',# + & , "678.8 # + 9 !,# (. *+,# !,# = & $ + & % !"# !,# + & & -*./0(2) !,',# + & ( 5 !,',#*% + & ) " ',# + & + "678.8 # + 9 !,#TCR is capital ratio and I.NSFR is my illiquidity measurement, Network statistics dummies (HNetw) include HIn-degree, HOut-degree, HBetweenness and HPageRank. Bi,t-1 is a vector of bank-level control. Cj is a vector of country-level control variables. Crises is a dummy variable for financial crises(2007)(2008)(2009)(2010)(2011)(2012). I include time and cross-section fixed effects in the regressions and I use the Huber-White estimator. Hansen Test is used. VIF test reports multicollinearity checks among all variables. All dependent and bank-level control variables are winsorized at 1% -99%. Standard errors are shown in parentheses. *, **, *** indicate significance at the 10%, 5%, and 1% level, respectively.

		1	2	3	4
	VARIABLES	HIn-Degree	HOut-	HBetweenness HPageRank
			Degree
	Capital Equation
	I.NSFR (1)	2.600***	2.339***	-0.103	1.989**
		(0.565)	(0.500)	(0.980)	(0.965)
	HNetw	5.532***	6.059***	4.532**	3.890**
		(1.461)	(1.421)	(2.260)	(1.618)
	HNetw*I.NSFR (2)	-2.634***	-2.347***	-0.155	-2.097**
		(0.571)	(0.502)	(0.984)	(0.976)
	Liquidity Equation
	TCR	0.276**	0.201**	0.223	0.313***
		(0.112)	(0.0980)	(0.287)
		-.0339	-.0079	-.258***	-.107**
	Observations	3,226	3,226	3,226	3,226
	No. Banks	506	506	506	506
	Hansen's J	4.67287	4.5542	2.91254	2.57347
	Hansen's J p-value	0.3225	0.3362	0.2331	0.4622
	Bank-level Controls	Yes	Yes	Yes	Yes
	Country-level Controls	Yes	Yes	Yes	Yes

Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

For robustness, I replace Tier 1 ratio with TCR (see

4.1).4 Following the literature, I consider the NSFR and not the LCR because, due to lack of data, the latter cannot be

A detailed description of the minimum-density algorithm applied for construction of the interbank network is provided in Appendix A. Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

The centrality measurements are calculated based on the methods of[START_REF] Bastian | Gephi: an open source software for exploring and manipulating networks[END_REF]. (6) Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

I also perform multicollinearity checks among all variables by running a VIF test. The results of the VIF test in Tables 4 and 5 indicate low multicollinearity. Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

The results are available upon request. Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

This algorithm has been constructed and run with a Matlab programme. The heuristic process that executes this method is fully described in[START_REF] Anand | Filling in the blanks: network structure and interbank contagion[END_REF]. (15) (16) (17) Documents de travail du Centre d'Economie de la Sorbonne -2020.22R (Version révisée)

Appendix A

Minimum Density

Introduced by [START_REF] Anand | Filling in the blanks: network structure and interbank contagion[END_REF], minimum density (MD) is an efficient and streamlined method and is consistent with total observed interbank lending and borrowing for each bank. The main assumption of MD is that producing and maintaining the extra linkages is costly for banks. The first step to conducting an MD approach is to solve the constrained optimisation problem:

c: Fixed cost for establishing the extra linkages.

Integer function 1: Equals 1 if and only if bank i lends to bank j. The bank's lending-borrowing capacity in this approach is constrained by the total amount of interbank loans (IL) and deposits (ID), which are considered marginals.

In the second step, the link-generating algorithm imposes a penalty upon each bank for deviations from the marginal borrowing and lending capacity:

Current deviation of bank i's from marginal lending and borrowing are measured by )3_: # BCD ):_: # respectively. Therefore, the model maximises the value of sparse matrix IE, which minimises marginal deviations when the above criterion is added to the objective function: This table presents a stylized bank balance sheet, together with the weights assigned to different assets and liabilities for the computation of the net stable funding ratio defined in [START_REF] Vazquez | Bank funding structures and risk: Evidence from the global financial crisis[END_REF]. This table presents regression results using GMM simultaneous equations model for an unbalanced panel of European commercial banks over the 2001-2013 period by introducing the interaction the network dummy and I.NSFR. I employ two steps GMM estimator with robust standard error:

Appendix C


domain: spi chim

text:
F Bourahima
email: fazati.bourahima@chpolansky.fr

T Baudin

M Rege

V Ji

F Brisset

A L Helbert

Energy attenuated by Ni powder distribution during laser cladding on Cu-Ni-Al substrate

Keywords: Laser cladding, Power attenuation, Powder distribution, Dilution zone

Laser metal deposition (LMD) is an alternative method to other cladding techniques such as Plasma Transfer Arc (PTA) or blowtorch for surface treatment in the glass industry. It aims to produce dense, high-quality coatings on a non-planar surface without affecting its thermal and mechanical properties. In this study, Ni based coatings were coated onto Cu-Ni-Al substrate using a 3 JET nozzle technique. During laser cladding, good metallurgical bonding is necessary to ensure the further surfacing process technique. A microstructural analysis was conducted, and the mechanical properties were then evaluated with microhardness analysis to link process parameters to coating bonding quality. A calculation of the power attenuation attempts to explain the impact of the powder distribution on the bonding. This work revealed that a chemical dilution zone exists between coating and substrate and is necessary for perfect metallurgical bonding. The heterogeneous bonding, observed through the section, along the curved interface coating/substrate, has been linked to the Gaussian distribution of the powder that attenuates the input power. The attenuated powder was measured all along the interface.

During bottle production, viscous glass is poured into Cu-Ni-Al molds at temperatures ranging from 700 to 1200°C. A glass mold must absorb the high glass temperature in order to cool it homogeneously. However, during this process, corrosion or even abrasion can appear on sensitive parts of the molds. Therefore, it is important to modify the mold surface properties in order to extend its lifespan.

Laser cladding is an innovative surfacing technique [START_REF] Steen | Laser Material Processing[END_REF] used to obtain well-bonded highquality materials, free of pores and cracks without affecting the substrate thermal properties. Many studies have investigated the Heat Affected Zone (HAZ), which is a zone belonging to the substrate that has not reached its fusion temperature *but observed a change in its structure due to the process.

In the mold industry, the aim is to limit this zone in order to prevent changes in the base material thermal properties. Currently, the main techniques used are Plasma Transferred Arc (PTA) and blowtorch. But these techniques create a very high HAZ. Some researchers have investigated the impact of laser cladding on the substrate microstructural change. In most of the studies, the HAZ is observed during laser cladding on cast iron substrates with the appearance of metastable phases like martensite or ledeburite [START_REF] Bourahima | Optimization of Microstructural Evolution during Laser Cladding of Ni Based Powder on GCI Glass Molds[END_REF][START_REF] Liu | Microstructural evolution and bonding characteristic in multi-layer laser cladding of NiCoCr alloy on compacted graphite cast iron[END_REF][START_REF] Jeshvaghani | Microstructural study and wear behavior of ductile iron surface alloyed by Inconel 617[END_REF][START_REF] Lin | Empirical alloys-by-design theory calculations to the microstructure evolution mechanical properties of Mo-doped laser cladding NiAl composite coatings on medium carbon steel substrates[END_REF][START_REF] Liu | Research on the transformation mechanism of graphite phase and microstructure in the heated region of gray cast iron by laser cladding[END_REF]. Arabi Jeshvaghani et al. [START_REF] Liu | Microstructural evolution and bonding characteristic in multi-layer laser cladding of NiCoCr alloy on compacted graphite cast iron[END_REF] have added that sometimes a complete graphite dissolution can be observed in the case of cladding on cast iron with spheroidal graphite. In some cases, this HAZ can lead to cracking behavior [START_REF] Liu | Microstructural evolution and bonding characteristic in multi-layer laser cladding of NiCoCr alloy on compacted graphite cast iron[END_REF].

Balu et al. [START_REF] Balu | Laser cladding of nickel-based alloy coatings on copper substrates[END_REF] have operated laser cladding with two kinds of Ni based powder on Cu substrate with and without pre-heating. They observed that the grain size in the HAZ is greater than it is further away from the interface. It has also been seen that this size increases at low laser scanning speed.

Hardness is lower in the HAZ than it is further away from the interface. They found that a pre-heating of the substrate leads to a better bonding due to a larger melting pool and HAZ.

Despite the limited HAZ, laser cladding often induces an area of local chemical dilution between the powder and the substrate. Authors have explained that in order to obtain dilution, a fusion between the substrate and the coating has to be present [7][8][9]. This dilution, dependent on the input laser [START_REF] Kim | Characterization of surface modification by laser cladding using low melting point metal[END_REF], has to be restricted in order to avoid a large HAZ but seems necessary to ensure good metallurgical bonding. There are two ways of describing the dilution (rate) [START_REF] Toyserkani | Laser Cladding[END_REF]: in geometrical terms, which consists in calculating the ratio between the melted area (including the HAZ) and the coating area. The second is in metallurgical terms, which consists in measuring the depth of the chemical mixture between the coating and the substrate. Liu et al. [START_REF] Liu | Research and development status of laser cladding on magnesium alloys: A review[END_REF] present a dilution rate by using the geometrical method during laser cladding of magnesium alloys. According to them, a too low dilution rate prevents good metallurgical bonding. Using the same definition of the dilution rate, Luo et al. [START_REF] Luo | Effect of NiCrBSi content on microstructural evolution, cracking susceptibility and wear behaviors of laser cladding WC/Ni-NiCrBSi composite coatings[END_REF] present the same observation during laser cladding of NiCrBSi. Balu et al. [START_REF] Balu | Laser cladding of nickel-based alloy coatings on copper substrates[END_REF] used the metallurgical method to measure the dilution rate during laser cladding of Ni based powder on a Cu substrate. They found that a high dilution can lead to a high HAZ. Pereira et al. [START_REF] Pereira | Mechanical and microstructural characterization of MCrAlY coatings produced by laser cladding: The influence of the Ni, Co and Al content[END_REF] observed during laser cladding of NiCoCrBSi on stainless steel, that a very low dilution zone (metallurgical method) at about 2 μm is enough to obtain a perfect bonding without affecting the mechanical properties of the substrate (no cracking behavior observed). O.S. Adesina et al. [START_REF] Adesina | Influence of phase composition and microstructure on corrosion behavior of laser based Ti-Co-Ni ternary coatings on Ti-6Al-4V alloy[END_REF] presented an EDS analysis to measure the dilution between two materials and chemical evolution of the materials' components [START_REF] Wang | Real-time monitoring of chemical composition in nickel-based laser cladding layer by emission spectroscopy analysis[END_REF].

The main challenge in laser cladding on a copper-based substrate is to obtain good bonding despite the low absorptivity of the substrate. Zhang et al. [START_REF] Zhang | Characterization on laser clad nickel based alloy coating on pure copper[END_REF] explained that it was impossible to obtain bonding without preheating, so they pre-heated the substrate to 300°C.

In the mold industry, the HAZ must be limited in order to avoid changing the molds' thermal properties and thus obtain a homogeneous glass surface. It appears that a compromise must be found to respect both the restricted HAZ and the dilution zone needed to guarantee the cohesion between the coating and the substrate. We should bear in mind that this cohesion is crucial to allow further machining of the repaired mold. Moreover, the clad has to be free of cracks and pores to retain a perfect glass surface.

The aim of this study is to obtain perfect bonding with a limited or non-existent HAZ. This can be achieved by identifying the relationship between the dilution zone (existence and thickness) and the bonding quality between coating and substrate with regards to the laser process parameters.

The laser cladding of Ni powder on a Cu based substrate will be performed without preheating and on a curved surface. A microstructural and chemical analysis of the coating/substrate interface will be conducted. Also, microhardness measurements will be presented. The influence of the Gaussian distribution of Ni powder on the laser power attenuation, and thus on the bonding, will be discussed.

Materials and experimental techniques

Laser cladding consists in melting an injected powder with a very thin surface layer of the substrate by a laser beam to produce a metallurgical bonding. During this study, a 4KW Nd: YAG laser with a wavelength of 1030 nm was used. An optical fiber with a diameter of 600 µm was used for guiding the beam. Ni powder was injected coaxially into the laser beam. A cross section of the Cu-Ni-Al mold of the bottle ring is shown in Figure 1a after cladding. Figure 1b describes the typical cross section that has been metallurgically studied.

During the process, the interaction distance, fs, between the laser and the powder is considered (Figure 2): At the focal plane (position B), the powder flow follows a Gaussian distribution. This has been proved by many research studies [START_REF] Ferreira | Experimental and numerical analysis of gas/powder flow for different LMD nozzles[END_REF][START_REF] Cheikh | Analysis and prediction of single laser tracks geometrical characteristics in coaxial laser cladding process[END_REF][START_REF] Pouzet | Additive layer manufacturing of titanium matrix composites using the direct metal deposition laser process[END_REF]. In the present study, the Gaussian distribution has been centered at the midpoint of the mold curved surfaces of the samples.

The chemical compositions of the powder and the substrate are listed in Table 1. The process parameters used during this study are presented in Table 2. These samples are extracted using Taguchi design of experiment. Only samples with a partial or perfect bonding have been selected.

X=0

Laser input

Powder To be characterized, samples were mechanically polished using SiC papers and then diamond polished to 1 µm for Optical Microscopy (OM), Scanning Electron Microscopy (SEM) and Energy Dispersive Spectroscopy (EDS) analyses. To operate Electron Backscatter Diffraction (EBSD), an extra OPS polishing was done. Vickers hardness tests were performed on the cross section, along a line perpendicular to the coating surface with a 10-gf load for 15 seconds. The measurements were taken from either side of the interface for all the 6 samples.

Results and discussion

Microstructure features and associated chemical and mechanical properties

Figure 3 describes the three different kinds of bonding behavior that can be observed: perfect bonding (Figure 3a), partial bonding (Figure 3b) where the bonding is partially present at the interface and no bonding (Figure 3c). samples (see Table 2). Figure 4a shows that CNC 5 presents perfect bonding (case of Figure 3a)

whereas Figure 4b shows a sample (CNC 6) with a discontinuous bonding (Figure 3b). Indeed, in the center of the curve, there is a lack of bonding. No analyses have been performed on samples with no bonding (Figure 3c) since the clad is easily separated from the substrate.

For both samples, the substrate is composed of a dendritic matrix. No heat affected zone Figure 5 shows an EBSD analysis of the microstructure and texture around the interface (black squares in 4a and b) for both samples (perfect or partial bonding) in order to observe the differences due to the bonding quality. It can be observed that microstructure is very similar in both cases (Figure 5a and5b).

The solidification features are in accordance with the literature [START_REF] Song | The relationship between convection mechanism and solidification structure of the iron-based molten pool in metal laser direct deposition[END_REF][START_REF] Peyre | Traitements de surface par laser et tenue à la corrosion aqueuse[END_REF][START_REF] Philibert | La solidification des métaux et des alliages[END_REF][START_REF] Debroy | Additive manufacturing of metallic components -Process, structure and properties[END_REF]. A nucleation phase, characterized by small equiaxed grains, is evidenced at the interface, then a columnar growth directed mainly parallel to the <001> direction of the dendrites is observed (Figure 5a).

A SEM-EDS analysis is presented (Figure 6) to observe the chemical impact of the laser cladding at the interface nucleation/substrate. A measurement of the dilution zone thickness was made by chemical analysis from the coating to the substrate (Figure 7). A diminution of the Ni element from the coating to the substrate can be observed. At the same time, the Cu composition progressively increases. Overall, Ni is at around 96% in the coating, then decreases down to 40% close to the interface and stays below 10% in the substrate matrix except inside the dendrites where the Ni increases to 39%. The dendrites also present a higher level of Al (the high concentration of Ni and Al inside the dendrites is in accordance with Figure 6). The dilution zone exhibits a chemical mixture over about 35µm for this sample.

Figure 8 shows the hardness measurements through the interface for four typical features of the microstructure: the columnar zone of the coating, the nucleation zone, the dilution zone and the substrate.

Coating Substrate It can be observed from Figure 8 that the micro-hardness of the dilution zone increased to 618

Dilution zone

HV compared to the coating and the substrate which are not higher than 317 HV. The high hardness in the dilution zone could be explained by the formation of an eutectic of Ni-Ni3B due to the low ratio of Si/B (≈1.3) in this area as shown by Hemmatie et al. [START_REF] Hemmati | Effects of the alloy composition on phase constitution and properties of laser deposited Ni-Cr-B-Si coatings[END_REF]. Indeed, they mention that this compound is of high hardness. No increase of hardness is noticed in the nucleation area despite the presence of small grains.

Since the dilution zone can be quantified using the EDS analysis it could be interesting to observe its evolution along the curved interface in order to corroborate its existence/thickness with the metallurgical bonding quality.

Dilution zone thickness along the curved interface

In order to state if the discontinuous bonding along the interface in the studied section is linked to the chemical dilution presence or thickness, a protocol of systematic DZ quantification has been implemented across the section of all samples and is presented in Figure 9. Five lines separated from each other by an angle of 22.5° along which the chemical composition have been measured from the Coating Substrate coating to the substrate within 400 μm, and where the DZ depth has been determined. Thanks to this method, it is possible to plot the DZ evolution along the curved interface (Figure 9). is minimum (zero in the case of no bonding at all). Moreover, for all the samples studied, the bonding is lower at the center of the curve section. Therefore, it is possible to draw a direct connection between DZ depth and bonding quality. Indeed, the deeper the DZ, the better the bonding will be. The discontinuous bonding is well highlighted by the DZ evolution along the curve of the cross section.

We would attempt to consider what phenomenon leads systematically to a weaker chemical dilution at the curve center of the section during laser cladding, regardless of process parameters.

Power attenuation

It is well known in the laser cladding process that powder particles contribute to attenuate the incident laser power. Tabernero et al. [START_REF] Tabernero | Modelling of energy attenuation due to powder flow-laser beam interaction during laser cladding process[END_REF] have described the impact of a shadow of particles during the interaction of the laser and the powder while cladding is in progress. They have assessed the attenuation undergone by the beam and characterized the density of energy that reaches the surface of the substrate. They were able to perform this calculation with a computational fluid dynamics (CFD) model, which has been experimentally validated. Moreover, the present authors used the calculation of the attenuated power to determine the minimum power necessary to ensure an accurate clad/substrate bonding [START_REF] Pereira | High temperature oxidation behavior of laser cladding MCrAlY coatings on austenitic stainless steel[END_REF]. Nevertheless, the local attenuation due to the Gaussian powder distribution has not been studied along the curve section.

El Cheikh et al. [START_REF] Cheikh | Analysis and prediction of single laser tracks geometrical characteristics in coaxial laser cladding process[END_REF] have discovered that at the focal position, fs, presented in Figure 2, a Gaussian distribution of powder can be observed. Qi et al. [START_REF] Qi | Numerical simulation of heat transfer and fluid flow in coaxial laser cladding process for direct metal deposition[END_REF] have proposed an equation of the powder distribution 𝑁(𝑥, 𝑧) depending on the radial distance x and the axial distance z in the case of a coaxial nozzle as shown:

𝑁(𝑥, 𝑧) = 𝑁 𝑚𝑎𝑥 (𝑧). 𝐸𝑥𝑝(-2𝑥 2 /𝑅 2 ) (1)
With 𝑁 𝑚𝑎𝑥 [𝑚 -3 ] the peak concentration at the center of powder flow (x=0), x[mm] the radial distance and R[mm] the powder stream.

To understand how the power is impacted, the Beer Lambert Law can be applied [START_REF] Qi | Numerical simulation of heat transfer and fluid flow in coaxial laser cladding process for direct metal deposition[END_REF][29]:

𝑃 ′ 𝑧 (𝑥, 𝑧) = 𝑃 𝑧 (𝑥). 𝐸𝑥𝑝(-𝛼𝑆𝑁𝑧)

With 𝑃 ′ 𝑧 [𝑊] the power attenuated by the powder flow, 𝑃 𝑧 [𝑊] the initial input power during laser cladding, S [𝑚𝑚 2 ] the particle section and α a factor which considers phenomena like the scattering effect, plasma generation... α was obtained from an experiment of the attenuated power during laser cladding. A power meter is placed under an optically neutral glass to protect it from the powder injected. During the test, an air flow was injected horizontally in order to deflect the projected powder. As an example, for an incident measured power of 1916 W, the measured attenuated power was 1498 W for a PFR of 28.5 g/min. This led to adjusting the α coefficient at a value of 6 (this result is in accordance with literature [START_REF] Tabernero | Modelling of energy attenuation due to powder flow-laser beam interaction during laser cladding process[END_REF]). If α is equal to 1, the attenuation is only due to particle shadow.

The present value of 6 indicates that other phenomena (scattering effect, plasma generation...) also contribute to the power attenuation.

By inputting the powder distribution on each position of x it is possible to observe the power attenuation on the radial distance for an incident power of 2800W as an example (Figure 10). Moreover, given that the absorptivity of Cu based materials is about 0.06, only 72W remains to reach the substrate (and probably melt it). Since the focal plane of the powder flow corresponds to the center of the mold curved surface, it can be assumed that this shadow effect explains the lack of dilution DZ observed at this position of the section. Therefore, this phenomenon may be at the origin of the absence of bonding for some cladding conditions with too weak incident laser power.

Conclusion

Ni based powder was deposited by laser cladding on a Cu based substrate with a curved surface. The cladding process parameters were widely explored in order to verify the bonding quality of the clad to the substrate. Three kinds of metallurgical bonding have been noted: lack of bonding, partial bonding or perfect bonding. In addition, a constant lack of bonding is observed in the center of the substrate curved surface. Moreover, the chemical dilution depth between Ni and Cu was measured at the interface clad/substrate depending on process parameters. It appears that the dilution depth is also lower at the same location of the curved surface. This could be explained by the Gaussian distribution of the powder flow around this point. Indeed, at the curved center the powder density is at maximum, so the input laser power is significantly attenuated. Less energy is then available for the substrate to melt and to ensure a perfect metallurgical bonding. The attenuated power evolution moving away from this central point would explain the discontinuous bonding along the interface in the cross section.

Tables caption

Table 1 : Chemical compositions of the substrate and powder 5 Table 2 : Process parameters 6

Figures caption

Figure 1 :Figure 2 :

 12 Figure 1: a) Half of the mold ring after Ni-laser cladding and b) studied cross section

Figure 3 :

 3 Figure 3: Three kinds of bonding behavior a) perfect bonding b) partial bonding and c) no bonding

(

  HAZ) seems to be present in the substrate because no microstructure modification is noticeable close to the interface with the clad. Concerning the Ni deposit, elongated grains are observed along solidification direction for both cases. Small grains can be seen close to the interface and at the extreme coating surface.

Figure 4 :

 4 Figure 4: Typical microstructure of the cross section of the samples a) CNC 5 with good bonding, b) CNC 6 with partial bonding obtained by SEM-BSE

Figure 5 :

 5 Figure 5: Crystallographic axes parallel to the solidification direction (black arrows). a) EBSD analysis of the black square area in Figure 4a and b) in Figure 4b.

Figure 6 :Figure 6

 66 Figure 6: EDS maps of the principal chemical elements composing the deposit and the substrate of sample CNC6 (a) Al, (b) Cu, (c) Ni Figure 6 clearly shows a local chemical mixture at the interface coating/substrate delimited

Figure 7 :

 7 Figure 7: Chemical composition of the dilution zone from the coating to the substrate (sample CNC 6)

Figure 8 :

 8 Figure 8: Hardness measurements corresponding to the characteristic features encountered from the coating to the substrate (red points)

Figure 9 :

 9 Figure 9: Measurement of the DZ depth over the entire cross section and DZ depth along the five red lines for all the samples

Figure 9

 9 Figure9reveals that the DZ evolution is the same across the section for all the samples studied.

Figure 10 :

 10 Figure 10: Distribution of the power attenuation on the substrate (x=0 is the focal point) for an input laser power of 2800 W Figure 10 indicates that, at the center of the powder flow (at the focal point fs Figure 2) the

4 Figure 2 : 5 Figure 3 : 7 Figure 4 : 7 Figure 5 : 8 Figure 6 : 9 Figure 7 :) 9 Figure 8 : 10 Figure 9 : 11 Figure 10 :

 425374758697981091110 Figure 1: a) Half of the mold ring after Ni-laser cladding and b) studied cross section 4 Figure 2: Powder distribution according to the focal position fs [17] 5 Figure 3: Three kinds of bonding behavior a) perfect bonding b) partial bonding and c) no bonding 7

Table 1 :

 1 Chemical compositions of the substrate and powder

	Elements (wt%)	Fe	Mn	Al	Ni	Zn	Pb	Sn	Si Cu	B Cr C
	Cu-Ni-Al	<1	0.5 8.5	15	8	<0.1 0.15 1	Bal	-	-	-
	Ni powder	1	0.1	-	Bal	-	-	-	2.5	-	1.7 0.3 0.5

Table 2 :

 2 Process parameters

	Sample	Power (W)	Speed	Powder Feeding Rate	Spot diameter
			(mm/s)	(g/min)	(mm)
	CNC 1	2400	6.5	26.5 g/mm	4 mm
	CNC 2	2600	10	24.5 g/mm	3 mm
	CNC 3	2800	6.5	32.5 g/mm	3 mm
	CNC 4	2800	8.5	24.5 g/mm	4 mm
	CNC 5	3200	8.5	28.5 g/mm	3 mm
	CNC 6	3200	10	30.5 g/mm	4 mm

Acknowledgments

The authors wish to thank C. Aboud, N. Barbosa and C. Lafarge from the ETS CHPOLANSKY for their help in the sample elaboration by laser cladding and their great skill in the field of this technology applied to non-planar surfaces. This research was supported by the ANRT (French National Agency for Research and Technology) and ETS Chpolansky.