Interbank market provides a transport platform that achieves the connection among banking institutions for the trading of liquidities. Credit institutions trade liquidities with a purpose to balance positions as well as to fulfil the average reserve required by the European Central Bank. In Italy banks trade liquidities on the Market for Interbank Deposits (e-MID) and most of the transactions are settled based on the overnight maturity. This paper is attempted to investigate the network topology of the Italian overnight money market by using the methods of statistical mechanism applied to complex networks. The objective is to assess the efficiency and the preferential trading relationship of the banking system, likewise to analyse the potential effect that the current institutional settings have on the network configuration. The data set is composed of all the overnight transactions recorded on the e-MID from January 1999 to September 2002. The evolution of the network structure is investigated through the maintenance period and over the years, and the activities of banks of different sizes are compared in order to understand the different business strategies each group of banks may adopt. Overall, evidence suggests that banks in the overnight market seem to manage liquidity efficiently and there is no obvious tendency towards preferential trading. Nevertheless, the banking system is highly heterogeneous with small number of large/medium banks borrowing from a high number of small banks. An even more connected configuration is driven by the current mechanism for reserve requirement as the end of the maintenance date approaches, which increases the systematic risk potential.
1．Introduction
A network (also called "graphs" in mathematics) is a representation of a set of items, which are called vertices, where some pairs of the vertices are connected by edges. Vertices are also called nodes, and edges are also called lines. Examples include the electrical network of interconnection between electrical elements, the internet, the social networks between individuals and organizations, the World Wide Web and many others. The edges may be directed or undirected. For example, the network of telephone messages between individuals would be directed since each message goes in only one direction; on the other hand, the friendship between two individuals is undirected. In addition, a network is incomplete if some pairs of vertices are not connecting directly with each other. That is, there is at least an intermediate vertex lying between two vertices. Set the World Wide Web as an example, the greater the incompleteness, the more intermediation is needed to achieve efficient information flowing, thus the more costly is the transfer.
According to Newman (2003), the simplest useful model of a network may be the random graph. Assuming there are n vertices, each undirected edge is placed between pairs of vertices at random so that each edge is present with independent probability p, and then the degree of the vertex (the number of edges incident to the vertex) follows a binomial distribution or a Poisson distribution in the limit of large n. However, real-world networks are normally non-random to the extent that they show strong clustering or transitivity properties which random graphs generally do not have. A network is said to show clustering if there is a high probability that two vertices, to which the same another vertex have been attached, are connected by an edge. More specifically, it node A is connected to node B and B is attached to node C, then it is highly probable that A and C is also connected by an edge. Another crucial difference is that the degree distributions of vertices of real-world networks usually do not follow the binomial distribution or Poisson distribution mentioned above, but follows either exponential or power-law distribution. Additionally, vertices in random graphs mostly have the same degree that equals to the mean vertex degree. This characteristic makes random graphs resilient to random removal of vertices, while the resilience of other networks is dependent on the distributions of vertices degrees.
Contrast to random graph, a scale-free network is vulnerable to targeted attack (Figure 1). A scale-free network is a network whose vertices degree distribution follows or asymptotically follows a power law. One remarkable characteristic of the scale-free network is that it has at least one vertex with a degree that greatly exceeds the average and these highest-degree vertices are called hubs. Hubs are tough as well as weak. If attack is taken randomly and only a small portion of nodes of low degree fail, the whole system is not likely to be affected due to its connectedness kept by remaining higher-degree hubs. Nonetheless scale-free system becomes extremely fragile after a few major hubs are taken out. Many real-world networks are considered to be scale-free, such as social networks and World Wide Web (Albert and Barabási, 2002).
Networks have also been studied extensively in economics and finance over the past few years. Battiston (2003) studies how the probability that the broad approves a strategy proposed by the Chief Executive Officer is affected by the size and the topology of the interlock network. Haldane（2009）researches both the structural vulnerabilities and robustness of financial system. He points out that the sub-prime crisis in 2008 provides a great evidence to apply network analysis in financial systems. He also explains that complexity and homogeneity are the two characteristics of financial network which had emerged over the past decade and overthrown the theory that complexity plus homogeneity spell stability but also spell fragility. Supported by Gai and Kapadia (2010), they develop a model of contagion in arbitrary financial networks to conclude that financial systems may present a robust-yet-fragile tendency. Additionally, Fricke (2012) analyses the trading strategies for members of the Italian interbank network and concludes that the preferential lending relationship on the micro-level lead to community structure on the macro-level.
There are so many different models or structures to which each network belongs, so it seems that they will play a significant role in determining the stability and functionality, especially when facing negative shocks. The topology and structure of interbank networks have been extensively analysed in recent years. Santos and Cont (2010) explores the structure and dynamics of interbank exposures of financial institutions in Brazil. They find that random graph cannot capture the behaviour of an interbank network effectively, but a directed scale-free graph with heavy-tailed degree and weighted distributions seems to adequately describe those properties. Imakubo and Soejima (2010), likewise, study the change of structure in the main interbank market of Japan. Their works reveals that the structure has transferred from the hub-and-spoke shape to a decentralized one. Different to Brazil interbank market, Japanese interbank network displays a small world network property (most vertices can be connected from every other by short paths through the network), where high clustering and short steps among exiguous nodes have also been reported. Another paper written by Georg (2011) analyses the effect of the interbank network structure on contagion and common shocks. It further concludes that money-centre networks are stronger than random networks with respect to financial distress and confirmed that the topology of the interbank does contribute to the assessment of banking system, especially in times of crisis. Georg (2011) also claimes that networks with large average path length are more resilient to financial attacks.
It is well known that credit institutions need to ensure adequate liquidity to meet their financial obligations and accomplish financial missions. To manage their temporary deficit or surplus of funds, institutions can trade liquidity through the interbank money markets. As interbank market allows credit institutions to smooth through temporal liquidity shortage, the structure and function of the market is essential to the efficiency and stability of the financial system. Institutions exchange liquidities with each other and a complex network is established with internal debts or loans. Each institution represents the vertex in terminology of networks and lending relationship can be expressed as oriented edges, with" A B" representing "A lends liquidity to B". Banks try to maximize their returns while they are managing liquidities. Hence it is expected that banks of different sizes may adopt different business strategies and play different roles in the network.
This paper, by applying the methods of statistical mechanics with respect to complex network, proposes to study the network topology of the Italian overnight interbank market. The money-market deposits traded on the Market for Interbank Deposits (e-MID) is investigated based on overnight maturity. It focuses on the evolution of the connectivity structure over the maintenance period as well as through the years. Likewise, the differences in the activities of banks of different sizes are analysed. The data set consists of all the overnight transactions recorded on the e-MID from January 1999 to September 2002. The main purpose of this paper is to assess the efficiency of the interbank system and to explore the potential influence that the current institutional settings have on the stability of the banking system.
The results suggest that the banking system is highly heterogeneous, where the distribution of the banks' degree is heavily tailed. To be more specific, a small group of large/medium banks borrow from a large number of small banks. This structure is particularly vulnerable to systematic risk. Especially approaching the end of the maintenance period, the banking system is pushed to a more connected configuration by the current institutional arrangements, which further increases the potential of contagion risk and systematic failure.
It is also indicated that the tendency towards preferential trading is not obvious and the interbank market is relatively efficient. Banks trade liquidity directly with counterparties without intermediaries, and it is only in the last few days of the maintenance period when a very few number of banks seem to act as intermediaries.
The remainder of the paper is organized as follows. Section two briefly explains the Italian interbank network, Italian overnight market and overnight interest rates and also introduces e-MID. In section three, data set is described and the metrics measuring the properties of networks are listed. Section four analyses the topology of the Italian interbank network and discusses the corresponding results. Section five has the conclusion.
2．Interbank market, Italian overnight market and e-MID
In order to achieve monetary policy objective, the European Central Bank (ECB) has conducted the principal tool--Open Market Operations (OMO) to expand or contract the amount of money in the banking system. The aim of the OMO is to steer and control Eurozone interest rates, manage the liquidity situation in the market and also to signal the stance of monetary policy. Broadly, the ECB controls liquidity in the banking system via refinancing operations, which basically mean repurchasing agreements. For example, banks receive a cash loan from the ECB by providing appropriate collaterals. Refinancing operations mainly contain four categories (main and longer-term refinancing operations, fine-tuning and structural operations) and they are conducted via an auction mechanism. Auctions are held on Mondays and settled on the following Wednesday. Until 2000, the ECB specified on the auction the interest rate at which it intended to lend money (fix rate tender). Since then a variable rate tender has been conducted where the ECB determines the minimum bid rate and banks bid against each other to access the available liquidity. In auctions of the fixed rate type, individual bank bids were allotted pro rata based on the ratio between total liquidity to be allotted and total bank bids. For the variable rate type, the highest rated bids are settled first, followed by other bids with lower interest rates, until all the available liquidity is allocated. In addition, to effectively stabilize interest rate and prevent liquidity shocks, the ECB requires the credit institutions in the Euro area to hold a minimum reserve over a certain period of time, which is determined in advance through the auction. The most efficient way for credit institutions to satisfy the reserve requirement is supposed to be the interbank market transactions.
Interbank Market
Interbank market provides a transport platform that achieves the connection among banking institutions for the trading of liquidity. Credit institutions are required to hold sufficient liquidity to manage their day-to-day running of operations. A shortage of liquidity can be solved by borrowing at the interbank market. In normal times, institutions who have liquidity surplus lend money to those who have not yet satisfied the reserve requirement, receiving interest on the lending. The reserve requirement (or cash reserve ratio), aimed at preventing liquidity shocks , is determined by the ECB who sets the minimum reserves and requires each credit institutions in the Euro area to hold with the respective national central bank (NCB). So during times of crises, central banks are able to reduce banks deficit by providing provisions so as to stabilize the financial system and ensure the functioning of the banking market. The current reserve coefficient required by the NCB has been 1% since January 18 of 2012 (including overnight deposits, deposits with agreed maturity or period of notice up to 2 years, debt securities issued with maturity up to 2 years and money market paper), and it was 2% all the time before that date. Reserve requirement is published in the ECB bulletin for each maintenance period [1] . Required by the ECB, compliance with the requirement is determined on the basis of the average of the daily balances in the reserve accounts over the maintenance period. Figure 2 shows the pattern of the average monthly excess reserves/deficiencies from 1999 to 2011 for the Euro area. It looks good that, on average, banks in the Euro area are generally capable of fulfilling the reserve requirement and the tendency seems to be upward.
Although banks' performance of fulfilling the reserve requirement is great, previous studies have indicated that the monthly reserves pattern within the maintenance period may not be that optimistic. Bank of England Publications (2000, cited from Iori, et al 2007) reportes that the market generally has a deficit at the start of each reserve maintenance period and the ECB need cover the deficit for that period with the first Main Refinancing Operation, which provides the majority of the liquidity required by the market. Furthermore, not all credit institutions actively manage their minimum reserves. Even though they may have to use the inevitable amount of liquidities for daily operations in the payments system, some of them (especially small institutions) would like to constantly keep their reserve account at the requisite level throughout the whole maintenance period (Gabbi, 1992).
Italian Overnight Market and Overnight Interest Rates
In Italy, most of the overnight transactions are settled on a central trading platform named e-MID. Though there are multiple maturities for which transactions can be settled, overnight loans account for the biggest part of all trades.
Raddent (2012) investigates the structure in the Italian overnight loan market and he concludes that over the last 12 years, the Italian interbank market has experienced significant changes. Besides local Italian institutions, a lot of foreign banks have been absorbed into the market; some smaller banks have disappeared in the market due to the activities of merge and acquisition, or because refinancing and liquidity management have been transferred to large credit institutions. Additionally, the overnight interbank market has become decayed through the years. Until 2007, banks would be more willing to lend to others on the overnight market because generally banks had good reputation. But after the sub-crime crises in 2008, the day-to-day monitoring of the counterparties became unavoidable as a consequence of an increased risk of default, and banks prefer to trade loan with a longer maturity.
The fulfilment of the credit institutions' reserve requirement and their borrowing/lending behaviours can be well illustrated by overnight interest rates, which represent an average of the rates at which banks borrow and lend to each other on the overnight market. Overnight interest rate is a measure of liquidity with sharp decline representing tranquil liquidity conditions and surging representing a shortage of liquidity. As a result, overnight interest rates can be a typical indication of the ECB's monetary policy stance.
Italian overnight interest rate is nearly bounded between two official interest rates set by the ECB. Some key interest rates are fixed by ECB's Governing Council for the Euro area, which include the interest rate on the main refinancing operations, the interest rate on the marginal lending facility and the interest rate on the deposit facility. The interest rate on the main refinancing operations, which is a signal of the ECB's monetary policy to the market, indicates the situations at which the ECB is going to intervene in the market by liquidity trading. Credit institutions can obtain overnight liquidity from the central bank at the rate of the marginal lending facility against eligible collateral or make overnight deposits with the central bank at the rate of the deposit facility. These two interest rates bound the overnight interest rate, with the former providing the ceiling and the latter providing the floor. This feature makes sure that the EBC monetary policy is effective.
An intra-day pattern can be identified in the Italian interbank market as a result of the features of the Italian payment system. It can be observed in figure 3 that the number of transactions peaks around 9 a.m. and then at 3 p.m. The causation of this pattern can be explained as: around 9 a.m., unsettled payments are settled automatically, the majority of which are composed by previous days' e-MID contracts; at approximately 12:00 a.m. the balance of the net payments are settled; around 12:30 p.m. banks settle the cash leg of the net securities settlement system; in the afternoon, banks primarily settle financial and interbank payments. Barucci et al (2004) specifies that the European settlement system TARGET adopts a similar trading pattern with that of the cross-border transactions, from which the adjustment of single liquidity positions, distribution of funds within corporate banking groups and inventory accumulation of reserves are implemented in the final hours of the operational day. Thus payments in the last hours of the day are also significant.
With respect to the monthly patterns of trading behaviours, both Barucci et al (2004) and Raddent (2012) observe an increase in the volatility of interbank loan interest rates and a growth in the number of transactions on the last few days of the maintenance period. However, they have two different views regarding the trading volume patterns within the maintenance month. Barucci et al (2004) suggest a decline in exchange volumes, whereas Raddent (2012) reports a slight increase in trading volume when approaching the 24th. The reason may be that they have adopted the data set at different times, with the former study using data during 1999-2001 and the latter one being from the period 1999-2010, during which the sub-prime crises happened. In addition, it is found out by Raddent (2012) that overnight interest rate peaks cyclically at each end of calendar month, while investigation carried out by Iori et al (2007) reported that overnight rates exhibit a big movement on the last few days of the maintenance period with either a sharp dip or a pronounced peak. The intra-maintenance period patterns during the period 1999-2002 are shown in figure 4 and the illustration of the average overnight interest rates is presented in figure 16 (discussed later).
e-MID
e-MID is the only electronic market for Interbank Deposits in the Euro Area and US. It provides a platform for the trading of Euros, US-dollars, Pound sterling and Polish zlotys. It was founded in Italy in 1990 for Italian Lira transactions and denominated in Euros in 1999. Both Italian and foreign banks participate in the market. Contracts are automatically settled through the TARGET2 system.
e-MID plays a leading role in liquidity transactions, with an ever-increasing number of international counterparts joining in the market. In August 2011, e-MID consisted of 192 members, of which 101 were Italian banks and 61were international banks. According to the "Euro Money Market Study 2006" published by the ECB, e-MID contributes 17% of total turnover in unsecured money market in the Euro Area.
The available maturities range from overnight to one year. 80% of the transactions are overnight. e-MID adopts a quote-driven mechanism where each quote contains the quoting bank's ID, the market side (bid or offer), the volume, the interest rate and the maturity. The minimum quote is 1.5 million Euros whereas the minimum trade size is only 50000 Euros. A quote is submitted if the ordering bank accepts the bid/offer. Before being carried out, the transaction has to be accepted by both counterparties as a result of facilitating credit line checking. The market also permits direct bilateral trades between counterparties [2] .
3. Data Set and Properties of Network
The empirical data set is based on all the overnight transactions on the e-MID platform from the period August 1999-September 2002 [3] . Each transaction gives the information about the date and the time of the trade, the quantity, the interest rate at which the transaction is settled and the names of the quoting and ordering banks. The number of participating banks was 215 in 1999, 196 in 2000, 183 in 2001 and 177 in 2002. The total number of the overnight transactions was 586,007.
The analysis of the network topology for the Italian overnight money market will be carried out through the methods of statistical mechanics applied to complex networks, so as to investigate the stability of the banking system as well as the trading relationships between (among) banks.
Before empirical analysis is conducted, the metrics that describe the properties of network will be introduced at first.
Properties of Network:
4.1 Vertex (Node)
A vertex is the fundamental unit of a network, and in this paper a vertex represents a credit institution. The total number of banks in the market is denoted as N and the number of active banks is .
4.2 Undirected/Directed Edges
The lending /borrowing relationship between two counterparties can be represented by an edge. A non-directed edge means that the relationship is bi-directional, which means banks lend money to each other. An oriented link indicates the direction of the flowing of money, such that a link is incoming to the borrower and outgoing from the lender. A directed network is more relevant for analysing the probability of contagion and system risk. Let denote the relationship between two vertices, with ( indicating that bank i lends to bank j (bank j lends to bank i) and () indicating no lending relationship from bank i to j (bank j to i).
4.3 Distance and Diameter
A geodesic path is the minimum paths between two vertices through the network. There may be more than one geodesic path from one vertex to another. The distance between two nodes is defined as the length (in number of edges) of their geodesic path. The average distance for a network is the average over all distances. The diameter of a network is defined as the length of the maximum geodesic path between any pairs. The diameter of a disconnected network, which is made up of several isolated components, is infinite. Presented by Iori et al (2007), there would be preferential paths between banks for money flow if the diameter of the banking network is substantially different from that of a random network.
4.4 Degree
The degree is defined as the number of edges connected to a vertex i, denoted as:
where the neighbourhood set γ(i) for a vertex i is defined as its immediately connected neighbours as :
A directed network has both an in-degree and an out-degree for each node, where in-degree , and out-degree . It measures the number of counterparties a bank borrows from (or lends to).
In a random graph, assuming the total number of vertices is n and each edge is presented with independent probability q, then the probability of a vertex having degree k is:
, where z=q(n-1) denotes the mean degree. Most real-world networks are not characterized by a Poisson distribution of degrees as random graphs, but characterized either by exponential distribution or power-law distribution for some constant exponent . Networks with power-law degree distributions are sometimes referred to as scale-free networks.
4.4 Resilience
Network resilience indicates the capability to defend itself against the removal of vertices. If vertices are removed from a network, the typical length of paths between some vertex pairs will increase, and if the number of removed vertices is significantly large the vertex pairs may ultimately become disconnected and the communication through the network may be destroyed.
Networks vary in their level of resilience with respect to vertex removal, which is generally determined by their degree distribution of vertices. Most networks are robust against random vertex removal but relatively vulnerable to targeted removal of the highest-degree vertices.
4.5 Transitivity and Clustering
One of the characteristics that random graphs do not possess is network clustering, sometimes also called transitivity. In terms of network topology, transitivity implicates the presence of an increasing number of triangles in the network. It can be quantified by defining a clustering coefficient, which is the average of the local clustering coefficient . In effect, clustering coefficient measures the fraction of triples of which a triangle is completed by the three edges. The local clustering coefficient for a vertex i is obtained by the proportion of links between the vertices within its neighbourhood divided by the number of links that could possibly exist between them. Hence the local clustering coefficient is given as
here for any i and j. For vertices with degree 0 or 1, . The local clustering coefficient indicates the probability that two vertices which have the same other vertex as a neighbour will themselves be linked with each other. In this case, it shows the probability that two banks have a trading relationship, if both of them are trading with the same other bank. Actually, this metric is obtained based on the non-directional operation where the direction of links is not considered.
The clustering coefficient for the whole network is given by the average of the local clustering coefficients of all the vertices N
A network is considered small-world if the average clustering coefficient C is significantly bigger than that of a random graph with the same vertices set. Small-world effect indicates the fact that most pairs of vertices in the network seem to be connected by a short path through the network. In addition, average clustering coefficients is able to implicate the situations of intermediaries, where small number of intermediaries may be explained by low clustering
4.6 Mixing Patterns (Affinity)
In a network, it is interesting to know which vertices pair up with which others and such kind of selective linking is called affinity. There may be a few different types of vertices and the probabilities of connection between vertices may depend on such types. In this case, vertices (banks) may be divided into different types based on the sizes of vertices (determined by trading volume).
Affinity can be quantified by an "assortative coefficient". Let be the number of edges connecting vertices of type i and j. Let E be a matrix with elements. Now define a normalized mixing matrix by
where means the sum of all the elements in matrix X. The element measures the fraction of edges that fall between vertices of types i and j. Then the assortative coefficient is defined as
This quantity is 0 in a randomly mixed network and 1 in a perfectly assortative one.
A more specific case of affinity refers to degree correlation. In this case, vertices (banks) may be divided into different types based on the degree of vertices. Degree correlation is able to give rise to some interesting network structure effects. For example, it discloses whether banks which have high vertex degrees are more likely to be connected to high-degree neighbours or low-degree neighbours.
Degree correlation can be quantified by a coefficient, which is defined as
If is increasing with k then high-degree vertices are more likely to have a majority of high-degree neighbours. If negatively correlated with k then high-degree vertices are more likely to be connected with low-degree vertices. The former property is also called assortative mixing and the latter one is called disassortative mixing. Degree correlation can be an indication of preferential trading relationship in the banking system.
4.7 Strength
The strength of links between vertices can be measured as the trading volume on the link, or the number of trades on the link or in terms of net flow.
The vertex weighted strength is defined as
where ( denotes the lending volume from bank i (bank j) to bank j (bank i). Similarly, the lending and borrowing strength can be separately indicated as
The vertex connectivity strength is defined as
where ( denotes the number of trades that bank i (bank j) lends to bank j (bank i).
The strength in terms of the vertex net flows can also be defined as
Note that a bank i is a net lender if Otherwise, it is a net borrower (.
4.8 Participation Ratio
Participation ratio can be utilized to measure the preferential relationships between banks. It measures the weights of edges, which can either be heterogeneously distributed or can be of the same order of magnitude. It is defined as
and
for lending and borrowing links. Or it is equivalently defined as
and
for lending and borrowing links.
If all the weights are of the same order of magnitude then but if the weights are heterogeneously distributed, of which a few number are dominant then is close to 1, indicating preferential relationships between banks [4] . The average participation ratio is obtained as
4． Analysis of the topology of Italian interbank network
This section primly focuses on the structure of the Italian overnight network and its evolution over the maintenance period. The behaviours of banks of different sizes will be analysed and the overall stability as well as the efficiency of this network will be assessed.
It has already been previously shown in section 2 with regard to the monthly patterns of trading behaviours that approaching the end of the maintenance periods, there is an increase in the number of transactions and a slight decrease in the average trading volume. This result suggests that banks start to trade actively on the last few days of the maintenance periods, in order to balance their positions or to adjust liquidity for fulfilling the reserve requirement. However, this fact implies a less optimistic situation that some banks are generally having inadequate liquidity before the end of the maintenance period, hence giving arise to a large number of borrowing behaviours to balance their reserves.
These findings are also supported by some further evidences shown in Figure 5, where the average number of active banks and the average degree through the maintenance period are plotted. It is clear from the figure that both the average number of active banks and the average degree are raising when approaching the end of the maintenance month. This reveals the fact that the increase in the number of transactions is due to the increasing trades from a higher-than-usual number of counterparties. It also suggests that the liquidity adjustments are achieved by a large number of transactions but with quite small volume. Additionally, the sharp increase in average degree may indicate that banks can adjust their liquidity quickly through directly trading with counterparties without more intermediaries, which indicates that the network is complete such that the Italian overnight market is efficient as expected.
The distribution of banks' degree is illustrated in figure 6. Regardless of the precise form of the fit, it is shown that the banking system is highly heterogeneous because the distribution is fat tailed, which indicates that a small number of banks trade with many others while the majority banks trade with few counterparties. A network seems to have power-law degree distribution if there is an approximately straight-line form on a log-log scale, or has an exponential degree distribution if there is an approximately straight-line form on a linear-log scale. So combining figure 6 and figure 16 (introduced later), the distributions appear to be somewhere in between power-law and exponential.
Figure 7 shows the affinity of the Italian overnight network. The illustration reveals a disassortative mixing property (negative degree correlation) of the system that higher-degree banks are more likely to be linked with lower-degree banks. It is not clear, however, what the explanation for this result is. But the property of affinity may indicate the potential tendency towards preferential trading between large-degree banks and low-degree banks. More detailed investigation about preferential tradings is discussed later.
In order to understand the structure of the Italian overnight network more deeply, the differences in the activities of banks of different sizes are investigated through classifying the banks by nationality and the amount of daily volume of transactions. It is aimed at finding the similar or different trading strategies of banks that have some properties in common. Following the Bank of Italy classification, banks are split into four groups: group 1 with foreign banks; group 2 with very small Italian banks from which the daily volume is in the range 1-23 million Euro; group 3 with small Italian banks from which the daily volume is in the range 23-70 million Euro; group 4 with large/medium Italian banks from which the daily volume is beyond 70 million Euro. The approximate proportion of each group in the banking system is separately counted as: 18.2% for group 1; 35.7% for group 2; 24.5% for group 3; 21.7% for group 4. Hence for the structure of the network, smaller banks account for a larger proportion.
A resulting network representation plotted on January 10 of year 2002 is shown in figure 8 [5] . It illustrates the connectivity among group 1 (black), group 2 (blue), group 3 (green) and group 4 (red) and the direction of the arrow is from the lender to a borrower and the empty circle identifies the lender bank. It can be identified that more central a vertex is located, higher degree it has. Foreign banks only have medium/small degree, indicating that a majority of the transactions are accomplished among Italian banks. Bank 123 is considered to trade with the most number of counterparties such that it is a hub of the graph. Other hubs can also be identified, for example banks 1, 6 and 11 which have relatively high degree. Hubs and nodes with high degree may be vulnerable to targeted attack, so banks which are more closer to the centre in the figure may play a more important role for stabilizing the banking system in times of crisis. It is notable that banks which have high degrees (near the centre) are generally borrower banks while those with low degree (on the periphery) are primly playing the role of lenders. It is not difficult to find out that most banks only play one role, either acting as a lender or as a borrower and a very few of them perform both roles (i.e. banks 31, 103 and 93).
Apart from the overview of the whole banking network on e-MID, statistical results provide stronger evidence to explain the network structure in terms of the banks of different sizes. Behaviours of both the borrower banks and lender banks are investigated separately for each group where Figure 9 illustrates the average degree and Figure 10 illustrates the strength measured by volume. By the end of the maintenance period, both the outgoing and incoming degrees increase while the strength has a slight decrease, which well explains the illustration in Figure 5 as for the growth in the number of degree and the decline in the total trading volume at the end of the period. It is clear that large/medium banks (group 4) definitely play a quite significant role in liquidity management. They have the highest incoming degree and the second highest outgoing degree, where incoming degree and outgoing degree separately represents the number of creditors and debtors. It is also remarkable that both the lending and borrowing degrees for large/medium banks increase the most when approaching the end of the period. Likewise, large/medium banks are both the largest lenders and the largest borrowers with the highest volume strength. As for very small banks (group 2) and small banks (group3), their performance is relatively stable. Diversification is quite small between lending and borrowing behaviours with regard to degree and strength. It is also notable that small Italian banks have the greatest number of debtors. From Figure 9, it is obvious that foreign banks (group 1) do not have very high incoming degree and indeed they have the lowest outgoing degree. So combined with figure 8 (that they are located far from the centre), it can be concluded that foreign banks only trade with very few number of counterparties. On the other hand, foreign banks are the second largest lenders and borrowers following the large/medium Italian banks with respect to the strength.
There may be various reasons that banks act as lenders while they are also acting as borrowers, and that can be an explanation as for the low number of intermediaries and system efficiency. But to generally specify the roles that banks of each group are playing (as lenders or borrowers), the time evolution of the strength measured by net flows (reminding that net flow is calculated as outgoing volume minus incoming volume) is considered in figure 11. A bank plays the role of net lender (borrower) on a typical day if it has a positive (negative) flow at the end of the day. From the illustration it can be observed that both the large/medium Italian banks and foreign banks have negative flows through the maintenance period so that they are primarily playing as the borrowers to the banking system. As both the red line and black line are tortuous it may be inferred that the number of transactions for the banks in group 1 and group 4 are fickle over the period. It is also interesting that the average flows for the banks in these two groups are negatively correlated and Iori et al (2007) explain this situation as that a significant amount of trading takes place between the foreign and the large/medium banks. Observing the top two lines it can be seen that small banks and very small banks have an average positive flow such that they are the net lenders. Compared with group 1 and group 4, the number of transactions coming from small banks is not that fluctuate. The distinction between them and the group 1 and 4 is reasonable. Banks can lend excess reserve to others in order to earn profit and at the same time reduce their own costs of borrowing. This is especially attractive for small banks because of the relatively low risk of earning interest. As for large/medium banks and foreign banks, they may have better channels for investing in more profitable projects or they probably possess adequate number of collaterals from their own debtors in the sense that they can make use of the collaterals to borrow more money for investments. It is notable that for all the four groups, average flow are approaching zero at the beginning of the maintenance period. This is reasonable because banks do not have to trade actively in a few days after their position is just balanced on 23rd.
To briefly sum up for above findings from figure 8 to figure 11, the banking system is highly heterogeneous where a relatively small number of large/medium banks borrow from high number of small banks and foreign banks in large volumes, while small banks lend to fewer number of banks in small volume. Especially approaching the end of the maintenance period when banks are generally short of liquidity, this heterogeneous configuration becomes more aggravated. This may increase the potential for contagion risk and systemic failure.
In regard to transitivity, it is rather difficult to directly deduce information from the evolution of the clustering coefficients of each day, resulting from the day-to-day changes in the number of links between banks. To solve the problem, methods and results from Iori et al (2007) are adopted, where they effectively construct random networks in two different ways and simultaneously compare the real clustering coefficients with those of the random networks. They define the relative clustering coefficient instead as the ratio of the clustering coefficient of the actual network to that of the two kinds of random network, separately denoted as. The first random network is a generic one constructed with the constriction that the number of vertices and edges should be the same as the real network for each day. The average clustering coefficient in this case is then given by [6] . The second type of random network is generated with the same number of vertices, edges and the same degree distribution for each day of the maintenance month. Because generally random networks follow a Poisson distribution in degrees, whereas in the second case the random network should have the same degree distribution with the real network, of which the degree is not Poisson distributed. As a result there is no proper analytical formula for its clustering coefficient. Hence 100 random networks of the second type are constructed and the random clustering coefficient can be calculated by averaging the 100 clustering coefficients of each random network.
Figure 12 shows the results of the relative clustering coefficients for the two cases, where the top shows that for case one and the bottom is for case two. The relative clustering coefficient of case one is about 1 over the maintenance period except that it is climbing beyond 1 as the end of the period approaches and reaches a high value near 1.8 ultimately at the last day. The coefficient of the second case has the same pattern as that in the first case and it is entirely below 1 throughout the period. Therefor it can be concluded that clustering coefficients of the actual network is generally similar or smaller than that of random networks, which indicates that the Italian overnight market has similar or smaller fraction of triples than random networks and also the small-world effect does not exist in the system. Reminding that random networks have quite small degree of clustering, the similar or even lower relative clustering coefficient may implicate the fact that clustering is also not significant in the real network. More specifically, low clustering may further explain the low number of intermediaries as mentioned before, confirming the efficiency of the Italian banking system. On the other hand, the ultimately increasing beyond unity for case one may imply the situation that when banks are urgent to balance positions, the need for intermediary trading will grow. It is noticeable that the clustering coefficients of this directed network are obtained via unidirectional operations to the extent that banks can either act as lenders or as borrowers on the same day, or they may act different roles at different day. The reasons are complicated and unknown because of the non-transparent information, so more advanced studies need to be carried out for more precise and rational results.
To assess the overall structure of the system, the network average distance is investigated instead of diameter. Since the number of active banks and links change on a day-to-day base, the average distance of the network also changes every day. Therefore the same method is adopted to calculate the relative distances and , which is the ratio of the average distance in the real network to that in a random network. The two types of random networks are constructed in the same way as mentioned above. The result is shown in Figure 13. It is clear that both ratios are fluctuating near the unity throughout the maintenance period, but the fluctuation is not significant. As the banking network almost has the same average distance as random networks, it can be inferred as expected that liquidity flows via short paths through the system to the extent that banks trade directly with another without intermediates. The average distance, however, can not be analysed separately from that of the random networks, thus it may not reflect the real organisation of the banking system. Fortunately from the topology side of the network, this information is substantial to reveal the liquidity-flowing efficiency through the entire banking system.
The participation ratio, which corresponds to the weights of the edges with regard to the number of transactions, as a function of the distance from the end of the maintenance period is plotted in figure 14 (top) for lending and borrowing transactions separately. It is clear from the illustration that the participation ratio for lending transactions is greater than that of the borrowing transitions over the maintenance period, whereas there is an exception of the sharp decline in lending transactions at the end of the period. The higher participation ratio for lending transactions is reasonable. Lender banks face high credit risk therefore they will prefer to lend to some few banks that may be capable to pay back money or that have sufficient collaterals. On the other hand, borrower banks probably do not take a risk as long as they have access for loans. Approaching the last few days of the maintenance period, banks have to balance their positions with the least delay so that the choice of counterparties becomes less important, which well explains the drop in the end. At the bottom of the figure, the participation ratio as a function of the banks' inverse degree is presented to identify the edges which have higher weights than others [7] . It can be observed that for a degree up to five, the weights of the links are of the same order of magnitude such that . A no-preferential-trading situation is indicated by the continuous black line as the benchmark. At higher degree the participation ratio diverges upward and towards unity, revealing the slight preferential relationship for both lending and borrowing transactions. The tendency towards preferential trading becomes stronger along with the increase of banks' degrees. This is rational because when the number of counterparties is bigger, banks may preferentially trade with counterparties which have good financial performance and reputations.
It is notable that the Euro was introduced into Italy at the beginning of 2001, thus the transaction patterns may have a change through the period 1999 to 2002. Figure 15 provides the cumulative number of vertices as a function of the banks' degrees and strength for the whole overnight network separately in 1999 and 2002. There is an obvious difference between the banks' maximum degree of each year for both outgoing and incoming links, where they decreases apparently in 2002. This can be supported with the fact that the number of e-MID members declines from 215 in 1999 to 177 in 2002, which may be the result of mergers or acquisitions. In 1999, no matter for strength or degree it is large lender banks which dominate the large borrower banks and the number of outgoing links greatly surpasses the number of incoming links. On the contrary, situation in 2002 is totally opposite where large borrower banks dominate the large lenders. This change of pattern is coincident with the fact discussed before that lending transactions should be more conservative to the extent that banks trade with a smaller number of counterparties when lending liquidity and a larger number when borrowing money.
More information may be needed to explore the reason of the divergence between 1999 and 2002, and there may be multiple reasons of which one possible reason is likely to be related to the change of economy in Italy. Table 1 presents the GDP growth rate in Italy during the four years. Compared with 1999, the economy grows at quite a slower rate of only 0.4% in 2002. During the period of strong economy growth the need for investments is high and banks do not have to face high credit risk because generally the financial performance is good, so banks may act mainly as the lenders in the interbank market. Over the time when GDP growth rate slows down, the decline in the frequency of both the strength and degree may be a consequence of the higher credit risk. In addition, after the introduction of the Euro, banks are likely to have more access for investing their liquidities and as a result they may experience liquidity shortages and mainly play the role of borrowers.
The changes of the pattern from 1999 to 2002 also have an effect on the evolution of the overnight interest rate through the maintenance period (figure 17). Reminding that towards the last few days of each maintenance month, a decrease in the overnight interest rates indicates excess liquidity while an increase implies a shortage of liquidity. Therefore figure 17 conveys the information that the banking system experience overall excess liquidity in 1999 while in 2002 an opposite situation comes out with an overall shortage of liquidity. Moreover, the relatively higher average overnight interest rate in 2002 may indicate a conservative money policy, which gives significant explanation for the low GDP growth rate.
5．Conclusions
This paper is attempted to analyse the network structure and function of the Italian overnight money market through methods of statistical computation applied to complex network. The overall pattern of the structural changes of the banking system during the maintenance periods and through the years is explored. The behaviours of banks of different sizes are additionally investigated with respect to their degrees and strength.
The structure of the banking system is considered as relatively efficient as expected based on the observations but likewise to be heterogeneous. The banks' degrees seem to have distributions lying somewhere between the power-law and exponential. An obvious pattern of the evolution of the network structure was identified towards the end of the maintenance period. Banks are generally short of liquidity at the end of the period, thus the average bank degree, the number of active banks and transactions are all increasing in order to fulfil the reserve requirement, but the average trading volume has a decreasing trends. Banks primarily play the role of either lenders or borrowers and a very few of them act the both roles. The overnight network shows a relatively random pattern. Even though the affinity metric shows that high-degree banks are more likely to be traded with lower-degree banks, the clustering participation ratio revealed that the preferential trading relationship barely exist and only the higher-degree banks will show slight tendency towards preferential trading. Since both the clustering coefficients and the average distance ratios of the banking network are similar to that of the random networks, the liquidity is able to directly flow between the lenders and borrowers without intermediaries. As for the activities of banks of different sizes, large/medium banks which are averagely of high degrees dominate the borrowing transactions while small banks mainly act as lenders. There is a significant change of pattern for the banking system, that in 1999 large lender banks dominated the large borrower banks and the entire banking system had overall liquidity excess, while in 2002 the opposite situation emerged and the system experienced an overall shortage of liquidity. The reason for the evolution during these years may be a consequence of the change in economy environment which is probably affected by the introduction of the Euro in 2001.
On the other hand, the current institutions arrangements which require banks to fulfil their average reserve requirement by a typical date is not likely to be beneficial for the structure development of the banking system. Large/medium banks account for a relatively smaller proportion among all the banks and their dominant role in borrowing transactions, such as high degrees and large trading volume, may make them vulnerable to targeted attacks. Especially towards the last few days of the maintenance period this heterogeneous configuration becomes aggravated, which increases the potential of contagion risk and systemic failure. Moreover, the increasing clustering approaching the end of the maintenance month may imply the growth in the number of banks that act as intermediaries, which indicates a reduction in trading efficiency. Therefore it is suggested to transform the current mechanism for reserve requirement into a more flexible one rather than the current one that is strictly regulated, for instance the single reserve settlement date could be replaced by multiple dates or by a settlement period.
Methodologically, further investigation needs to be carried out in the future with respect to the structure of the Italian overnight network and some current metrics should be improved for a more precise and rational statistical results. Given the non-transparent and bilateral nature of the transactions in the banking system, behaviours of banks can not be fully and certainly explained. For example, the causation of banks' role changing, that they act as either the lenders or the borrowers at different days or they act the both roles on the same day, is still waiting to be explored. Additionally, the results shown by affinity that high-degree banks are more likely to trade with the low-degree banks, are still can not be fully explained. Moreover, the clustering coefficients are obtained based on the un-directional operation, whereas this paper analyses a directional network. Thus more accurate and specific metrics are required to be derived for different types of network. At last, this paper focuses on the Italian overnight banking system during the period 1999 to 2002, and advanced research for recent years can be carried out through topological analysis with the objective to identify the evolution of the network over a decade.
Figures
（1）
（2）
Figure : Random Network (1) and Scale-free Network (2). Dark nodes are hubs.
Vertices in random graphs mostly have the same degree that equals to the mean vertex degree. Scale-free networks have at least one vertex with a degree that greatly exceeds the average
Figure : Average monthly Excess Reserves/Deficiencies in Euro Area (EUR billions)
Figure : Average numbers of transactions during a day over the period 1999-2002. The number of transactions peaks around 9 a.m. and then at 3 p.m.
Average daily number of transactions preceding the 24th with descending distance over the period 1999-2002. There is an increase at the end of the maintenance period.
Average daily trading volume preceding the 24th with descending distance over the period 1999-2002. There is a decline at the end of the maintenance period
Figure : Intra-maintenance period patterns during the period 1999-2002
Average number of active banks preceding the 24th with descending distance over the year 2002. There is a pronounced peak towards the end of the period.
Banks' average degree preceding the 24th with descending distance over the year 2002. The sharp increase in degree indicates efficient connectivity through the interbank network.
Figure : Average number of active banks (top) and average degree (bottom) in 2002
Figure : Cumulative distribution function of banks' degree in 2002. The banking system is highly heterogeneous because the distribution is fat tailed, which indicates that a small number of banks trade with many others while the majority banks trade with few counterparties.
Figure : Affinity in the Italian overnight network. It shows a disassortative mixing property (negative degree correlation) of the system that higher-degree banks are more likely to be linked with lower-degree banks.
Group 1
Group 2
Group 3
Group 4
Figure : Banking network on January 10, 2002. The direction of the arrow is from the lender to the borrower, and the empty circle indicates the lender bank. More central a vertex is located, higher degree it has.
Figure : Outgoing (top) and incoming (bottom) average degree per bank preceding the 24th with descending distance over the period 1999-2002. The small Italian banks have the highest outgoing degree and the large/medium Italian banks have the highest incoming degree.
Figure : Outgoing (top) and incoming (bottom) volume strength per bank preceding the 24th with descending distance over the period 1999-2002. Followed by foreign banks, the large/medium Italian banks are both the largest lenders and the largest borrowers.
Figure : Net flows per group of banks in 2002 preceding the 24th with descending distance. Both the large/medium Italian banks and foreign banks are primarily playing as the borrowers to the banking system. The average flows for the banks in group 1 and 4 are negatively correlated, which indicates that a significant amount of trading takes place between the foreign and the large/medium banks. Banks in group 2 and 3 are the net lenders.
Figure : Relative clustering coefficients and preceding the 24th with descending distance. The clustering property of the real network is similar to that of random networks, indicating no intermediaries.
Figure : relative distances and preceding the 24th with descending distance. Liquidity flows along short paths through the banking system without intermediaries.
Figure : Participation ratio preceding the 24th with descending distance for borrowing and lending transactions (top). Participation ratio as a function of the banks' inverse degree for borrowing and lending transactions (bottom). The oblique line represents the benchmark of no preferential trading. The participation ratio is higher for lending transactions. A slight tendency towards preferential trading exists for higher degrees.
Figure : The cumulative number of banks as a function of the banks' degrees (top) and strength (bottom) for the whole overnight network in 1999 and 2002.
Figure : Average overnight interest rate in 1999 (top) and 2002 (bottom) preceding the 24th with descending distance. The banking system experiences overall excess liquidity in 1999, while in 2002 an opposite situation comes out with an overall shortage of liquidity.
Tables
1999
2000
2001
2002
1.3%
2.7%
1.6%
0.4%
Table : Real GDP growth rate in Italy
