Significant ties: Identifying relationship lending in
temporal interbank networks
Teruyoshi Kobayashi
Taro Takaguchi

August 2017
Discussion Paper No.1717

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANSignificant ties: Identifying relationship lending in temporalinterbank networks
Teruyoshi Kobayashi1 and Taro Takaguchi2,3,4
1 Department of Economics, Kobe University, Kobe, Japan
2 National Institute of Information and Communications Technology, Tokyo, Japan
3 National Institute of Informatics, Tokyo, Japan
4 JST, ERATO, Kawarabayashi Large Graph Project, Tokyo, Japan

August 29, 2017

Abstract
Relationship lending is conventionally interpreted as a strong partnership between a lenderand a borrower. Nevertheless, we still lack consensus regarding how to quantify a lending relationship while simple statistics such as the frequency and volume of loans have been frequently
used. Here, we propose the concept of a significant tie to statistically evaluate the strength of a
relationship. Application of the proposed method to the Italian interbank networks reveals thatthe percentage of relationship lending among all bilateral trades is consistently around 20%–30%
and that their trading properties are distinct from those of transactional trades.
Keywords: Relationship lending, interbank markets, temporal networks
JEL Classifications: G10, G21

1Introduction

The role of a strong relationship between a lender and a borrower, the so-called relationshiplending (or relationship banking), is one of the most widely discussed issues in theoretical andempirical studies of banking. Many empirical studies investigate the economic impact of relationship lending on the terms of loans, such as interest rates and the amount of funds lent,aiming to test the theoretical implications that have been provided since the early 1990s (Sharpe,
1990; Rajan, 1992; Elyasiani and Goldberg, 2004; Freixas and Rochet, 2008). In particular, relationship lending is considered to play an important role in providing liquidity to borrowers
facing credit constraints by reducing the extent of information asymmetry between lenders and1borrowers. On the other hand, borrowers in relationship trades could be “locked-in” by lenders
due to their exclusive acquisition of private information, leading to a hold-up problem (Petersen
and Rajan, 1995; Von Thadden, 2004; Freixas and Rochet, 2008).
Although a large fraction of the previous studies on relationship lending investigated thebilateral relationship between a bank and a non-financial firm (Sette and Gobbi, 2015; Kysucky
and Norden, 2015), other studies explored the role of relationship lending in the interbank
market, where banks lend to and borrow from each other (Furfine, 1999; Cocco et al., 2009;
Aﬃnito, 2012; Craig et al., 2015; Hatzopoulos et al., 2015; Bräuning and Fecht, 2017). For
example, Cocco et al. (2009) showed that in the Portuguese interbank market, bilateral trades
made by banks with stronger relationships tend to exhibit lower interest rates. In Italy, Aﬃnito(2012) found that relationship lenders played an essential role as liquidity providers, especiallyin the midst of the global financial crisis of 2007–2009. Bräuning and Fecht (2017) argued thatduring the financial crisis, relationship lenders in Germany oﬀered lower interest rates to theirclose partners.
The results of the previous analyses, however, are based on ad-hoc and simple measures ofrelationship lending, and their simplicity may cause a mismeasurement error, especially when
there is heterogeneity in banks’ activities. A naı̈ve measure of relationship lending is the numberof transactions between two banks conducted during a certain period of time (Furfine, 1999;
Bräuning and Fecht, 2017). Another widely used measure is the degree of concentration in
lending (Cocco et al., 2009; Afonso et al., 2013), measured by the share of funds lent to aparticular counterparty. These two measures are expected to straightforwardly capture thestrength of a bilateral relationship in the interbank market; a bank pair engaging in relationshiplending would trade more frequently and devote a larger share of their total trading volume
to the trades between them than to trades with other banks. However, these measures might
misinterpret the strength of lending relationships. First, the number of trades with a particularcounterparty may merely reflect a bank’s need to trade in the interbank market. For instance, if
two banks have strong needs to provide and obtain overnight liquidity in the interbank market,
respectively, these banks are likely to trade by chance even if they have no preferences for trading
partners. Second, the degree of concentration in lending can be aﬀected by the diﬀerence inthe balance-sheet size of counterparties. For example, suppose that a large bank demands agreater amount of funds than smaller banks do. If a bank lends to the large bank, the degreeof concentration in lending may appear to be large, even though the lending bank has nopreference for partners. The share of lending volume to a particular partner could correctly
capture relationship lending if all the counterparties had the same liquidity demands. Given
these limitations, we need a more carefully designed measure of relationship lending that willallow us to control for these factors.
In this work, we propose the concept of a significant tie as a quantitative definition ofrelationship lending. Two banks are said to be connected by a significant tie if the number2

of trades between them is too large to be explained by random chance after controlling fortheir intrinsic activity levels. We control for the activity levels of banks by employing a simple
network-generative model as the null model. The so-called fitness model, one of the standard
network-generative models in network science (Caldarelli et al., 2002; De Masi et al., 2006;
Musmeci et al., 2013), considers a situation in which the probability of matching two banks isgiven by a function of their activity levels (i.e., fitnesses) independently of the history of theirtransactions. This history-independent property enables us to explicitly compute the theoretical
distribution of the number of bilateral trades under the null hypothesis that there is no preference
for partners, thus allowing for statistical tests. In this paper, we regard a bank pair connectedby a significant tie as engaging in relationship lending. This definition would eliminate thepossible mismeasurement of relationship lending due to the diﬀerence in banks’ activity levels,
which should reflect their liquidity demands and balance-sheet sizes.
We apply the proposed identification framework to data on over one million interbank transactions conducted in the Italian interbank market (e-MID) during 2000–2015. The results reveal
important facts about relationship lending, some of which can be summarized as follows. First,
throughout the data period, the percentage of relationship lending among all bilateral transactions is around 20%–30%, although the percentage slightly increased at the time of financial
distress (e.g., the terrorist attacks in 2001 and the global financial crisis in 2008–2009). Second,significant ties tend to last for longer periods than non-significant ties do, which is consistent
with the conventional notion of relationship lending. Interestingly, the duration of relationships
has a decreasing hazard rate (i.e., the probability of ending a relationship is decreasing in duration). This implies that the value of relationships in the interbank market increases in time,
contrary to the finding of Ongena and Smith (2001) on bank–firm relationships. Third, theinterest rates for relationship lending are indistinguishable from those for transactional lending
before and after the global financial crisis, but in the midst of the crisis the borrowers of relationship lending paid considerably higher interest rates than the average. This suggests thatsome banks faced a hold-up problem at the time of financial distress. Fourth, the chance thata bank pair is connected by a significant tie is aﬀected little by the nationality of the banks,suggesting the absence of home-country bias in building trading relationships.
The rest of the paper is organized as follows. In section 2, we explain the data on interbank
transactions. The method for identifying relationship lending is described in section 3, and theresults are shown in section 4. Section 5 concludes.

2

Data

We use time-stamped data on interbank transactions conducted in the Italian online interbank
market (e-MID) between September 2000 and December 2015.1 The data contain detailed
1

The e-MID data is commercially available from e-MID SIM S.p.A based in Milan, Italy (http://www.e-mid.it/).

3

800
150
600
100

400

50

0

200

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

0

years

Figure 1: Evolution of interbank networks. Black solid and blue dotted denote the number of active banks N (leftaxis) and the number of edges E (right axis) of a daily network, respectively. The largest (November 23, 2000), a
middle-sized (October 29, 2008) and the smallest networks (August 15, 2013) are visualized.

information on every bilateral transaction: date and time (e.g., “2000-09-04 09:12:40”), thenationality and IDs of banks (e.g., “IT0002”, where “IT” denotes Italy), maturity, interest rates,
and trade amount. In this work, we only use the overnight transactions of unsecured Euro
deposits labeled as “ON” (i.e., overnight) or “ONL” (i.e., overnight large, namely overnighttransactions no less than 100 million Euros), which comprise the great majority of transactions(> 86%) in the e-MID market. An advantage of focusing on overnight trades is that we canconstruct a sequence of snapshots of daily interbank networks having banks as nodes and lendingborrowing relationships as edges (Fig. 1). An edge is created when a loan is made. If there
are multiple transactions between two banks during a day, we represent the trading relationshipas one unweighted edge. As a result, the number of edges over the whole data period totals
1,033,349.
From Fig. 1, it is evident that interbank networks constantly change their size on a daily
basis, and there is a common downward trend in the numbers of active banks N and edges E.
Here, “active” banks in a daily network are defined as banks that had transactions at least once
between 9:00 and 18:00. Downward spikes in N and E are mostly due to national holidays in
Italy.2 On the other hand, the presence of a long-term downward trend could be attributed tomultiple factors such as the onset of the global financial crisis, the Greek sovereign debt crisis,
and the introduction of highly expansionary monetary policies of the the European Central
Bank (ECB) (and possibly other central banks). Summary statistics of the time series of dailyinterbank networks are presented in Table 1.
2

Weekends are not included since the market is closed.

4

Table 1: Summary statistics of the daily interbank networks. Symbol x denotes the average of variable x over thecorresponding period, and ⟨k⟩ is the daily average degree. Subscripts “max” and “min” represent the maximum andminimum values, respectively.

All

2000–2006

2007–20092010–2015# days

3,922

1,618

767

1,537

N

95.80

130.40101.67

56.45

Nmax

161

161

144

89

Nmin

13

56

48

13

E

262.96402.16266.00114.91

Emax

662

662

461

265

Emin

15

122

76

15

⟨k⟩

2.54

3.07

2.57

1.97

3

Model and methods

3.1

Fitness model

As a baseline framework for the subsequent statistical analysis, we introduce here a simple
model of daily interbank networks that describes how a lender and a borrower are matched.
Our model is a variant of the fitness model (Caldarelli et al., 2002; De Masi et al., 2006). The
fitness model has been frequently used in the field of network science to explain the mechanismof dynamic network formation, in which the probability that two agents are connected depends
on the fitness of the agents. In the context of interbank markets, fitness corresponds to theintrinsic activity level of a bank, such as the demand for short-term liquidity if the bank is apossible borrower and the willingness to supply funds if the bank is a possible lender. In spiteof its simplicity, the fitness model has been shown to explain many rich properties that emerge
from the evolution of interbank networks (De Masi et al., 2006; Kobayashi and Takaguchi, 2017).
In this study, we regard daily interbank networks as undirected (i.e., we ignore the direction ofedges) because our main focus is on identifying and analyzing the role of the bilateral relationshipbetween banks. We assume that the probability u that bank i trades with bank j on a given
day is expressed by the product of their activity levels:
u(ai , aj ) ≡ ai aj ,(1)

where ai > 0 represents the activity level (or fitness) of bank i.3 The model nests a wide variety
of well-known network generating models, depending on the specification of {ai }. For example,
3

In Kobayashi and Takaguchi (2017), we used a matching function of the form u(ai , aj ) = (ai aj )α . In the current

5

if ai = a ∀ i, then the model is equivalent to an Erdős-Rényi random graph with constant√matching probability u = a2 (Erdős and Rényi, 1959). If ai = ki / 2M , where ki and M are
the degree of bank i and the total number of edges in a daily network, respectively, then thematching probability is given by u = ki kj /(2M ), resulting in the configuration model (Newman,2010).4
We first estimate the activity vector a ≡ (a1 , . . . , aN ), assuming that every element of a isconstant during an aggregate period consisting of τ consecutive business days. In other words,
daily networks in an aggregate period are regarded as independent realizations from the fitness
model with estimated a. In short, we are extracting a N × 1 vector of bank activity levels
from the observed network structure containing N × N elements of information on bilateral
trades (i.e., adjacency matrix). This dimensionality reduction obviously discards the structural
information of a network. In return, the resultant estimates enable us to infer the extent towhich a random matching between banks can explain the empirical network structure, avoiding
an over-identification problem. Based on the estimates of a, we identify the existence of relationship lending by testing whether the observed number of transactions between two banks issignificantly larger than the value expected by the null hypothesis (i.e., the fitness model).

3.2

Maximum likelihood estimation of activity levels

We split the daily data set into aggregate periods, each consisting of τ business days, andperform a maximum likelihood estimation of a period by period. Aggregate periods are indexedby t′ = 1, . . . , t′max , where t′max ≡ ⌊tmax /τ ⌋ and tmax denotes the total number of business daysin the data. For the sake of simplicity, we omit subscript t′ in the rest of this section.
If trading pairs are independently matched each day according to probability u(a, a′ ), then
the number of trades between banks i and j conducted over τ business days follows a binomial
distribution with parameters τ and u(ai , aj ). For a given activity vector a, the joint probability
function of the number of trades in an aggregate period then leads to()∏τ
p({mij }|a) =u(ai , aj )mij (1 − u(ai , aj ))τ −mij ,mij
i,j:i̸=j(2)

where mij ≤ τ denotes the number of trades (i.e., edges) between i and j observed in an
aggregate period. The log-likelihood function is thus given by
L(a) = log p({mij }|a)∑=[mij log (ai aj ) + (τ − mij ) log (1 − (ai aj ))] + const.,(3)

i,j:i̸=jmodel, we can set α = 1 without loss of generality because the case of α ̸= 1 can be recovered by redefining theactivity parameter as aα .
4
The configuration model is a network model that generates a random network having a prescribed degree sequence{ki }. See Newman (2010) for details.

6where “const.” denotes the terms that are independent of a. Let N denote the number ofactive banks that have at least one transaction during a given aggregate period. The maximumlikelihood estimate of a is the solution for the following N equations:
Hi (a∗ ) ≡∑ mij − τ (a∗i a∗j )j:j̸=i1 − (a∗i a∗j )= 0, ∀ i = 1, . . . , N,

(4)

The first-order condition (4) is obtained by diﬀerentiating the log-likelihood function Eq. (3)with respect to ai . The system of nonlinear equations, H(a) = 0, can be solved by using astandard numerical algorithm.5 Hereafter, the computed solution (i.e., the maximum likelihood
estimate) of a is denoted by a∗ ≡ (a∗1 , . . . , a∗N ). By repeating this process period by period, we{}
obtain the estimates of activity vectors a∗1 , a∗2 , . . . , a∗t′max .

3.3

Statistical tests for relationship lending

Here we present two sorts of statistical tests; one is for identifying bank pairs engaging in
relationship lending and the other is for detecting relationship-dependent banks. In the samemanner in which estimate the activity levels of banks (section 3.2), we split the daily data set′
aggregate periods and implement the tests period by period.
into tmax

3.3.1

Edge-based test for relationship lending

If bank i has no preference for trading partners and thereby finds a partner in a random manneras suggested in the fitness model, then the number of bilateral transactions between banks iand j during a given period, mij , should obey the following binomial distribution:(g(mij |a∗i , a∗j ) =

τ
mij

)
u(a∗i , a∗j )mij (1 − u(ai∗ , a∗j ))τ −mij , ∀ i, j = 1, . . . , N.

(5)

In contrast, if bank i has a strong (i.e., non-random) partnership with bank j, then thecdistribution of mij will deviate from a binomial distribution. Let mij
denote the c-th percentile(0 ≤ c ≤ 100) of g(mij |a∗i , a∗j ) (i.e., c/100 = G(mcij |a∗i , a∗j )), where G is the cumulative distribution function (CDF) of g(mij |a∗i , a∗j ). If mij > mcij for a c value close to 100, then the empiricalnumber of transactions is too great to be explained by random chance under the null hypothesis,
indicating the presence of relationship lending. We call this test the edge-based test since this isa test for the significance of edges in interbank networks. If mij > mcij , then we say that banks
i and j are connected by a significant tie and engaging in relationship lending. We set c = 99
5

We solved the problem by using the Matlab function fsolve, which is based on a modified Newton method,

called the trust-region-dogleg method. The initial values of a are given by the configuration model, ai =√ ∑∑2 i<j mij /τ , where the numerator and the denominator represent the daily means of bank i’s
j:j̸=i (mij /τ )/degree and the doubled number of total edges, respectively. There are a few cases in which the estimated activity
values ai and aj indicate u(ai , aj ) > 1. In such cases, we assume u = 1.

7

Random matchingbank i

+

=

+

X

2 1

mij = 5

j:j6=i2bank jRelationship lending

+

=

+

1

3
1

t=1

t=2

t=3

Figure 2: Schematic of a significant tie. For illustrative purposes we set τ = 3. The size of circle represents theactivity level of a bank. If banks are matched randomly according to the fitness model, then banks with higher
activity levels will receive larger number of edges on average. If the number of trades between bank i and bank j is
too great to be explained by random chance, then the two banks are considered to be connected by a significant tie
and engaging in relationship lending.

(i.e., 99% significant level) throughout the paper. A schematic of a significant tie is presented
in Fig. 2. Importantly, the number of bilateral trades in a given period does not necessarily
indicate the presence of a significant tie. Under a random matching (the upper row of Fig. 2),bank i trades twice with each of the two counterparties having the largest activity levels, whichshould be a natural consequence given the high matching probabilities. By contrast, banks iand j trade three times in the bottom row of Fig. 2, which is unexpected based on their small
activity levels. Therefore, bank i is considered to engage in relationship lending with bank j
but not with the other three.

3.3.2

Node-based test for relationship-dependent banks

Since we have random matching probabilities u(a, a′ ), we can also test the extent to which abank depends on a limited number of partners. The probability function of aggregate degree
Ki is given asf (Ki |a∗ ) =∑
Aij , Ki g(mij = 0)1−Aij (1 − g(mij = 0))Aij × δ ∑ ∏{Aij } j:j̸=i

=

∑ ∏

j



∑
Aij , Ki  , (6)(1 − u(a∗i , a∗j ))τ (1−Aij ) (1 − (1 − u(a∗i , a∗j ))τ )Aij × δ {Aij } j:j̸=ij8

where Aij is the (i, j)-element of the aggregate adjacency matrix; Aij = 1 if there is at least
one transaction between banks i and j during an aggregate period, and Aij = 0 otherwise.
δ(x, y) denotes the Kronecker delta which equals one if x = y and zero otherwise. Note that thesecond equality follows from relation g(mij = 0) = (1 − u(a∗i , a∗j ))τ (Eq. (5)). In fact, Eq. (6)is equivalent to the distribution of the sum of N − 1 random variables drawn from a Bernoulli
distribution with parameter {1 − (1 − u(a∗i , a∗j ))τ }j:j̸=i , or a Poisson binomial distribution. Here
we would like to compute the CDF of f (Ki |a∗ ) to evaluate the significance of empirical Ki .
However, exact calculation of the CDF of a Poisson binomial distribution is notoriously diﬃcult(N )because one must compute K
number of terms (Steele, 1994). Thus, we instead approximateithe probability distribution of Ki to a Poisson distribution (Le Cam, 1960):∗

f (Ki |a∗ ) ≈where λ∗i ≡

∑

∗ ∗ τ
j:j̸=i [1 − (1 − u(ai , aj )) ].

i −λi
λK
i e
≡ fe(Ki |a∗ ),
Ki !

(7)

An error bound for this Poisson approximation isprovided by an extended version of the Le Cam’s theorem (Le Cam, 1960; Barbour and Eagleson,
1983; Steele, 1994):
∞
∑

i −λi
λK
2(1 − e−λi ) ∑ 2i epij , ∀ i, j,
<
Ki !
λ∗i
∗

∗

f (Ki |a∗ ) −

Ki =0

(8)

j:j̸=iwhere pij ≡ 1 − (1 − u(a∗i , a∗j ))τ .
The Poisson approximation enables us to formally test the null hypothesis that the empirical′aggregate degree Ki is explained by random chance. Let Kic denote the c′ -th percentile (0 ≤c′ ≤ 100) of fe(Ki |a∗ ). In other words, c′ /100 = Fe(Ki |a∗ ), where Fe(Ki |a∗ ) is the CDF of′fe(Ki |a∗ ). If the data reveal that Ki < Kic for c′ close to zero, then bank i has a significantly

smaller number of trading partners than random chance would suggest. If this is the case, it
indicates a significant dependence of bank i on relationship lending. Hereafter we call this typeof test the node-based test, and we set c′ = 1.

3.4

Selection of aggregate length τ

Before applying the model and statistical tests described in the previous sections to empirical
data, we must determine parameter τ , the length of an aggregate period. In fact, varying τ
would cause trade-oﬀs between approximation accuracy and the stability of aggregated data.
On the one hand, the choice of τ would directly aﬀect the accuracy of the Poisson approximation
through its influence on λ∗i and pij in Eq. (8). The average error bound (Eq. (8)) increases withτ as limτ →∞ pij = 1, ∀ i, j (Fig. S1a in Supplementary Information). Taking into account thispositive relationship between the error bound and τ , τ should be set as small as possible. On theother hand, employing a smaller value of τ would also aﬀect the stability of statistical results as
the aggregate networks could become more unstable because the number of active banks wouldchange drastically period to period (see Fig. 1). This necessarily reduces the stability of the

9

0.8
0.7

activity

0.6
0.5
0.4
0.3
0.2
0.1
0

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

years

Figure 3: Maximum-likelihood estimates of activity. Solid line represents median, and lower and upper dotted lines
respectively denote the 5th and 95th percentiles of the estimated activity distribution in each aggregate period.

data to be examined. Figure S1b illustrates that the average and the standard deviation of theabsolute changes in N , denoted by ∆Nt′ ≡ |Nt′ −Nt′ −1 | (t′ = 2, . . . , t′max ), take minimum values
around τ = 12. Judging from these observations, we employ τ = 10 as a benchmark value. We
note that all the qualitative results shown in this paper are quite robust and not sensitive tothe choice of τ .

4

Results

4.1

Estimation results: Activity level

The distribution of the estimated activity levels, a∗ , is shown in Fig. 3. The distribution has
been relatively stable throughout the data period. Based on those estimates, we can infer
how many transactions would be conducted under the null hypothesis in which the matchingprobability is given by u(a∗i , a∗j ) ∀ i, j. The empirical number of transactions in an aggregate
period, denoted by M , is given as
M=

∑

mij .

(9)

i<j

The expected number of transactions under the null hypothesis M ∗ is given asM∗ = τ∑u(a∗i , a∗j ).

(10)

i<j

Figure 4a illustrates the relationships between N and M in the empirical data and the estimatedmodel. The almost perfect fit between the estimated values of M ∗ and the empirical data
indicates that the maximum likelihood estimation works fairly well; the estimated activity level

10

a 10

b 10

4

2

Real data
Model

103

101

50

100

150

200

50

100

150

200

Figure 4: Comparison between the model and the empirical values of M and K. Each dot corresponds to an aggregate
period. (a) The total number of edges predicted by the model (Eq. (10)) well fits the empirical data. (b) The numberof unique partners predicted by the model (Eq. (12)) overestimates the empirical data.

accurately captures the actual bank activity in terms of the total number of trades. In Kobayashi
and Takaguchi (2017), we showed that there is a clear superlinear relationship between thenumbers of banks and edges at the daily scale (i.e., τ = 1) using the same data. Figure 4a infact reveals that a similar scaling relation arises even at the aggregate level of τ = 10 businessdays.
On the other hand, if we take the presence of relationship lending as a given, the empiricalnumbers of trading partners should be smaller than the estimate values under the null hypothesis. To see this, Fig. 4b shows the average of aggregate degree K, the number of unique trading
partners in an aggregate period:
K=

1 ∑
Aij .
N i,j

(11)

Under the null hypothesis, the average aggregate degree is computed as
K∗ =

1 ∑
[1 − (1 − u(a∗i , a∗j ))τ ].
N i,j

(12)

As shown in Fig. 4b, K ∗ overestimates K, meaning that in the real world banks tend to be more
selective than a random matching would suggest. In the next section, we identify the presenceof relationship lending by statistically testing the extent of deviation from the null model.

4.2

Identification results: Significant ties and relationship-dependentbanks
Figures 5a and 5c, respectively, show the number and fraction of significant ties identified by theedge-based test. We also checked the robustness of the results to the choice of τ in SI (Fig. S2).
Overall, while the number of significant ties has been decreasing along with the downward trend

11

a

b

Edge-based test

Node-based test

# rel.-dependent banks

100

400
300
200
100
0

c

d

frac. significant ties

1
0.8
0.6
0.4
0.2
0

80
60
40
20
0

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

frac. rel.-dependent banks# significant ties

500

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

1
0.8
0.6
0.4
0.2
0

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

years
Figure 5: Statistical identification of relationship lending. The number of (a) identified pairs by the edge-based testand (b) nodes identified by the node-based test. The fraction of the identified (c) pairs and (d) nodes. The results
are at 1% significant level in both tests (i.e., c = 99 in the edge-based test and c′ = 1 in the node-based test).

of E (see Fig. 1), the percentage of significant ties among all ties is consistently around 20%–
30% with some fluctuations over time. More precisely, in the early 2000s, fewer than 20% ofbilateral relationships are considered significant ties, and the percentage gradually rises until
2009, when the financial markets suﬀered from a fear of liquidity shortage. In the midst of
the global financial crisis, more than 30% of bank pairs are connected by significant ties. The
percentage then gradually decreased until around 2014. In June 2014, the ECB lowered its key
policy rate to a negative value for the first time in history, which may have aﬀected the trading
patterns of banks such that they became more likely to trade with particular partners than withrandomly selected ones.
Figures 5b and 5d show the number and share of relationship-dependent banks identified by
the node-based tests, respectively. As in the case of significant ties, the share of relationshipdependent banks increased drastically after the September 11 terrorist attacks in 2001 and after
the failure of Lehman Brothers in September 2008. We note that the results of the node-based
tests should be treated with care; the fraction of relationship-dependent banks increases withτ while the fraction of significant ties is almost unaﬀected (Fig. S2). A possible reason for thisdependence on τ is a deterioration in the accuracy of the Poisson approximation (Eq. (8)) as
described in section 3.4. Although the absolute values of the fraction of relationship-dependent

12

a

b

1

0.8

0.8

0.6

0.6

0.4

0

0.4

among all ties
among significant ties

0.2

Fraction

10.2
001 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

c

d

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

0

years

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

among all banks
among relationship-dependent banks
01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

Figure 6: Fraction of (a) Italian-Italian, (b) Italian-foreign, and (c) foreign-foreign bank pairs. Black solid andred dotted lines indicate the fractions of the corresponding pairs among all pairs and among relationship pairs,
respectively. (d) Fraction of Italian banks among all banks (black solid line) and among relationship-dependentbanks (blue dotted lines).

banks vary with τ , the relative trends over the data period appear similar.
As we saw in Fig. 5, the proposed methods (section 3.3) allow us to statistically identify bankpairs engaging in relationship lending and relationship-dependent banks. It is worth noting that
this would not be possible without an appropriate null model, which was missing in previous
studies (see section 4.4 for an evaluation of the previous measures for the strength of relationshiplending).
Information regarding banks’ country IDs enables us to investigate the correlations between
banks’ nationality and the existence of a significant tie and between nationality and the chance ofbeing a relationship-dependent bank. Since Italian banks occupy a great majority in the e-MID
market, we split all ties into three combinations of nationalities: Italian-Italian, Italian-foreign,and foreign-foreign pairs.6
As shown in Fig. 6, the fraction of Italian-Italian pairs among all pairs was close to one
in the early 2000s, yet it considerably decreased toward the onset of the global financial crisis
in 2007–2008. At the same time, Italian-foreign and foreign-foreign pairs started to increase
their presence over the pre-crisis period. The fraction of Italian-Italian pairs began to increaseagain shortly after the financial crisis occurred, gradually returning to its pre-crisis level. This6

The list of all countries is as follows (the number of banks is in parenthesis): Austria (2), Belgium (6), Switzerland(6), Germany (23), Denmark (1), Spain (7), Finland (1), France (10), Great Britain (14), Greece (6), Ireland (5),Italy (213), Luxembourg (4), Holland (4), Norway (1), Poland (1), and Portugal (4).

13April 23, 2001
— May 7, 2001

June 12, 2007
— June 25, 2007

June 23, 2014
— July 4, 2014

All ties

Significant ties

convertRelationshipdependent banks

Figure 7: Visualization of aggregate networks. In the upper and middle rows, red and black circles represent
Italian and foreign banks, respectively. In the bottom row, relationship-dependent banks are denoted by blue circles.
The visualization is done by igraph package for Python (http://igraph.org/python/), using the Kamada-Kawai
algorithm (Kamada and Kawai, 1989).

seems to suggest that an Italian bank tends to trade with other Italian banks when the market
is under stress. However, the share of Italian-Italian significant ties among all significant ties
moved in sync with the fraction of Italian-Italian pairs among all pairs, suggesting the absence ofhome-country bias in creating significant ties. Somewhat counterintuitively, Italian-Italian pairs
are less likely to form significant ties than Italian-foreign and foreign-foreign pairs are. When it
comes to the fraction of relationship-dependent banks (Fig. 6d), the trend over the data period
is similar to that of the fraction of Italian-Italian pairs (Fig. 6a). In particular, the percentage
of Italian banks among all relationship-dependent banks is no less than 80% throughout thedata period. The deviation between the two lines in Fig. 6d suggests that the probability of
becoming a relationship-dependent bank has been higher for Italian banks than for non-Italianbanks at least until around 2012.
Figure 7 presents a visualization of networks observed in diﬀerent aggregate periods. In theearly 2000s, there is no clear cut of groups since most active banks are Italian and they are

14

a

b

c

d

e

f

Figure 8: Duration of a lending relationship. (a)–(c): red triangle (blue circle) denotes the complementary cumulative
distribution function (CCDF) of the length of consecutive periods in each of which a bank pair is connected by asignificant tie (a non-significant tie). In panel (a), the slope of the CCDF is also shown (black solid), which isestimated by the maximum-likelihood method proposed by Clauset et al. (2009). (d)–(f): histogram of the total
number of periods in which a bank pair is connected by a significant tie (or a non-significant tie).

well connected to each other. We observe a similar situation when we construct the networkof significant ties only. By contrast, in a period shortly before the financial crisis, apparently
there exist two tightly connected groups of banks, one formed by Italian banks and the otherby foreign banks. This observation is explained by the result shown in Fig. 6; the fraction offoreign-foreign pairs reached its peak in 2007 while the fraction of Italian-foreign ties began todecrease in 2006. The two groups can be seen more clearly if we leave significant ties only since
just a few significant ties connect Italian and foreign banks in this period. In 2014, the network
looks similar to that in 2001, but the numbers of active banks and edges are much smaller in2014 than in 2001. In addition, the share of relationship-dependent foreign banks is relatively
larger in the period during the crisis than in the pre- and post-crisis periods, although the vast
majority of relationship-dependent banks are still Italian banks.

4.3

Role of relationship lending

The previous sections confirmed the existence of significant ties in the empirical data. In thissection, we explore the diﬀerence in the outcomes of significant and non-significant ties in termsof their duration, trading conditions, and structural characteristics.

15

4.3.1

Duration and the value of partnership

If relationship lending is understood as a long-lasting relationship between banks, the durationof significant ties should be longer than that of non-significant ties. Here, the duration of a(non-)significant tie between two banks is defined as the length of consecutive periods in each ofwhich these banks form a (non-)significant tie between them. In fact, the duration distributionof significant ties has a fatter tail than that of non-significant ties (Fig. 8). The duration
distribution of significant ties follows a power law at least in the pre-crisis period (2000–2006).
This fat-tail behavior indicates that the longer the duration length, the more likely the current
partnership will continue (i.e., the hazard rate is decreasing). To see this, let P (d) = 1−(κ/γ)d−γ(κ > 0) be a continuous approximation of the CDF of duration length d. The hazard rate λ, orthe probability that a bank pair terminates their d−period relationship, leads toλ(d) =p(d)γ= ,1 − P (d)d(13)

where p(d) is the probability density function of d. It follows that during the pre-crisis period,
the hazard rate at duration length d is given by λ(d) ≈ 2.17 d−1 .
The decreasing hazard contrasts with the previous result for bank–firm relationships shownby Ongena and Smith (2001). They found that the probability of terminating a relationshipincreases in duration, arguing that the value of relationships decreases over time. Our result indicates that the opposite holds true for the interbank market; the value of interbank relationships
may increase over time. This is consistent with the traditional theory of relationship lendingthat supports the benefit of a long-term relationship (Freixas and Rochet, 2008), suggesting thatthe longer the duration of a partnership, the greater the extent of private information owned bya lender (Sharpe, 1990).
One might argue that the long duration of significant ties simply comes from the fact that relationship pairs tend to trade more frequently than non-relationship pairs do. However, Fig. 8d–f
reveals that the number of periods in which non-relationship pairs trade is larger than that ofrelationship pairs. Thus, the long duration of a significant tie is not attributed to the highfrequency of the pair’s trades.

4.3.2

Terms of trades and the substitutability of trading partners

In this section, we analyze the impact that the presence of a significant tie has on trade conditions(i.e., interest rates and the amount of loans). To control for the influences of shifts in the policy
rate and variations in the trading volume, we define the weighted average of detrended interest
rates on bilateral transactions between banks i and j as∑rawt∈Dt′ (rt,ij − ⟨rt ⟩)wt,ij∑
rt′ ,ij ≡,t∈Dt′ wt,ij

16

(14)

a

b

years

years

Figure 9: Impact of relationship lending on (a) interest rates and (b) trade amount. Solid line and red shading
indicate the average and the 95% confidence interval, respectively. In each panel, the diﬀerence is calculated bysubtracting the values for non-significant ties from those for significant ties.

where

∑
⟨rt ⟩ ≡

i<j
∑

raw
rt,ij
wt,ij

i<j wt,ij

,

(15)

raw
and rt,ij
is the raw interest rate. wt,ij is the total volume of funds traded between banks i andj on day t. Set Dt′ represents the set of dates t that belong to aggregate period t′ . The averageamount of loans per trade between banks i and j is defined as
W t′ ,ij ≡∑ wt,ij,
mt′ ,ij

(16)

t∈Dt′

where mt′ ,ij denotes the total number of trades between banks i and j during period t′ .
Figure 9 shows the diﬀerences in rt′ ,ij and W t′ ,ij between significant ties and non-significant
ties, calculated by subtracting the values for non-significant ties from those for significant ties.
The weighted interest rates are higher for relationship trades than for transactional trades byaround three to six basis points during the global financial crisis. This fact implies the presence ofimperfect substitutability of trading partners and that relationship lending played an importantrole in the management of liquidity (Aﬃnito, 2012).7 In interbank markets, it is occasionallyobserved that banks trying to meet urgent liquidity needs accept high interest rates to avoid
stigma even if they can borrow from the central bank at lower rates (Ashcraft et al., 2011; Ennis
and Weinberg, 2013). The result shown in Fig. 9a implies that those banks that played a role
as “lenders of last resort” were connected with their borrowers by significant ties.
The upward spike in the diﬀerence in interest rates observed around January 2012 is considered to be caused by a “longer-term refinancing operation (LTRO)” introduced by the ECB.
As pointed out by Barucca and Lillo (2015), the introduction of LTRO suddenly reduced the7

A price discrimination could occur if the maturity structures were diﬀerent between relationship and transactional

trades, but in this work we focus only on overnight transactions.

17

a

b 1
sig
fraction

T2=
T2 + T3

0.8

sig
sig
sig

+ sig

sig
sig

0.6
0.4
0.2
0

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

years

Figure 10: Trilateral relationship and significant ties. (a) Schematic of the probability Pnonsig ≡ T2 /(T2 + T3 ) thata triangle having two significant ties has a non-significant closing tie. Red edges labeled “sig” represent significant
ties. (b) Time series of Pnonsig and the fraction of {Tℓ }ℓ=0,1,2,3 among all triangles.

number of active banks and the volume of loans in the e-MID market. The decrease in the number of active banks might have undermined the substitutability of trading partners by limiting
the number of possible partners, leading to an increase in the price of loans for relationshipdependent banks.

4.3.3

Significant ties in trilateral relationshipIn the literature of social network analysis, it has been widely recognized that there is a tendency
that “friends of friends are friends” (Wasserman and Faust, 1994). This is called a triadic
closure since the two individuals having a friend in common often close the triangle (Easley and
Kleinberg, 2010). Many studies have revealed that triadic closure plays an important role inachieving social cooperation (Hanaki et al., 2007), determining the spread of a behavior acrossties (Centola, 2010), and understanding the long-term evolution of network structure (Lewis
et al., 2012), to name a few. Analogously, the purpose of this section is to see whether triadic
closures are also ubiquitous in the “friendship” network of banks. To be more precise, thequestion we address here is whether a significant tie is more likely to close a triangle of trading
relationships (i.e., trilateral relationship) than a non-significant tie, provided that the triangle
has at least two significant ties. This question is motivated by the well-known fact that triangles
in social networks are mostly made of three strong ties (Granovetter, 1973; Onnela et al., 2007;
Easley and Kleinberg, 2010). If a significant tie is more likely to close a triangle, it would
indicate a previously unknown similarity between financial and social networks. In contrast,
if a non-significant tie is more likely to close a triangle, then it would shed light on a uniquecharacteristic of financial networks.
To answer this question, we first need to count the numbers of triangles in the aggregate
networks having diﬀerent numbers of significant ties (see Appendix for the procedure of calculation). Let Tℓ denote the number of triangles having ℓ significant ties (ℓ = 0, 1, 2, 3) in an

18

aggregate network. The quantity we want to compute is schematically visualized in Fig. 10a;
if Pnonsig ≡ T2 /(T2 + T3 ) is significantly larger than the fraction of non-significant ties in thewhole network (i.e., the probability of placing a non-significant tie by chance), then the closing
tie of a trilateral relationship is more likely to be a non-significant tie than random chance would
suggest. Since the percentage of significant ties is roughly 20%–30% throughout the data period∑(Fig. 5), the fraction of non-significant ties, denoted by Snonsig ≡ |Inonsig |/ i<j Aij , where
Inonsig is the set of non-significant ties, turns out to be around 0.7 − 0.8, which becomes thebaseline for evaluating Pnonsig .
Figure 10b shows that Pnonsig is always above 0.9 except for a few aggregate periods, meaningthat a trilateral relationship having at least two significant ties tends to have a non-significant
tie as the closing tie. This observation is statistically verified by the t-test for the null hypothesis that the means of Pnonsig and Snonsig are equal, which is rejected with p-value < 0.001.
Figure 10b also illustrates the time series of {Tℓ }ℓ=0,1,2,3 normalized by T , the total number oftriangles in each period. The order T0 > T1 > T2 > T3 consistently holds true throughout thedata period. In addition, we see some trends in their relative shares; the share of T0 and T1
roughly move in opposite directions while the shares of T2 and T3 remain stable.
The result suggests that the local dynamics of tie formation in financial networks is quite
diﬀerent from that in social networks. While triangles of three strong ties are ubiquitous in
networks formed by human interactions, interbank networks do not exhibit such a property.
This is somewhat surprising because Kobayashi and Takaguchi (2017) reported that social andfinancial networks share many local- and global-scale properties. The current analysis thus
provides new evidence regarding how economic transaction patterns of banks are essentiallydiﬀerent from social interaction patterns of humans.

4.3.4

Intraday trading patterns

In the previous sections, we observed that bank pairs connected by significant ties exhibit
diﬀerent behaviors than other transactional pairs at a τ -day aggregate scale. In this section, we
explore intraday trading patterns to see if the existence of a significant tie has any impact on
trades at higher frequencies.
In Fig. 11, we observe subtle diﬀerences in the timing of intraday trading. A bank pairengaging in relationship lending tends to conduct a larger fraction of trades at early hours (9:00–
11:00) and a smaller fraction of trades after 15:00 than a bank pair engaging in transactional
trading (Fig. 11a). This diﬀerence in the timing of trades does not seem to have a considerable
impact on interest rates, but late-hour relationship trades resulted in slightly higher interest
rates than those of transactional trades until the crisis period (Fig. 11b).8 Nevertheless, we
still see a downward sloping term structure of intraday interest rates, which has been reported
8raw
The interest rate on the trade between banks i and j at time θ on day t is defined as rθ,t,ij = rθ,t,ij− ⟨rt ⟩, where

the superscript “raw” denotes the raw interest rate (before detrending) and ⟨rt ⟩ is defined in Eq. (15).

1909-10am 10-11am11-12am 12-1pm 1-2pm 2-3pm 3-4pm 4-5pm 5-6pm

0
9-10am 10-11am11-12am 12-1pm 1-2pm 2-3pm 3-4pm 4-5pm 5-6pm

time

50

50

150
100

non-significant tie
significant tie

200

-10
9-10am 10-11am11-12am 12-1pm 1-2pm 2-3pm 3-4pm 4-5pm 5-6pm

10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00

100

150

200

-10
9-10am 10-11am11-12am 12-1pm 1-2pm 2-3pm 3-4pm 4-5pm 5-6pm

-5

-5

5

10

0
9:00

0

non-significant tie
significant tie

10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00

0

5

10

0
9:00

0.01

0.01

0.03

0.04

0.02

non-significant tie
significant tie

0.02

0.03

0.04

10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00

0
9-10am 10-11am11-12am 12-1pm 1-2pm 2-3pm 3-4pm 4-5pm 5-6pm

50

100

150

200

-10
9-10am 10-11am11-12am 12-1pm 1-2pm 2-3pm 3-4pm 4-5pm 5-6pm

-5

0

5

10

0
9:00

0.01

0.02

0.03

0.04

amount of trade. In panels (a) and (b), the error bar indicates the 95% confidence interval.

and non-significant ties, respectively. (a) Frequency of trades in each time interval. (b) Mean deviation of interest rates from the daily mean. (c) Average

Figure 11: Impact of relationship lending on intraday interest rate and trade amount. In each panel, red dotted and black solid lines denote significant ties

c

b

a

Frequency

interest rate (bp)amount (million Euros)20

previously (Baglioni and Monticini, 2010, 2008; Abbassi et al., 2017).
It is evident from Fig. 11c that the positive diﬀerence in the trade amount between relationship and transactional lending tends to get larger as the market-closing time approaches. These
gaps in the interest rate and amount of trades may reflect the fact that those banks that mustobtain or release liquidity at the end of the market tend to rely on their partners to which they
are connected by significant ties.

4.4

Comparison with previous measures

In closing this section, we assess the previously proposed measures of relationship lending bycomputing the extent to which they are able to detect significant ties. A naı̈ve measure oflending relationship is the frequency of interactions between two banks (Furfine, 1999; Kysucky
and Norden, 2015; Bräuning and Fecht, 2017):
RLij,t′ ≡ log(1 + mij,t′ ),(17)which denotes the logarithm of the number of transactions between banks i and j conductedduring an aggregate period t′ . The second and more widely used measure is the lender–preference
index (LPI) (Cocco et al., 2009; Aﬃnito, 2012; Craig et al., 2015; Bräuning and Fecht, 2017):∑
t∈Dt′ wij,t∑
LPIij,t′ ≡ ∑.
(18)
j:j̸=it∈Dt′ wij,t
LPI captures the degree of concentration of lending to a particular partner.9 If the fraction offunds lent to a particular partner is high, then it would indicate the existence of relationshiplending. These two conventional measures are usually employed as explanatory variables oflinear regression models.
Now let us assess the accuracy of RL and LPI in terms of their detectability of significant ties.
Let Isig denote the set of significant ties identified by our edge-based test, which is treated as the“ground truth.” The fraction of significant ties among all ties in a given period is represented∑
as Ssig ≡ |Isig |/ i<j Aij . For each x = RL, LPI, let Ix be the set of bank pairs whose score of
x is ranked top Ssig % in the corresponding period. If measure x correctly reflects the strengthof a bilateral relationship, then the following Jaccard index will take a value close to one:Jx =|Ix ∩ Isig |, x = RL, LPI.
|Ix ∪ Isig |(19)

Figure 12 reveals that, somewhat surprisingly, RL outperforms LPI, although the definitionof LPI seems more sophisticated than that of RL as a measure of a lending relationship. Thismay be due to the fact that the degree of concentration of lending to or borrowing from aparticular bank does not necessarily relate to the number of trades with the bank, whereas ourdefinition of a significant tie generally favors a bank pair conducting a large number of trades.
9Note that the amount lent is equal to the amount borrowed in our setting since we only treat undirected networks.

21

1
0.8

LPI
RL

0.6
0.4
0.2
0

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

years

Figure 12: Comparison with the conventional measures of relationship lending. Black solid (red dashed) line represents
the “accuracy” of LPI (RL), defined by the Jaccard index (Eq. (19)). RL and LPI are given by Eqs. (17) and (18),respectively.

RL just reflects the number of bilateral trades regardless of their volume, but it is closer to ouridea of significant ties than LPI is. Of course, just counting the number of trades is not enoughbecause one must take into account the diﬀerence in banks’ activity levels. An observation of
repeated trades between two banks does not necessarily lead to the presence of a significant
tie because such repetitive trades may be explained by random chance if the two banks exhibit
high activity levels. Nevertheless, Fig. 12 shows that RL is far more appropriate than LPI as ameasure of relationship lending, although JRL ∼ 0.6 does not mean that RL is very accurate.

5

Conclusion and discussion

This work proposed a statistical test for identifying bank pairs that are engaging in relationshiplending by introducing the concept of a significant tie. The proposed identification test wasapplied to the Italian daily interbank networks formed by overnight transactions. The point of
our identification method is whether or not the number of trades between two banks can be
explained by random chance after controlling for the intrinsic activity levels of those banks. If
the number of trades is statistically significant (i.e., cannot be explained by random chance),then we say that the two banks are connected by a significant tie. We showed that the percentage
of significant ties among all ties has been constantly 20%–30% over the past years, while thenumber of significant ties itself has been declining along with the total number of trades in theinterbank market.
We found several important properties that distinguish relationship lending from other transactional lending. First, the duration of a significant tie is, on average, longer than that of anon-significant tie. We argued that this property indicates that the value of continuing a relationship increases in duration, as suggested by many theoretical studies (Freixas and Rochet,22

2008). Second, a trilateral relationship of three significant ties is quite rare. In fact, the lack of
triadic closures is a unique feature of interbank networks, which will distinguish the structuralcharacteristic of interbank networks from that of social networks, such as friendship networksof humans (Onnela et al., 2007; Easley and Kleinberg, 2010). Third, in the midst of financial
distress, banks in need of liquidity tend to rely on banks to which they are connected by significant ties even at the cost of high interest rates. This may be evidence that relationship lenders
play a role as the “lender of last resort” during financial turmoil while they impose high interest
rates, implying the existence of a hold-up problem. Fourth, there is no home-country bias increating significant ties.
We remark that our statistical tests for significant ties are quite general and thereby not limited to the use of identification of relationship lending in interbank networks. For example, they
could be applied to trading networks in the corporate bond market (Maggio et al., 2017) and themunicipal bond market (Li and Schurhoﬀ, 2014), searching for a hidden structure of significant
ties between dealers. Our methods could also be implemented on temporal social networks to
identify “friends” from a record of face-to-face interactions (Cattuto et al., 2010). The proposed
identification method could, in principle, be applied to various “friendship” networks in social
as well as economic systems.
The current work also provides a temporal-network analysis of the interbank market, whichis still scarce in the field of network science and in economics, with a few exceptions (Barucca
and Lillo, 2015; Kobayashi and Takaguchi, 2017). Understanding the dynamic formation ofinterbank networks is quite important because the structure of networks formed by overnightbilateral transactions drastically changes day to day, accordingly the risk of financial contagion
varies over time. While most of the studies on financial systemic risk are based on static
networks (Gai and Kapadia, 2010; Cont et al., 2013; Brummitt and Kobayashi, 2015; Nireiet al., 2016), in the real world the risk of financial contagion emerges on networks with timevarying structures. We hope that our work will advance our knowledge about the mechanism of
temporal financial networks, which could contribute to the real-time management of financial
stability.

Appendix: Counting the number of triangles
In section 4.3.3, we counted the number of triangles in an aggregate network to investigate therole of significant ties in a trilateral relationship. Computing the number of triangles having ksignificant ties, Tk , is straightforward if we exploit the power of adjacency matrix. First, the∑3
total number of triangles in the whole network is given by T = k=0 Tk = tr(A3 )/6, where tr(·)
denotes trace. This equality is based on the fact that the (i, j) element of An represents the23

number of paths from i to j that can be reached at exactly n steps. Therefore, the diagonal
elements of A3 contain the numbers of triangles. Second, the number of triangles formed bythree significant ties is given as T3 = tr(A3sig )/6, where Asig is an adjacency matrix of the network
consisting only of significant ties. Third, the number of triangles formed only by non-signifiant
ties leads to T0 = tr((A − Asig )3 )/6. Fourth, the number of triangles having exactly one andtwo significant ties, T1 and T2 , are obtained as follows:
1. Create a “signed” adjacency matrix Asigned , where (Asigned )ij = 1 if i and j are connectedby a significant tie, −1 if connected by a non-significant tie, and 0 otherwise.
2. Compute Tsigned = tr(A3signed )/6. This is equal to the diﬀerence between the number oftriangles having an odd number of significant ties and the number of triangles having aneven number of significant ties (i.e., Tsigned = (T1 + T3 ) − (T0 + T2 )).
∑3
3. Derive T1 and T2 by substituting T , T0 , and T3 into equations T = k=0 Tk and Tsigned =(T1 + T3 ) − (T0 + T2 ).
This procedure gives us Tk for k = 0, 1, 2, 3.

Acknowledgements
Kobayashi acknowledges financial support from the Japan Society for the Promotion of Science
Grants no. 15H05729 and 16K03551. Takaguchi was supported by JST ERATO Grant Number
JPMJER1201, Japan.

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27

Supplementary Information:“Significant ties: Identifying relationshiplending in temporal interbank networks”
Teruyoshi Kobayashi and Taro Takaguchi

1error bounda⌧ = 10

⌧ =5
2

2

1.5

1.5

1

1

0.5

0.5

0

0

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16⌧ = 15⌧ = 20

2

2

1.5

1.5

1

1

0.5

0.5

0

0

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

years

b

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

5

mean
std

4.5
4
3.5
3
2.5
2

0

10

20

⌧

30

40

50

Figure S1: Eﬀects of varying aggregate length τ . (a) An increase in τ makes it less accurate the Poisson approximationof a Poisson binomial distribution. Solid line represents the error bound indicated by the RHS of Eq. (8). Upper andlower dotted lines denote the maximum and minimum error bounds, respectively. (b) Mean and standard deviationof absolute changes in N .

2⌧ = 2001 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

frac. rel.-dependent banks

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

frac. rel.-dependent banks

1
0.8
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0

1
0.8
0.6
0.4
0.2
0

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

1
0.8
0.6
0.4
0.2
0

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

frac. rel.-dependent banks

frac. significant ties

Node-based test

frac. rel.-dependent banks⌧ = 15

frac. significant ties

⌧ = 10

frac. significant ties

⌧ =5

frac. significant ties

Edge-based test
1
0.8
0.6
0.4
0.2
0

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

1
0.8
0.6
0.4
0.2
0

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

1
0.8
0.6
0.4
0.2
0

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

1
0.8
0.6
0.4
0.2
0

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

years
Figure S2: Eﬀects of varying aggregate length τ . Left and right columns present the fraction of significant ties andthe fraction of relationship-dependent banks, respectively.

3