 Regular article
 Open Access
 Published:
Social dynamics of financial networks
EPJ Data Science volume 7, Article number: 15 (2018)
Abstract
Recurrent interactions between agents play an essential role in the organization of a dynamic complex system. While intensive researches have been done on social systems formed by human interactions, dynamical rules are not well understood in economic systems. Here we study the evolution of financial networks and show that repeated interactions between financial institutions taking place at the daily scale are characterized by social communication patterns of humans emerging at higher time scales. The “social” dynamics of financial interactions are highly stable and little affected by external shocks such as the occurrence of the global financial crisis. A dynamic network model based on random pairwise matching accurately explains the observed daily dynamical patterns. The observed similarity between social and financial interactions gives us previously unknown stylized facts about a financial system, which could lead to a deeper understanding of the fundamental source of systemic risk.
Introduction
Financial systemic risk is one of the most serious threats to the global economy. The global financial crisis of 2007–2009 showed that a failure of one bank can lead to a financial contagion through a complex web of financial linkages, which are created by everyday transactions among financial institutions [1–6]. Even after the crisis, many countries have experienced a prolonged recession, the socalled Great Recession, showing that the social cost of a financial crisis can be enormous [7, 8]. Evaluating and controlling systemic risk has therefore been recognized as one of the greatest challenges for interdisciplinary researchers across different fields of science [9–12].
In the modern financial system, interbank markets play a fundamental role, in which banks lend to and borrow from each other (hereafter, we refer to all types of financial institutions as “banks” for brevity). Lending and borrowing in interbank markets are necessary daily tasks for banks to smoothen their liquidity management [13], but at the same time, they also form the center of a global web of interconnected risk; shocks to interbank markets may spill over to other parts of the global financial system through the financial linkages to which banks are connected [10, 14, 15]. Many previous studies attempt to assess systemic risk by simulating different scenarios of cascading bank failures on both real [16–20] and synthetic interbank credit networks [15, 21–26]. Studies of financial cascades based on synthetic networks often assume a particular static structure, such as random [22, 23], bipartite [15, 27], and multiplex structures [25, 26], successfully revealing that the structural property affects the likelihood of default cascades to a large extent. However, since the great majority of realworld interbank transactions are in fact overnight [13, 28], interbank networks should be treated as dynamical systems with their structure changing on a daily basis. This temporal nature of real interbank networks inevitably limits the practical usefulness of the conventional static approach to systemic risk. Nevertheless, we still have little knowledge about how the structural characteristics of daily networks evolve over time. It has long been believed that the dynamics of interbank networks is random and thus has no meaningful regularity at the daily scale [29]. How the current network structure is correlated with the past structures, or more specifically, how banks choose current trading partners based on their trading history, is still unknown. The ambiguity of the structural dynamics of interbank networks itself may also become a source of systemic risk by veiling the complexity of interconnectivity [30]. The current lack of studies on the mechanics of real interbank networks is in stark contrast to the abundance of research on their static property [16, 19, 31–34].
The main aim of this work is to uncover fundamental dynamics governing real interbank networks at both local and systemwide scales. For this purpose, we first seek dynamical regularities that would characterize interaction patterns between individual banks by examining millions of overnight transactions conducted in the Italian interbank market during 2000–2015 [35]. We discover that there exist explicit interaction patterns that rule daily banktobank transactions, which turn out to be essentially the same as the patterns characterizing human social communication; that is, banks trade with each other in the same way that people interact with friends through phone calls and facetoface conversations [36, 37]. In fact, those “social” interaction patterns of banks have been surprisingly stable over time, even amid the global financial crisis. On top of local interactions between banks, there emerges a systemwide scaling relationship between the numbers of banks and transactions, just as the number of phonecall pairs scales superlinearly with the size of population [38].
To explain the origin of social dynamics in interbank networks, we develop a model that generates a sequence of synthetic daily networks from which the observed dynamical patterns simultaneously emerge at both local and network scales. Our discovery of the fundamental mechanism underpinning the daily evolution of interbank networks will enhance the predictability of systemic risk and provide an important step toward the realtime management of financial stability.
Results
The dataset to be analyzed in this work is the time series of daily networks identified from the timestamped data of interbank transactions conducted in the Italian interbank market during 2000–2015 (see Methods for data description). The daily interbank networks have directed edges originating from lending banks to borrowing banks. One may regard the amount of funds transferred from a lender to a borrower as edge weights, but here we regard the daily networks as unweighted, since we found that the dynamics of edge weights can be understood independently from the edge dynamics themselves (see Methods for the analysis of edge weights).
An important general observation regarding the dynamics of daily networks is that both the network size N and number of edges M have followed downward trends (Fig. 1(a) and (b)). This led the networks closer to a bipartite structure between pure lenders and pure borrowers [39] (Fig. 1(c) and Table 1), entailing a nonnegligible turnover in the set of banks participating in each daily network (see the turnover rate in Table 2 and Fig. S1(a) and (b) in supplementary materials (SM)). Table 2 summarizes the basic statistics, in which we divide the entire sample period into three subsample periods to observe whether a structural change around the global financial crisis in 2007–2009 is present.
Dynamical patterns of daily networks
The downward trends in N and M, along with the intermittent spikes, left a broad range of daily combinations \((N,M)\), which allows us to reveal a superlinearity, \(M\propto N^{1.5}\) (Fig. 2(a)). This suggests that the average degree of a daily network increases with order \(\sqrt{N}\), since \(\langle k\rangle=M/N\propto\sqrt{N}\). It should be noted that the fact that M is given as a powerlaw function of N is similar to a widely observed phenomenon in social networks, called superlinear scaling, in which the number of edges scales superlinearly with the number of nodes across different locations [38, 40–42].
In addition to the macroscopic dynamics of N and M, we also find microdynamics of individual edges: edge duration and interval time. We define duration τ as the number of successive business days on each of which a lender and a borrower performs at least one transaction (i.e., directionality is taken into account). Aggregated over all trading pairs, τ follows a powerlaw distribution whose complementary cumulative distribution function (CCDF, see Table 3 for the list of abrreviations hereafter) has an exponent between 2.5 and 2.9 (Fig. 2(b) and Fig. S2 in SM. The exponents are estimated using the method proposed in Refs. [43, 44]). Similar powerlaw distributions are observed when we redefine τ as the duration of individual banks’ successive trading activity either for lending, borrowing, or both (Fig. S3 in SM).
On the other hand, the interval time Δτ for a bank pair is defined as the interval length between two consecutive transactions during which the bank pair performs no transactions. In contrast to τ, Δτ does not follow a powerlaw distribution while it still shows a longtailed behavior (Fig. 2(c)). The interval distribution fits well with a Weibull distribution up to a certain cutoff level (Fig. 2(c), Inset. See Methods for details on the fitting method [45, 46]).
We observe that the distributions of τ and Δτ have been quite stable throughout the whole data period. This observation is notable, not only because N and M continually fluctuate at the daily resolution over the course of the decreasing trends (Fig. 1(a)), but also because a large fraction of the set of participating banks changes day to day (Fig. S1(a) and (b)). The high metabolism of the interbank market suggests that the stationarity of τ and Δτ is not necessarily attributed to the presence of steady relationships between particular banks.
The fact that banks’ behavior is characterized by heavytailed distributions of duration and interval time might imply that banks are social. Another property that reinforces the social aspect of banks is found in the growth rate of aggregate degree \(K(t)\), denoted by the average cumulative number of unique trading partners up to time t [37, 47]. The normalized aggregate degree, defined by \(K(t)/K(T)\), grows sublinearly in time (Fig. 2(d)), meaning that the rate at which banks find a new partner tends to decrease over time. Note that such sublinear growth patterns have been reported for the mobility pattern of mobilephone users [47] and for contact networks of human individuals formed via facetoface interactions [37].
Model of daily network evolution
The above findings show that the dynamical patterns of interbank transactions are robust across different periods, which leads us to consider that a universal mechanism generating daily interbank networks might exist. Here, we show that the emergence of these regularities can be reconstructed by a dynamical generalization of the fitness model [48, 49].
First, we show that variations in the system size of a simple fitness model can explain the empirical superlinear relation \(M \propto N^{1.5}\). For ease of exposition, suppose for the moment that the networks are undirected. In the fitness model, fitness value \(a_{i} \in[0,1]\) is assigned to bank i \((1 \leq i \leq N_{\mathrm { P}})\), where \(N_{\mathrm { P}}\) represents the potential market size, given by the number of banks that may perform transactions during a day. In the context of interbank transactions, fitness value \(a_{i}\) can be interpreted as the activity level, or willingness, of bank i to trade. The probability that an edge is formed between i and j is given by \(p_{ij} = (a_{i} a_{j})^{\alpha}\) \((\alpha\geq1)\). For each network generated by this rule, N and M denote the number of active banks with at least one edge (thus \(N \leq N_{\mathrm { P}}\)) and the total number of edges, respectively.
By generating model networks with \(N_{\mathrm { P}}\) varying from 20 to 300 for a given \(\alpha\in[2,8]\), there arises a scaling relation \(M \propto N^{\beta}\) with \(1 < \beta< 2\) (symbols in Fig. 3). In previous studies [48–50], a theoretical analysis of the fitness model predicted \(M \propto N^{2}\), which differs from both our empirical observations (Fig. 2(a)) and numerical simulations (Fig. 3). In fact, this discrepancy is explained by the presence of isolated banks. In this model, the probability of a bank being isolated (i.e., no edges attached), defined by \(q_{0}\), is given by a function of \(N_{\mathrm { P}}\):
where \(\mathrm{ B}(x,y)\) and \(\mathrm{ B}_{z}(x,y)\) are beta and incomplete beta functions, respectively (see Methods for derivation). Consequently, N and M are given by
Since \(q_{0}(N_{\mathrm { P}}) \to0\) as \(N_{\mathrm { P}}\to\infty\), \(N \simeq N_{\mathrm { P}}\) and \(M \simeq \langle(a_{i} a_{j})^{\alpha}\rangle N(N1)/2 \propto N^{2}\) hold true for a sufficiently large \(N_{\mathrm { P}}\), which recovers the quadratic scaling shown in the previous studies [48–50]. However, for the range of network sizes observed from the data, \(q_{0}(N_{\mathrm { P}})\) is not negligible. A combination of \((N,M)\) derived from (2) for given values of \(N_{\mathrm { P}}\) perfectly fits the simulation result (lines in Fig. 3).
While the empirical superlinear relation \(M \propto N^{1.5}\) can be explained by variations in \(N_{\mathrm { P}}\), this simple model is unable to reproduce the distributions of τ and Δτ (Fig. 2(b) and (c)) and the sublinear growth of \(K(t)\) (Fig. 2(d)), because these characteristics come from the effect of memory in the formation of links between banks [28]. To capture the memory effect, we now introduce a fluctuation in fitness \(a_{i}\). The dynamic network formation in our model proceeds by repeating the following two steps: (i) edge creation between banks and (ii) update of each bank’s activity level \(a_{i,t}\). We consider three bank types: pure lenders, pure borrowers, and bidirectional traders. Pure lenders (pure borrowers) are the banks that may lend to (borrow from) but never borrow from (lend to) other banks. To take into account the fact that the interbank structure is almost perfectly bipartite when the network size is small (Fig. 1(c)), we assume that bidirectional traders may lend only to pure borrowers and borrow only from pure lenders. The fraction of each bank type is given as \((f_{\mathrm{ L}}, f_{\mathrm{ B}}, f_{\mathrm{ D}})=(0.56, 0.34, 0.1)\) for pure lenders, pure borrowers, and bidirectional traders, respectively, based on the empirical average (Table 1). The type assigned to each bank is fixed throughout the simulation.
At the beginning of the edgecreation stage in day t, the interbank system comprises \(N_{\mathrm { P}}\) isolated banks without any edges. Bank i lends to bank j on day t with probability \(p_{ij,t}\), given by
where L, B, and D denote the sets of pure lenders, pure borrowers, and bidirectional traders, respectively. After applying this procedure to every combination of \((i,j)\), we remove all the edges and move on to day \(t+1\). At the beginning of day \(t+1\), the activity level of bank i is updated as follows. With probability \(h(a_{i,t})\), \(a_{i,t+1}\) is reset to a random value drawn from the uniform distribution on \([0,1]\), where \(h(a_{i}) \equiv c_{1}^{1}a_{i}^{c_{2}}\). This reset probability is intended to capture the metabolism of the interbank market, in which some banks exit after continuous transactions while other banks enter after a long resting period. With probability \(1h(a_{i,t})\), the activity level is updated according to a random walk process on the unit circle, given by
where \(\theta_{i,t+1}\) is a randomwalk variable that describes the angle on the unit circle (see Fig. 4 for a schematic). Since an activity level must be on \([0,1]\), \(a_{i,t+1}\) is given by the absolute values of \(\cos{\theta_{i,t+1}}\). \(\varepsilon _{i,t+1}\) is a random variable uniformly distributed on \([0.002,0.002]\). The introduction of fluctuations in \(\varepsilon_{i}\) (and therefore in \(\theta_{i}\) and \(a_{i}\)) would capture variations in money demand that banks are facing on a daily basis in practice. We note that assuming a random walk on the real line would not be able to replicate the empirical duration of bilateral transactions. The volatility of ε is set such that the simulated bank turnover would well match the empirical one (Fig. S1). The initial value for the angle is set such that \(\theta_{i,0} = \arccos(a_{i,0})\), where \(a_{i,0}\) is drawn from the uniform distribution on \([0,1]\). The above two steps, edge creation and activity updating, are repeated until reaching the predefined terminal date T. Initial 5000 simulation periods are discarded to eliminate the influence of the initial conditions. We find that the combination \((\alpha, c_{1}, c_{2})=(4, 2000,2)\) gives the best fit to the empirical properties. The influence of parameters \(c_{1}\) and \(c_{2}\) on the slope of superlinear scaling is illustrated in Fig. S11.
The simulated distributions of duration τ and interval Δτ for pairwise transactions replicate the empirical distributions for a given \(N_{\mathrm { P}}\) (see lines in Fig. 2(b)–(c)). In addition, the growth pattern of the normalized aggregate degree \(K(t)/K(T)\) is also successfully reproduced (Fig. 2(e)). We further evaluate the model fit for other dynamical properties such as the degree distribution and the strength as a function of degree [51] (Figs. S7–S9). We confirmed that the model can robustly reproduce the duration and interval distributions under different parameter settings (Figs. S4–S6).
Fitting model to the data
It should be noted that while the activity level \(a_{i,t}\) fluctuates with time independently from other banks’ activity levels, the model ensures that the size and the structure of generated networks are stationary for a given \(N_{\mathrm { P}}\). In reality, however, the evolution of daily networks show a decreasing trend (Fig. 1(a) and (b)) and the network size varies from day to day owing to various external factors (e.g., shifts in monetary policy [39] and the seasonality of money demand due to the national holidays and/or the reserve requirement system [13]). In the model, the averages of the network size \(\langle N\rangle\) and the number of edges \(\langle M\rangle\) are controlled by tuning parameter \(N_{\mathrm { P}}\) (Fig. 5(a)). Therefore, we need to estimate the daily sequence of \(N_{\mathrm { P}}\) to reconstruct time series of empirical daily networks. Here, we take the following steps. First, we generate sufficiently many instances of synthetic networks for a given \(N_{\mathrm { P}}\) to compute the histogram of \((N, M)\) (Fig. 5(b)). Generating networks over a sufficiently broad range of \(N_{\mathrm { P}}\) provides a conditional probability function \(f(N, M N_{\mathrm { P}})\) that would cover the range of N and M observed in the empirical networks (Fig. S10). Second, for a given empirical daily network with \((N^{\prime}, M^{\prime})\), we choose a daily estimate of \(N_{\mathrm { P}}\), denoted by \(N_{\mathrm {P},\mathrm {ML}}\), such that \(N_{\mathrm {P},\mathrm {ML}}= \operatorname{argmax}_{N_{\mathrm { P}}}f(N^{\prime}, M^{\prime} N_{\mathrm { P}})\).
We find that simulated instances of \((N, M)\) concentrate tightly along the regression line of \(M\propto N^{1.5}\) when we vary \(N_{\mathrm { P}}\) from 50 to 350 (Fig. S10). This agreement holds true even under alternative parameter values within a reasonable range of variation (Figs. S4(a) and S11). The sequence of daily estimates of \(N_{\mathrm {P},\mathrm {ML}}\), each of which is based on the empirical combination \((N^{\prime}, M^{\prime})\) of a day, exhibit a longterm downward trend consistent with the empirical sequence of N and M (Fig. 6(a)). Specifically, \(N_{\mathrm {P},\mathrm {ML}}\) is proportional to N and increases nonlinearly in M with saturation at \(N_{\mathrm {P},\mathrm {ML}}\approx300\) (Fig. 6(a), Inset).
A time series of model networks based on the daily estimates of \(N_{\mathrm {P},\mathrm {ML}}\) reproduces the tendency toward a perfect bipartite structure (Fig. 6(b)), although we have not explicitly modeled how the network structure should change with \(N_{\mathrm { P}}\). The tendency toward a bipartite structure may reflect the fact that the average degree of generated networks becomes smaller as the size of network shrinks. Indeed, the relationship between N and M across the fitted daily networks explains the emergence of superlinearity (Fig. 6(c)), which indicates that \(M \propto N^{1.5}\) or \(\langle k\rangle\propto\sqrt{N}\).
Discussion
The time series of daily networks reveal many dynamical regularities encoded in millions of financial transactions. An important finding is that the transaction patterns between banks are similar to the social communication patters of humans, which have been observed at higher temporal resolutions (typically 20∼60 seconds) than a daily resolution. For instance, a powerlaw scaling in the distribution of the interaction duration has been found in human contact networks, such as facetoface conversation networks of individuals [36, 37]. We note that the general economic meaning of longterm relationships between a lender (e.g., a bank) and a borrower (e.g., a firm) has long been studied in microeconomics, and some studies pointed out that there also exist longterm relationships between banks in interbank markets [28, 52–55]. This would imply that there could be common mechanisms that govern social behavior of humans and economic behavior of financial institutions. Although the social activity patterns have been found only in social systems so far, given our findings, it would be reasonable to speculate that similar activity patterns might be found in other complex systems as well as in financial systems.
The sublinear growth pattern of aggregated degree has also been reported in the mobility pattern of mobilephone users [47]. In addition, superlinear scaling at the network level (called “urban scaling” [40]) emerges in various social contexts, such as the relationship between the number of mobile connections and the population size of cities [38]. The discovered dynamical patterns are quite robust and hold true even amid the global financial crisis, suggesting that there is a universal mechanism connecting financial and social dynamics.
The contribution of our work is not limited to the findings on the transition patterns of interbank networks. The model we propose here can be used as a generator of synthetic networks for studies of financial systemic risk. As is often the case, inaccessibility to empirical data on financial transactions forces academic researchers to use synthetic networks with limited empirical properties [15, 25] or to infer real network structure based on available partial information [18, 56–62]. Our model provides a way to easily generate synthetic time series of networks that exhibit dynamical properties characterizing the daily evolution of real interbank networks. We hope that the current work will not only deepen our knowledge about the dynamic nature of interbank networks, but also help to improve the conventional approach of systemicrisk studies toward a more dynamic analysis.
We leave three remaining issues that need to be addressed in future research. First, while the observed dynamical patterns are quite robust and seem universal given the similarity to social network dynamics, it is worth investigating whether those findings hold true in other countries as well. Second, we might be able to find other dynamical patterns at different time resolutions such as intraday, weekly, and monthly. If that is the case, we need to see if those dynamical patterns found in different time scales are consistently explained by the current model. Finally, our finding reveals an independence of local interaction patterns of banks from a globalscale network evolution, such as the decreasing trend in network size. This implies that there is no feedback loop between micro and macroscopic phenomena, meaning that banks are not adaptive to their environments. Further research is needed to explain why such a decoupling phenomenon takes hold in financial networks.
Methods
Data description
The original timestamped data are commercially available from eMID SIM S.p.A based in Milan, Italy [35]. The data contain all the unsecured eurodenominated transactions between financial institutions made via an online trading platform, eMID. We focus on overnight (labelled as “ON”) and overnightlarge (“ONL”) transactions. ON transactions refer to contracts that require borrowers to repay the full amount within one business day from the day the loans are executed. ONL transactions are a variant of the ON transactions, where the amount is no less than 100 million euros.
The data processing procedure is as follows. First, we extract transactions made between September 4, 2000 and December 31, 2015. The choice of the initial date is based on the introduction of the ONL category [13]. This leaves us with 1,119,258 ON and 73,480 ONL transactions, which comprise 86% of all the transactions during that period. Next, we transform all the ON and ONL transactions into a sequence of daily networks by applying the daily time window of 8:00–18:00 [28]. We then extract the transactions that belong to the largest weakly connected component of each daily network, which account for 99.3% of all the daily transactions on average (the minimum is 78%). We referred to this component as daily network throughout the analysis. Multiple edges between two banks are simplified. In the end, we have 1,187,415 transactions conducted by 308 financial institutions over 3922 business days.
Fitting procedure for the interval distribution
As shown in Fig. 2(c), the empirical distribution of transaction interval Δτ for each bank pair does not follow a power law. We instead find that the interval distribution nicely fits a Weibull distribution for \(1 \leq\Delta\tau < \Delta\tau_{\mathrm{ u}}\), where \(\Delta\tau_{\mathrm{ u}}\) denotes a cutoff value.
The complementary cumulative distribution function (CCDF) of a Weibull distribution [46] is given by
where \(c > 0\) and \(\lambda>0\) are parameters. Distribution \(P_{c}(x)\) can also be written as \(n_{x}/N_{X}\), where \(N_{X}\) is the total number of interval values observed, and \(n_{x}\) is the rank of interval length x (i.e., \(n_{x}\) is the number of observed interval values such that \(\Delta\tau\geq x\)). By taking the logarithm of \(n_{x}/N_{X} = \exp { \{  ( x / \lambda )^{c} \}}\), we obtain the following expression [45]:
where \(x_{n}\) represents the interval length whose rank is n (i.e., \(x_{1} > x_{2} > \cdots> x_{N_{X}}\)), and β is defined as \(\beta \equiv\lambda^{c}\). We use Eq. (7) to find β and c that give the best fit to a Weibull distribution. We introduce n̂, the logged rank of cutoff value \(\Delta\tau _{u}\), and estimate parameters \((\beta, c)\) for a subset of the observed values of Δτ, in a similar way as is done in the standard estimation procedure for a powerlaw exponent [43]. The cutoff \(\Delta\tau_{\mathrm{ u}}\) corresponds to the \(e^{\hat{n}}\)th largest interval length. Parameters β, c, and n̂ are determined as follows.

1.
For a given pair \((c, \hat{n})\), estimate β in (7) by the ordinary least squares (OLS). Repeat this for sufficiently many values of \(c \in[0,1)\) (we set \(c < 1\) because the tail of the empirical distribution of Δτ is apparently heavier than that of an exponential distribution). The estimate of β is denoted by \(\beta^{*}(c, \hat{n})\).

2.
For n̂ given in step 1, find the optimal value of c, denoted by \(c^{*} ( \hat{n} )\), such that the coefficient of determination \(R^{2}\) for the OLS regression is maximized, in which case \(\beta= \beta^{*}(c^{*}(\hat{n}), \hat{n})\). Let \(R^{2}(\hat{n})\) denote the maximum of \(R^{2}\) for a given n̂.

3.
By repeating steps 1 and 2 for all the predefined values of n̂, find the optimal cutoff value \(\hat{n}^{*} \equiv \operatorname{argmax}_{\hat{n}}{R^{2}(\hat{n})}\). In the end, the estimates of the parameters are given by \(\hat{n} = \hat{n}^{*}\), \(c = c^{*}(\hat{n}^{*})\), and \(\beta= \beta^{*}(c^{*}(\hat {n}^{*}), \hat{n}^{*})\).
Figure 7(a) illustrates the determination of the optimal logrank cutoff \(\hat{n}^{*}\). The inset of Fig. 2(c) in the main text shows the OLS fit to (7) when \(\hat{n} = \hat{n}^{*}\) (note that \(x_{n}\) corresponds to Δτ in that figure). Once \(\hat{n}^{*}\) is determined, it is straightforward to obtain the corresponding cutoff \(\Delta\tau_{u}\). Figure 7(b) verifies the goodness of fit between the empirical CCDF and the estimated Weibull distribution.
Analytical solution for the fitness model with a finite size effect
Relationship between N and M. As we described in the main text, we assume that initially there are \(N_{\mathrm { P}}\) many isolated nodes. Node i \((1 \leq i \leq N_{\mathrm { P}})\) is assigned a fitness \(a_{i} \in[0,1]\) which is drawn from density \(\rho(a)\).
The probability of edge formation between two nodes i and j is denoted by \(u(a_{i}, a_{j})\). We define N as the number of nodes connected with at least one edge and M as the total number of edges in a network. We express N and M as functions of \(N_{\mathrm { P}}\):
where \(q_{0}(N_{\mathrm { P}})\) is the probability of a randomly chosen node being isolated (i.e., no edges attached) and \(\overline{k}(N_{\mathrm { P}})\) is the average degree over all nodes including isolated ones. Thus, to obtain the functional forms of N and M, we need to get the functional forms of \(q_{0}(N_{\mathrm { P}})\) and \(\overline{k}(N_{\mathrm { P}})\). In the following, we first derive the functional forms of \(q_{0}(N_{\mathrm { P}})\) and \(\overline{k}(N_{\mathrm { P}})\) in a general setting. Then, we will restrict our attention to the case with \(\rho(a) = 1\) (i.e., a uniform distribution) and \(u(a_{i}, a_{j}) = (a_{i} a_{j})^{\alpha}\) to explain the empirical superlinear relation between N and M in the same specification as in the main text.
Average degree of networks including isolated nodes, \(\overline{k}(N_{\mathrm { P}})\). Given the fitnesses of all nodes \(\boldsymbol {a} = (a_{1}, a_{2}, \ldots, a_{N_{\mathrm { P}}})\), the probability that node i has degree \(k_{i}\) is
where \(c_{ij} \in \{ 0, 1 \}\) is the \((i,j)\)element of the adjacency matrix and \(\boldsymbol {c}_{i} = (c_{1i}, c_{2i}, \ldots, c_{N_{\mathrm { P}}i})^{\top}\) is the ith column vector. Function \(\delta(x, y)\) denotes the Kronecker delta. Let us redefine a product term in the square bracket of (9) as
Since \(g(k_{i}  \boldsymbol {a})\) is the convolution of \(\{ f_{j}(c_{ij}; a_{i}, a_{j}) \}_{j}\), its generating function
is decomposed as
where \(\hat{f}_{j}\) is the generating function of \(f_{j}(c_{ij}; a_{i}, a_{j})\), given by
Degree distribution \(p(k_{i}; N_{\mathrm { P}})\) is defined by the probability that node i has degree \(k_{i}\) and is related to \(g(k_{i}  \boldsymbol {a})\) such that
where we define \(\rho(\boldsymbol {a}) \equiv\prod_{i} \rho(a_{i})\) and \(d \boldsymbol {a} \equiv\prod_{i} \,da_{i}\). Therefore, differentiation of \(\hat {g}_{i}(z  \boldsymbol {a})\) with respect to z gives the average degree \(\overline{k}(N_{\mathrm { P}})\):
From Eqs. (10) and (13), we have \(\hat {f}(z; a_{i}, a) = \sum_{c_{ij}} z^{c_{ij}} f(c_{ij}; a_{i}, a) = (z1)u(a_{i}, a) + 1\). It follows that
Substituting these into Eq. (15) leads to
It should be noted that (18) is equivalent to Eq. (21) of Ref. [50].
Probability of node isolation, \(q_{0}(N_{\mathrm { P}})\). From (14), the probability of a node being isolated, \(q_{0}(N_{\mathrm { P}}) \equiv p(k_{i} = 0; N_{\mathrm { P}})\), is given by
Special case: \(\rho(a) = 1\) and \(u(a, a^{\prime}) = (a a^{\prime})^{\alpha}\). Substituting \(\rho(a) = 1\) and \(u(a, a^{\prime}) = (a a^{\prime})^{\alpha}\) into Eq. (18) gives
Similarly, substituting the same conditions into Eq. (19) gives
By rewriting the integrand as \(x = 1  \frac{1}{\alpha+1} a_{i}^{\alpha}\), we have
where \(\mathrm{ B}(x,y) \equiv\int_{0}^{1} t^{x1} (1t)^{y1} \,dt\) is the beta function and \(\mathrm{ B}_{z}(x,y) \equiv\int_{0}^{z} t^{x1} (1t)^{y1} \,dt\) (\(0 < \operatorname{Re}(z) \leq1\)) is the incomplete beta function. Combining these results with Eq. (8), we end up with
If \(N_{\mathrm { P}}\) is sufficiently large, then \(q_{0}(N_{\mathrm { P}}) \simeq0\) and thereby \(N \simeq N_{\mathrm { P}}\) and \(M \simeq(1/\alpha+1)^{2} N(N1)/2 \propto N^{2}\). Therefore, the solution is consistent with that of the previous studies [48–50] in the absence of the finitesize effect.
Dynamics of weights
Empirical observation. On top of the edge dynamics that we discussed in the main text, the dynamics of edge weights also exhibits specific patterns. Let us define the weight of an edge, \(w_{ij,t}\), as the total amount of funds transferred from bank i to j on day t. We define the growth rate of edge weights as \(r_{ij,t} \equiv\log {(w_{ij,t+1}/w_{ij,t})}\) for bank pair \((i, j)\) such that \(w_{ij,t+1}w_{ij,t}>0\) [51]. The distribution of \(r_{ij,t}\), aggregated over all pairs and all t, exhibits a symmetric triangular shape with a distinct peak at 0 (Fig. 8(a)). The shape of the distribution indicates that a large fraction of bank pairs do not change the amount of funds when they keep trading, and if they change the amount, the rate of change will be typically small. A similar sort of triangularshaped distribution of the growth rate of weights has also been found in networks of email exchanges [63], airlines [51] and cattle trades between stock farming facilities [64].
Model of weight dynamics. To reproduce the dynamics of edge weights, we add the following step to the model. Let us consider the edge between i and j formed in day t. If there is an edge from i to j in day \(t1\), then the edge weights in day t is given by
where random variable \(\nu_{ij,t}\) takes different values across bank pairs and are assumed to follow a powerlaw distribution with exponent η to maximize the fit to \(P(r)\) (Fig. 8) and the empirical weight distribution (Fig. 9(a)–(c)). Positive constant κ is introduced to match the scale of edge weights with that of the data (i.e., millions of euros). On the other hand, if there is no edge from i to j in day \(t1\) but there is in day t, then
Any nonadjacent pairs \((i,j)\) has \(w_{ij,t}=0\).
We set the weight parameters as \((q, \kappa, \eta) =(0.5, 80, 3.3)\) to fit \(P(r)\) and the simulated weight distributions with the empirical ones, respectively. Figures 8(b) and 9(d)–(f) show that our model of weight dynamics successfully replicates the empirical distributions.
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Acknowledgements
The authors thank Naoki Masuda for useful comments on the manuscript and Research Project “Network Science” organized at International Institute for Advanced Studies for providing an opportunity to initiate the project.
Availability of data and materials
The original data are commercially available from eMID SIM S.p.A based in Milan, Italy [35]. MATLAB code for generating synthetic networks is available from Teruyoshi Kobayashi’s website: https://terukobayashi.wordpress.com/.
Funding
TK acknowledges financial support from the Japan Society for the Promotion of Science Grants no. 15H05729 and 16K03551. TT was supported by JST ERATO Grant Number JPMJER1201, Japan.
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TK and TT conceived the research. TK conducted the empirical and numerical analysis. TT processed the data and conducted the theoretical analysis. TK and TT discussed the results and wrote the manuscript. All authors read and approved the final manuscript.
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Correspondence to Taro Takaguchi.
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Keywords
 Financial networks
 Systemic risk
 Temporal networks
 Scaling
 Fitness model