4.1 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.
4.2 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
$$\begin{aligned} P_{c}(x) = \exp{ \biggl\{  \biggl( \frac{x}{\lambda} \biggr)^{c} \biggr\} }\quad \text{for $x>0$}, \end{aligned}$$
(6)
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]:
$$\begin{aligned} (x_{n})^{c} =  \beta(\log{n_{x}} \log{N_{X}}), \end{aligned}$$
(7)
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.
4.3 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 \(N_{\mathrm { P}}\):
$$\begin{aligned} \textstyle\begin{cases} N = (1 q_{0}(N_{\mathrm { P}})) N_{\mathrm { P}},\\ M = \frac{\overline{k}(N_{\mathrm { P}}) N_{\mathrm { P}}}{2}, \end{cases}\displaystyle \end{aligned}$$
(8)
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
$$\begin{aligned} g(k_{i}  \boldsymbol {a}) = \sum _{\boldsymbol {c}_{i}} \biggl[ \prod_{j \neq i} u(a_{i}, a_{j})^{c_{ij}} \bigl(1 u(a_{i}, a_{j})\bigr)^{1c_{ij}} \biggr] \times\delta \biggl(\sum _{j \neq i} c_{ij}, k_{i} \biggr), \end{aligned}$$
(9)
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
$$\begin{aligned} f_{j}(c_{ij}; a_{i}, a_{j}) &\equiv u(a_{i}, a_{j})^{c_{ij}} \bigl(1 u(a_{i}, a_{j})\bigr)^{1c_{ij}}. \end{aligned}$$
(10)
Since \(g(k_{i}  \boldsymbol {a})\) is the convolution of \(\{ f_{j}(c_{ij}; a_{i}, a_{j}) \}_{j}\), its generating function
$$\begin{aligned} \hat{g}_{i}(z  \boldsymbol {a}) \equiv\sum_{k_{i}} z^{k_{i}} g(k_{i}  \boldsymbol {a}) \end{aligned}$$
(11)
is decomposed as
$$\begin{aligned} \hat{g}_{i}(z  \boldsymbol {a}) = \prod_{j \neq i} \hat{f}_{j}(z; a_{i}, a_{j}), \end{aligned}$$
(12)
where \(\hat{f}_{j}\) is the generating function of \(f_{j}(c_{ij}; a_{i}, a_{j})\), given by
$$\begin{aligned} \hat{f}_{j}(z; a_{i}, a_{j}) \equiv\sum _{a_{ij}} z^{a_{ij}} f_{j}(a_{ij}; a_{i}, a_{j}). \end{aligned}$$
(13)
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
$$\begin{aligned} p(k_{i}; N_{\mathrm { P}}) = \int g(k_{i}  \boldsymbol {a}) \rho(\boldsymbol {a}) \,d \boldsymbol {a}, \end{aligned}$$
(14)
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 \(\overline{k}(N_{\mathrm { P}})\):
$$\begin{aligned} \overline{k}(N_{\mathrm { P}}) &= \sum_{k_{i}} k_{i} p(k_{i}; N_{\mathrm { P}}) \\ &= \sum_{k_{i}} k_{i} \int g(k_{i}  \boldsymbol {a})\rho(\boldsymbol {a}) \,d\boldsymbol {a} \\ &= \frac{d}{dz} \int\hat{g}_{i}(z  \boldsymbol {a}) \rho(\boldsymbol {a}) \,d \boldsymbol {a} \Big_{z=1} \\ &= \frac{d}{dz} \int\rho(a_{i}) \,da_{i} \prod _{j \neq i} \int\hat {f}_{j}(z; a_{i}, a_{j}) \rho(a_{j}) \,da_{j}\Big_{z=1} \\ &= \int\rho(a_{i}) \,da_{i} \frac{d}{dz} \biggl[ \int\hat{f}(z; a_{i}, h) \rho(h) \,dh \biggr]^{N_{\mathrm { P}}1} \Big_{z=1} \\ &= (N_{\mathrm { P}}1) \int\rho(a_{i}) \,d a_{i} \biggl[ \int d a \rho(a) \hat{f}(z; a_{i}, a) \biggr]^{N_{\mathrm { P}}2} \int d a \rho(a) \frac{d}{dz} \hat{f}(z; a_{i}, a) \Big_{z=1}. \end{aligned}$$
(15)
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
$$\begin{aligned} & \int \,da \rho(a) \hat{f}(z; a_{i}, a) = (z1) \int \,da \rho(a) u(a_{i}, a) + 1, \end{aligned}$$
(16)
$$\begin{aligned} & \int \,da \rho(a) \frac{d}{dz} \hat{f}(z; a_{i}, a) = \int \,da \rho(a) u(a_{i}, a). \end{aligned}$$
(17)
Substituting these into Eq. (15) leads to
$$\begin{aligned} \overline{k}(N_{\mathrm { P}}) = (N_{\mathrm { P}}1) \int \int d a d a^{\prime}\rho(a) \rho \bigl(a^{\prime}\bigr) u \bigl(a, a^{\prime}\bigr). \end{aligned}$$
(18)
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
$$\begin{aligned} q_{0}(N_{\mathrm { P}}) &= \int g(k_{i} =0  \boldsymbol {a})\rho(\boldsymbol {a}) \,d \boldsymbol {a} \\ &= \int d a_{i} \rho(a_{i}) \biggl[ 1  \int u(a_{i}, a) \rho(a) \,d a \biggr]^{N_{\mathrm { P}}1}. \end{aligned}$$
(19)
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
$$\begin{aligned} \overline{k}(N_{\mathrm { P}}) &= \biggl( \frac{1}{\alpha+ 1} \biggr)^{2} (N_{\mathrm { P}}1). \end{aligned}$$
(20)
Similarly, substituting the same conditions into Eq. (19) gives
$$\begin{aligned} q_{0}(N_{\mathrm { P}}) &= \int d a_{i} \biggl( 1  \frac{1}{\alpha+1} a_{i}^{\alpha}\biggr)^{N_{\mathrm { P}}1}. \end{aligned}$$
(21)
By rewriting the integrand as \(x = 1  \frac{1}{\alpha+1} a_{i}^{\alpha}\), we have
$$\begin{aligned} q_{0}(N_{\mathrm { P}}) &= \frac{1}{\alpha} (\alpha+ 1)^{\frac{1}{\alpha}} \int _{1\frac{1}{\alpha+1}}^{1} (1x)^{\frac{1}{\alpha}1} x^{N_{\mathrm { P}}1} \,dx \\ &= \frac{1}{\alpha} (\alpha+ 1)^{\frac{1}{\alpha}} \biggl[ \mathrm{ B} \biggl( N_{\mathrm { P}}, \frac{1}{\alpha} \biggr)  \mathrm{ B}_{1\frac {1}{\alpha+1}} \biggl( N_{\mathrm { P}}, \frac{1}{\alpha} \biggr) \biggr], \end{aligned}$$
(22)
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
$$\begin{aligned} \textstyle\begin{cases} N = N_{\mathrm { P}}[ 1 \frac{1}{\alpha} (\alpha+ 1)^{\frac{1}{\alpha }} ( \mathrm{ B} ( N_{\mathrm { P}}, \frac{1}{\alpha} )  \mathrm{ B}_{1\frac{1}{\alpha+1}} ( N_{\mathrm { P}}, \frac{1}{\alpha} ) ) ],\\ M = ( \frac{1}{\alpha+ 1} )^{2} \frac{N_{\mathrm { P}}(N_{\mathrm { P}}1)}{2}. \end{cases}\displaystyle \end{aligned}$$
(23)
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.
4.4 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
$$\begin{aligned} w_{ij,t} \equiv \textstyle\begin{cases} w_{ij,t1} & \text{with probability $1q$}, \\ \kappa\nu_{ij,t}p_{ij,t} & \text{with probability $q$}, \end{cases}\displaystyle \end{aligned}$$
(24)
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
$$\begin{aligned} w_{ij,t} \equiv \kappa\nu_{ij,t}p_{ij,t}. \end{aligned}$$
(25)
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.