We investigate network entropy of dynamic banking systems, where interbank networks analyzed include random networks, smallworld networks, and scalefree networks. We find that network entropy is positively correlated with the effect of systemic risk in the three kinds of interbank networks and that network entropy in the smallworld network is the largest, followed by those in the random network and the scalefree network.
There exist financial connections in the interbank market, which make it possible for the interbank market to be represented as a network. It is important to study the financial connections in the interbank market from the network perspective. The reason for this is that the financial connections can become a channel for propagation and amplification of shocks, which is directly linked to the stability of economic/financial systems [
According to the above literature, we can know that banking systems can be modeled as the complex networks, which are useful to investigate systemic risk. In the realm of complex networks, the entropy has been adopted as a measure to characterize properties of the network topology [
Based on the above analysis, it can be seen that entropy measures have been rarely adopted to analyze interbank networks and banking systemic risk. And the single study only adopts network entropy to measure diversity of highly connected global banking networks. Besides, there are a lot of researches on adopting the entropy to investigate complex networks (e.g., [
Therefore, in this paper, we apply the measure of network entropy to the dynamic banking systems, where interbank networks analyzed include random, smallworld, and scalefree networks. In the context of the analysis of interbank networks, we transform adjacency matrices into stochastic matrices and then apply the concept of entropy. In this paper, we find that network entropy is positively correlated with the effect of systemic risk in the three kinds of interbank networks and that network entropy in the smallworld network is the largest, followed by those in the random network and the scalefree network.
The rest of the paper is organized as follows. Section
The modeling of dynamic banking systems in this paper is based on the study of Lux [
The first phase is the update of liquid assets and net worth. At the beginning of time
The second phase is default settlement. After the first phase, bank
The third phase is the credit lending. We assume that there is a threshold (
In this paper, we assume that the funds do not transfer from lenders to borrowers until borrowers’ demand for liquidity is satisfied. Banks’ balance sheets will be updated according to the actual borrowing or lending. If potential borrowers’ demand for liquidity is not satisfied, they default. For the sake of simplicity, the total number of banks in the banking system is constant over time. Therefore, in this paper, we assume a simple mechanism of entryexit: a default bank is replaced by a new one. The balance sheet structure of the new bank is the same as the initial balance sheet structure of the default bank. This can be interpreted as the entry of new banks into the interbank market. In fact, this mechanism is present in the existing literature, such as the study of Gatti et al. [
According to the above modeling of dynamic banking systems, we can obtain dynamic interbank networks, which can be represented by adjacency matrix
Given a stochastic matrix
Moreover, we can obtain Shannon entropy (
According to the studies [
Benchmark parameters of the model.
Parameter  Description  Benchmark value  Range of variation 


Number of banks  100  Positive integer 

Range of values of initial assets 



Pareto distribution parameter  1.2  Positive number 

Initial proportion of investments  0.9 


Initial proportion of net worth  0.08 


Proportion of 
0.08 


Interest rate parameter  0.01  Positive number 

Scaling parameter for the level of deposit fluctuations  0.1  Positive number 

Probability of connection between any two nodes in random networks  0.3 


Number of nearest neighbors of a node in smallworld networks  30  Positive integer 

Probability of randomly rewiring each edge of the lattice for smallworld networks  0.01 


Initial number of nodes in scalefree networks  25  Positive integer 

Number of edges of a new node in scalefree networks  15  Positive integer 
Demetrius and Manke [
Network entropies and effects of systemic risk. (a), (b), and (c) correspond, respectively, to the results of random, smallworld, and scalefree networks.
Moreover, we adopt Pearson’s correlation to investigate the correlation between network entropies and the effects of systemic risk. Table
Pearson’s correlation coefficients between network entropies and systemic risk.
Parameter  Random network  Smallworld network  Scalefree network 


0.3077  0.2729  0.5523 

0.3108  0.3236  0.5508 

0.1992  0.2845  0.5119 

0.2556  0.2934  0.4834 

0.3077  0.2729  0.5523 

0.2922  0.3176  0.5516 

0.3077  0.2729  0.5523 

0.3098  0.2106  0.4747 

0.2644  0.1734  0.4688 
We now investigate the difference of network entropies in the three kinds of interbank networks. Figure
Network entropies in the three kinds of interbank networks.
According to the above model, we can see that
Network entropies in the three kinds of interbank networks under different parameter values. (a)–(f) are the results when
In this paper, we first construct artificial banking systems and then investigate network entropy of dynamic banking systems, where the three kinds of potential interbank networks are analyzed, namely, random networks, smallworld networks, and scalefree networks. First, simulation analysis shows that the change trend of network entropies is similar to that of systemic risk and that network entropy is positively correlated with the effect of systemic risk in the three kinds of interbank networks. Besides, we find that the value of network entropy in the smallworld network is the largest among the three kinds of interbank networks, followed by those in random and scalefree networks.
In this paper, we analyze the network entropy in known network topologies. However, several works in the systemic risk and banking literatures show that financial networks are organized in a coreperiphery structure. Therefore, we believe that more research needs to be done in order to understand how network entropy behaves in financial networks. For example, how does network entropy behave in coreperiphery structures? Moreover, is the entropy dependent on the network core size? Or does it show the same pattern regardless of the periphery and core sizes? Similar to most of the literature in this field, we define systemic risk as the number of defaulting banks. In the future, we would consider the total loss of capitalization of the banking system as a robustness indicator.
The authors declare that they have no conflicts of interest.
This research is supported by NSFC (no. 71201023, no. 71371051, and no. 71671037), Social Science Fund Project of Jiangsu Province (no. 15GLC003), Humanities and Social Science Planning Foundation of the Ministry of Education of China (no. 16YJA630026), and Teaching and Research Program for Excellent Young Teachers of Southeast University (no. 2242015R30021).