# A Deep Learning Approach to Dynamic Interbank Network Link Prediction

## Abstract

**:**

## 1. Introduction

- Inspired by Chen et al. (2021), the model is proposed to combine the advantages of the Graph convolutional network (GCN), which obtains valuable information and learns the internal representations of the network snapshots, with the benefits of the Long short-term memory model (LSTM), which is effective at identifying and modeling short and long-term temporal relationships embedded in the sequence of data.
- To handle the network sparsity and the fact that we care more about the existing links than nonexisting links; we design a loss function that adds a penalty to nonexisting links.
- On test data, the proposed model is assessed and compared with two traditional statistical baseline models using the metrics Area Under the ROC Curve (AUC) and Precision–Recall Curve (PRAUC). They are the Discrete autoregressive model and Dynamic latent space model. The findings indicate that our proposed model beats the two models in predicting future links in both precrisis and crisis periods for the top 100 Italian trading dataset and European core countries dataset.

## 2. Literature Review

#### 2.1. Financial Contagion

#### 2.2. Interconnectedness Network Models

## 3. Materials and Methods

#### 3.1. Problem Definition

#### 3.2. GC–LSTM Framework

#### 3.2.1. Graph Convolutional Network

#### 3.2.2. Long Short-Term Memory

**Forget Gate**: The forget gate decides what information should be kept or removed from the cell state.**Input gate**: The input gate decides what information should be added to the cell state.**Output gate**: The output gate decides what the next hidden state should be.

#### 3.2.3. GC–LSTM Model

#### 3.2.4. Decoder Model

#### 3.3. Loss Function and Model Training

## 4. Experiments and Results

#### 4.1. e-MID Dataset

**Degree**: The degree of the network is defined as the number of connections as a proportion of all possible links inside the network (Boss et al. 2004). A low value of the degree might indicate a low level of liquidity in the e-MID interbank market.**Clustering coefficient**: The clustering coefficient is a measure of how closely nodes in a network cluster together (Soramäki et al. 2007).**Centrality**: In this part, we introduce three kinds of centrality, which are degree, betweenness, and Eigen centrality. For the degree centrality, it is defined as the number of links incident upon a node (Temizsoy et al. 2017). Since only the node’s immediate ties are considered when calculating degree centrality, it is a local centrality measure. For between centrality, which is introduced by Freeman (1978), it is defined as the number of times a node functions as a bridge along the shortest path between two other nodes, since it focuses on a node’s distance from all other nodes in the network and is a measure of global centrality in this sense. The last centrality measure we introduce is Eigen centrality (Negre et al. 2018). Eigen centrality calculates a node’s centrality based on its neighbors’ centrality, which is a measure of the influence of a node in a network. The score of the Eigen centrality of a bank is between 0 to 1, where higher values indicate more essential banks for interconnection.**Largest strongest connected component**: A strongly connected component is the portion of a directed graph where each vertex has a route to another vertex. The fraction of banks connected to other banks via directed edges on the network scaled by the total number of banks in the network is defined as the largest strongest connected component of the graph. If the value of the largest strongest connected component is close to 1, it means that the network is highly connected, and if the value is close to zero, the network is much more fragmented.

#### 4.2. Baseline Methods

**Dynamic latent space model**: Dynamic latent space model is a model based on the distance idea in social networks (Hoff et al. 2002). The model assumes that the link probability between any two nodes depends on the distance between the latent position of the two nodes. A dynamic latent space model is proposed by Sewell and Chen (2015) and is used on the interbank network model by Linardi et al. (2020).**Discrete autoregressive model**: To avoid systemic risk, the information of the counterparty plays an important role to decide who to trade with. The past trading relationship, which is also seen as link persistence, is documented in the paper Papadopoulos and Kleineberg (2019). The relationship is defined as preferential trading and allows banks to ensure liquidity risk in the presence of market frictions such as information and transaction cost (Cocco et al. 2009; Giraitis et al. 2012). Based on the preferential trading theory, the link formation strategy of the Discrete autoregressive model (Jacobs and Lewis 1978) is that the value of a link between bank i and bank j at time t is determined by past value at time $t-1$ and the ability to create new links. Therefore, the model could be described as follow:$${A}_{i,j,t}={\theta}_{i.j,t}\ast {A}_{i,j,t-1}+(1-{\theta}_{i,j,t}){X}_{i,j,t}$$

#### 4.3. Evaluation Metrics

#### 4.4. Parameter Sensitivity

**The penalty $\mathbf{\lambda}$ index:**As exiting links are much more important than the nonexisting links, we add a penalty to the nonexisting links with a different ${\lambda}_{i,j}$ from 1 to 4. Additionally, we set the ${\lambda}_{i,j}$ value for the existing links to be 1. If the penalty value is the same for both the existing links and nonexisting links, then we treat the two kinds of links with no difference. The results shown in Figure 3 indicate that a larger penalty could lead to slightly larger AUC and PRAUC. This suggests we choose a higher penalty score for nonexisting links in the following model parameter settings.**The window size****l****:**In most cases, a larger historical interbank network snapshots input might improve the performance in link prediction. In our case, we use a range of window sizes from 5 to 20 with a regular interval of 5, and the results for both AUC and PRAUC follow a similar pattern. By choosing the window size to be 10, we could achieve both the highest AUC and PRAUC. The results are shown in Figure 4.**The****K****-hop neighborhood:**The K-hop neighborhood idea comes from social network analysis. The larger the size of K, the more information a node utilizes from its neighborhood. In our interbank network, a larger K does not help in link prediction. It means that if a bank i trades with another bank j, even if bank j has a close relationship to bank z, bank i will not preferentially trade with bank z. The results are shown in Figure 5.

#### 4.5. Link Prediction

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**K-hop neighborhood. The blue node is the source node, the area that covers the yellow nodes is the 1-hop neighborhood, the area that covers the yellow and green nodes are the 2-hop neighbors, and the area that covers the yellow, green and red nodes is the 3-hop neighborhood.

**Figure 2.**Architecture of Long short-term memory model. The notations of the graph are shown as follows. ${f}_{t},{i}_{t},{o}_{t}$ are the forget gate, input gate and output gate. ${C}_{t}$ is the cell state, ${\tilde{C}}_{t}$ is the new candidate values, ${x}_{t}$ is the input at time t and ${h}_{t}$ is the hidden state. × is the pointwise multiplication operation, + is the addition operation, $\sigma $ is the sigmoid function and $tanh$ is the hyperbolic tangent function.

**Figure 3.**The evaluation metrics with different penalty scores. In (

**a**,

**b**), we use the window size $l=10$ and 1-hop neighborhood GCN units. We set the penalty from 1 to 4, and the performance of AUC and PRAUC scores are shown in (

**a**,

**b**).

**Figure 4.**The evaluation metrics with different historical time periods. In (

**a**,

**b**), we use the penalty value equal to 4 and 1-hop neighborhood GCN units. We set the window size from 5 to 20, and the performance of AUC and PRAUC scores are shown in (

**a**,

**b**).

**Figure 5.**The evaluation metrics with different K values. In (

**a**,

**b**), we use the window size $l=10$ and the penalty value for nonexisting links are 4. We set the K-hop neighborhood for the GCN units from 1 to 4, and the performance of AUC and PRAUC scores are shown in (

**a**,

**b**).

Notation | Description |
---|---|

${A}_{t}$ | the adjacency matrix of the interbank network snapshot at time t |

l | the window size for prediction |

N | the number of banks (nodes) in the network |

d | the number of hidden layers in the GC–LSTM model |

${T}_{k}$ | Chebyshev polynomial function |

$\widehat{{A}_{t}}$ | the output probability matrix at time t |

${b}^{f}$, ${b}^{c}$, ${b}^{i}$, ${b}^{o}$ | the bias terms in gate function |

${W}_{z}^{f}$, ${W}_{z}^{c}$, ${W}_{z}^{i}$, ${W}_{z}^{o}$ | the weight terms in gate function |

$GC{N}^{K}$ | the graph convolutional operation |

${\lambda}_{i,j}$ | penalty parameter in Equation (9) |

**Table 2.**Summary statistics of weekly aggregated e-MID interbank network in top 100 Italian banks. The average degree in each network is referred to as Degree. The clustering coefficient is denoted as the Clustering coefficient. The three different centrality measures are degree centrality, betweenness centrality and Eigen centrality. Additionally, the fraction of nodes in the largest strongly connected component is the largest strongest connected component. The significance levels of 10% (*), 5% (**) and 1% (***) are used to assess the mean difference between the crisis and the precrisis period with the t-test.

Time Period | Interconnectedness Statistics | Mean | Standard Deviation |
---|---|---|---|

All data results | Degree | 0.0670 | 0.0090 |

Clustering coefficient | 0.1157 | 0.0296 | |

Betweenness centrality | 0.0045 | 0.0024 | |

Eigen centrality | 0.0506 | 0.0039 | |

Degree centrality | 0.1340 | 0.0180 | |

Largest strongest connected component | 0.1892 | 0.1241 | |

Precrisis | Degree | 0.0682 | 0.0081 |

Clustering coefficient | 0.1207 | 0.0269 | |

Betweenness centrality | 0.0048 | 0.0023 | |

Eigen centrality | 0.0508 | 0.039 | |

Degree centrality | 0.1365 | 0.0162 | |

Largest strongest connected component | 0.2006 | 0.1229 | |

Crisis | Degree | $0.0621$ *** | 0.0058 |

Clustering coefficient | $0.0888$ *** | 0.0248 | |

Betweenness centrality | $0.0025$ *** | 0.0015 | |

Eigen centrality | $0.0491$ * | 0.0036 | |

Degree centrality | $0.1242$ *** | 0.0115 | |

Largest strongest connected component | $0.1267$ ** | 0.1085 |

**Table 3.**AUC score for three models in the top 100 Italian banks. The significance level of 1% (***) is used to assess the mean difference between the benchmark models (DAR or Latent space model) and the GC–LSTM model with the t-test.

Time Period | Methods | Mean AUC | Standard Deviation |
---|---|---|---|

All data results | DAR | $0.695$ *** | 0.036 |

Latent Space Model | $0.777$ *** | 0.023 | |

GC–LSTM | 0.895 | 0.016 | |

Precrisis | DAR | $0.703$ *** | 0.034 |

Latent Space Model | $0.784$ *** | 0.018 | |

GC–LSTM | 0.893 | 0.016 | |

Crisis | DAR | $0.660$ *** | 0.018 |

Latent Space Model | $0.746$ *** | 0.019 | |

GC–LSTM | 0.905 | 0.013 |

**Table 4.**PRAUC score for three models in the top 100 Italian banks. The significance level of 1% (***) is used to assess the mean difference between the benchmark models (DAR or Latent space model) and the GC–LSTM model with the t-test.

Time Period | Methods | Mean PRAUC | Standard Deviation |
---|---|---|---|

All data results | DAR | $0.390$ *** | 0.054 |

Latent Space Model | $0.152$ *** | 0.021 | |

GC–LSTM | 0.431 | 0.038 | |

Precrisis | DAR | $0.401$ *** | 0.049 |

Latent Space Model | $0.158$ *** | 0.017 | |

GC–LSTM | 0.432 | 0.038 | |

Crisis | DAR | $0.349$ *** | 0.032 |

Latent Space Model | $0.125$ *** | 0.017 | |

GC–LSTM | 0.426 | 0.035 |

**Table 5.**AUC score for three models in the core country banks. The significance level of 1% (***) is used to assess the mean difference between the benchmark models (DAR or Latent space model) and the GC–LSTM model with the t-test.

Time Period | Methods | Mean AUC | Standard Deviation |
---|---|---|---|

All data results | DAR | $0.666$ *** | 0.051 |

Latent Space Model | $0.717$ *** | 0.095 | |

GC–LSTM | 0.782 | 0.054 | |

Precrisis | DAR | $0.670$ *** | 0.048 |

Latent Space Model | $0.709$ *** | 0.039 | |

GC–LSTM | 0.779 | 0.056 | |

Crisis | DAR | $0.650$ *** | 0.057 |

Latent Space Model | $0.753$ *** | 0.025 | |

GC–LSTM | 0.795 | 0.042 |

**Table 6.**PRAUC score for three models in the core country banks. The significance level of 1% (***) is used to assess the mean difference between the crisis and the precrisis period with the t-test.

Time Period | Methods | Mean PRAUC | Standard Deviation |
---|---|---|---|

All data results | DAR | $0.175$ *** | 0.040 |

Latent Space Model | $0.094$ *** | 0.023 | |

GC–LSTM | 0.275 | 0.074 | |

Precrisis | DAR | $0.177$ *** | 0.042 |

Latent Space Model | $0.092$ *** | 0.024 | |

GC–LSTM | 0.432 | 0.075 | |

Crisis | DAR | $0.170$ *** | 0.030 |

Latent Space Model | $0.105$ *** | 0.013 | |

GC–LSTM | 0.426 | 0.065 |

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**MDPI and ACS Style**

Zhang, H.
A Deep Learning Approach to Dynamic Interbank Network Link Prediction. *Int. J. Financial Stud.* **2022**, *10*, 54.
https://doi.org/10.3390/ijfs10030054

**AMA Style**

Zhang H.
A Deep Learning Approach to Dynamic Interbank Network Link Prediction. *International Journal of Financial Studies*. 2022; 10(3):54.
https://doi.org/10.3390/ijfs10030054

**Chicago/Turabian Style**

Zhang, Haici.
2022. "A Deep Learning Approach to Dynamic Interbank Network Link Prediction" *International Journal of Financial Studies* 10, no. 3: 54.
https://doi.org/10.3390/ijfs10030054