# DII-GCN: Dropedge Based Deep Graph Convolutional Networks

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Preliminary

**Definition**

**1**

**.**Given a graph G with added self-loops, the adjacency matrix A, degree matrix D, and identity matrix I, respectively. An applicable Laplacian matrix normalization method is given by:

**Definition**

**2**

**.**Given a directed graph G, the eigenvalues and corresponding eigenvectors of Laplacian matrix L are given by,

**Definition**

**3**

**.**The convolution formula of a simplified GCN model can be defined by:

**Definition**

**4**

**.**For a graph G, ${e}_{ij}$ is an edge from node ${v}_{i}$ to ${v}_{j}$ in G, and A = ${a}_{ij}$, D = ${d}_{ij}$ are the adjacency matrix and degree matrix of G, respectively. Suppose J $\in {R}^{n}$, an n-dimensional vector on G, then the ${e}_{ij}$ edge derivative at the ${v}_{i}$ node is calculated as follows:

## 4. Proposed Model

#### 4.1. Residual Network and Cross-Layer Connection

**Definition**

**5**

**.**The basic convolution formula of a GCN model with residuals is defined by:

#### 4.2. DII-GCN Model Design

**Definition**

**6**

**.**For graph G, suppose the Dropedge probability is p, then we use ${A}_{p}$ to represent the normalized adjacency matrix after Dropedge:

## 5. Experiment and Analysis

#### 5.1. A Comparative Experiment of Learning and Classification Accuracy

- (1)
- P (Positive) and N (Negative): represent the number of positive and negative examples in the training sample, respectively.
- (2)
- TP (True Positives): The number of positive examples that are correctly classified. The number of samples that are positive and classified as positive by the model.
- (3)
- FP (False Positives): The number of positive examples incorrectly classified. The number of samples is negative but classified as positive by the classifier.
- (4)
- FN (False Negatives): The number of false negatives. The number of positive samples but classified as negative by the classifier.
- (5)
- TN (True negatives): The number of negative examples correctly classified. The number of negative samples is classified as negative one.

- (i)
- Consider G separately. The typical algorithm GCN used in the experiment.
- (ii)
- Consider G+D. Rong (Rong et al., 2019) puts forward the reasons for Dropedge in GCN and demonstrates some effects. Therefore, we perfected G+D and named D-GCN for the comparative experiments in this paper.
- (iii)
- Consider G+R. GCNII is a typical representative model.
- (iv)
- Consider G+D+R. The DII-GCN model proposed in this paper belongs to this category.

- (1)
- The DII-GCN model has the highest accuracy on the three standard data sets.
- (2)
- The GCN model can obtain better learning accuracy on the 2-layer network. However, as the depth increases, the learning accuracy drops sharply, and not it is not easy to use the Dropedge method (corresponding to the G+D model) to support deep GCN construction.
- (3)
- The DII-GCN and GCNII models can support the construction of deep graph convolutional networks, and the learning accuracy of DII-GCN improved from GCNII on the three standard data sets.

- (1)
- On the Cora dataset, for the first three classes (class identifiers are 0, 1, 2), the classification accuracy after Dropedge (p = 0.9) is better than the GNN without Dropedge (p = 1) at all iteration stages; For label 2, we found that the algorithm we proposed with other algorithms showed a decreasing trend in accuracy, but DII-GCN was still better than several others. The reason for their all decreasing may be the presence of numerous missing values in this category of data.
- (2)
- For class label 3, the previous Dropedge effect is not very satisfactory, but after 270 Epochs, the Dropedge appears to have a good effect. However, in a class with poor classification accuracy, the learning process improves by an appropriate number of iterations.

#### 5.2. Dropedge Effectiveness Analysis

- (1)
- For the Cora dataset, Dropedge coefficient p = 0.9 or 0.8, and the number of layers is over 16. Therefore, the DII-GCN model accuracy is around 84%. Furthermore, DII-GCN can deepen the network by setting the appropriate Dropedge coefficient p. As a result, the level can obtain a stable and higher learning accuracy.
- (2)
- When p is below 0.6, the DII-GCN model accuracy is not high, indicating the Dropedge effect is not ideal because too many Dropedges cause the graph data structure to be lost. It also illustrates the scientific meaning of the GNN. Using the associated information of the node can improve the node evaluation effect.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Dataset | Cora | Citeseer | Pubmed |
---|---|---|---|

Number of nodes | 2708 | 3327 | 19,717 |

Number of edges | 5429 | 4732 | 44,338 |

Number of class | 7 | 6 | 3 |

Number of Feature | 1433 | 3703 | 500 |

Dataset | Weight Control $\mathit{\lambda}$ Coefficient | Shear Output Coefficient Dropout | Dropedge Coefficient p | Residual Coefficient $\mathit{\alpha}$ | Learing Rate |
---|---|---|---|---|---|

Citeseer | 0.5 | 0.6 | 0.9 | 0.1 | 0.01 |

Cora | 0.5 | 0.6 | 0.9 | 0.1 | 0.01 |

Pubmed | 0.5 | 0.6 | 0.98 | 0.1 | 0.01 |

Dataset | Model | Layer | ||||
---|---|---|---|---|---|---|

2 | 16 | 32 | 64 | 128 | ||

Cora | GCN | 81.1 | 64.9 | 60.3 | 28.7 | — |

D_GCN | 82.8 | 75.7 | 62.5 | 49.5 | — | |

GCNII | 80.4 | 84 | 84.5 | 84.5 | 84.3 | |

DII_GCN | 80.6 | 84.2 | 84.5 | 84.5 | 84.6 | |

Dataset | Model | Layer | ||||

2 | 16 | 32 | 64 | 128 | ||

Citeseer | GCN | 70.8 | 18.3 | 25 | 20 | — |

D_GCN | 72.3 | 57.2 | 41.6 | 34.4 | — | |

GCNII | 65.9 | 71.7 | 72.2 | 72.1 | 72 | |

DII_GCN | 66.2 | 72 | 72.5 | 72.2 | 72.3 | |

Dataset | Model | Layer | ||||

2 | 16 | 32 | 64 | 128 | ||

Pubmed | GCN | 79 | 40.9 | 22.4 | 35.3 | — |

D_GCN | 79.6 | 78.5 | 77 | 61.5 | — | |

GCNII | 78.1 | 79.2 | 79.3 | 72.2 | 72.1 | |

DII_GCN | 77.8 | 79.3 | 79.7 | 79.3 | 79.3 |

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**MDPI and ACS Style**

Zhu, J.; Mao, G.; Jiang, C. DII-GCN: Dropedge Based Deep Graph Convolutional Networks. *Symmetry* **2022**, *14*, 798.
https://doi.org/10.3390/sym14040798

**AMA Style**

Zhu J, Mao G, Jiang C. DII-GCN: Dropedge Based Deep Graph Convolutional Networks. *Symmetry*. 2022; 14(4):798.
https://doi.org/10.3390/sym14040798

**Chicago/Turabian Style**

Zhu, Jinde, Guojun Mao, and Chunmao Jiang. 2022. "DII-GCN: Dropedge Based Deep Graph Convolutional Networks" *Symmetry* 14, no. 4: 798.
https://doi.org/10.3390/sym14040798