# Graph Learning for Attributed Graph Clustering

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## Abstract

**:**

## 1. Introduction

- We develop a similarity graph learning approach for graph clustering. It effectively digs the interplay between the topology and the node features. Meanwhile, it retains the initial high-order relationships;
- A node sampling strategy is applied to choose some important nodes, which makes our method scalable to a large graph;
- Extensive experimental results demonstrate the effectiveness and efficiency of our algorithm with respect to many state-of-the-art methods, including several recent deep models.

## 2. Related Work

## 3. Methodology

**Notations:**Define a non-directed graph as $\mathcal{G}=(\mathcal{V},E,X)$, where $\mathcal{V}=\{{v}_{1},{v}_{2},\cdots ,{v}_{n}\}$ is the set of n nodes, $X={\{{x}_{1},\cdots ,{x}_{n}\}}^{\top}\in {\mathcal{R}}^{n\times d}$ denotes the attribute matrix, and E represents the edge set denoted by an adjacency matrix $\tilde{A}=\{{\tilde{a}}_{ij}\}\in {\mathcal{R}}^{n\times n}$. If there is an edge between ${v}_{i}$ and ${v}_{j}$, ${\tilde{a}}_{ij}=1$; otherwise, ${\tilde{a}}_{ij}=0$. With degree matrix D, symmetrically normalized adjacency matrix can be written as $A={D}^{-\frac{1}{2}}(\tilde{A}+I){D}^{-\frac{1}{2}}$, where a self-loop to each node is applied [3] and I is an identity matrix with a proper size. In essence, ${a}_{ij}$ denotes the transition probability of a single step random walk between ${v}_{i}$ and ${v}_{j}$. Then, graph Laplacian $L=I-A$. Partitioning the n nodes into g distinct groups is the goal of graph clustering.

#### 3.1. Graph Learning

#### 3.2. Scalable Graph Learning

#### 3.3. Anchor Selecting with Graph Mining

Algorithm 1 GLGC |

Input: The attribute matrix $X\in {\mathcal{R}}^{n\times d}$, the affinity matrix A, parameters k, $\gamma $, P and $\alpha $, number of anchors m, cluster number g.Output: g partitions |

## 4. Experiments

#### 4.1. Datasets

#### 4.2. Comparison Methods

#### 4.3. Evaluation Metric

#### 4.4. Experimental Setup

#### 4.5. Clustering Results

- GLGC outperforms the most recent deep methods GMM-VGAE, DAEGC, DNENC-Att, and DNENC-Con in most cases. Our improvements over DAEGC, DNENC-Att, and DNENC-Con are considerable. For Pubmed, our method and GMM-VGAE produce comparable results. Facing the systematic use of complex deep learning methods, our method is very competitive and attractive;
- With respect to AGC, our method obtains much better performance. Although they both use graph filtering to process the data, our method follows a graph learning approach. This verifies the advantage of automatic graph construction;
- We can see that the methods that only utilize one type of information generate inferior performance. By contrast, the methods that employ both structure and attribute information generally perform better. This verifies the importance of developing methods that incorporate both types of information. Beyond this, it would be crucial to fully explore the interplay of them;
- GLGC consistently outperforms other GCN-based clustering methods: GAE, VGAE, MGAE, ARGE, ARVGE. GLGC’s performance of: ACC and NMI on Cora and Citeseer, NMI on Pubmed, all metrics on Wiki, ACC on Large Cora, all metrics on Coauthor Phy are the best compared with those methods, while other metrics are close to the best one. Other methods learn latent representations and then construct a graph for spectral clustering. The built graph might not be optimal for downstream clustering. By contrast, our method directly outputs a graph for spectral clustering. This is a crucial distinction between our approach and the currently used approaches;
- In most cases, degree sampling performs better than core sampling. Apart from Large Cora, degree sampling has advantages in all the cases on other five datasets. It confirms that the sampling strategy is also data-specific. In particular, it is perhaps related to the size of graph, especially the edge density;
- GLGC produces better performance than FGC in most cases. In fact, FGC is the unsampled version of GLGC. Sampling has the bonus of mitigating some negative effects of noise nodes and edges.

#### 4.6. Time Comparison

#### 4.7. Ablation Study

#### 4.8. Parameter Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The influence of parameters k and $\alpha $ on results of Cora (first row) and Large Cora (second row) datasets in FGC.

**Figure 4.**The influence of m on the results of the Cora dataset in GLGC. The error bar denotes the standard deviation of 10 tests.

Dataset | Nodes | Edges | Features | Classes |
---|---|---|---|---|

Cora | 2708 | 5429 | 1433 | 7 |

Citeseer | 3327 | 4732 | 3703 | 6 |

Pubmed | 19,717 | 44,338 | 500 | 3 |

Wiki | 2405 | 17,981 | 4973 | 17 |

Large Cora | 11,881 | 64,898 | 3780 | 10 |

Coauthor Phy | 34,493 | 247,962 | 8415 | 5 |

**Table 2.**Clustering performance on all datasets. The top and second-best outcomes are marked in blue and are underlined, respectively.

Methods | Input | Cora | Citeseer | Pubmed | Wiki | Large Cora | Coauthor Phy | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ACC% | NMI% | F1% | ACC% | NMI% | F1% | ACC% | NMI% | F1% | ACC% | NMI% | F1% | ACC% | NMI% | F1% | ACC% | NMI% | F1% | ||

Spectral-g | Graph | 34.19 | 19.49 | 30.17 | 25.91 | 11.84 | 29.48 | 39.74 | 3.46 | 51.97 | 23.58 | 19.28 | 17.21 | 39.84 | 8.24 | 10.70 | - | - | - |

DNGR | Graph | 49.24 | 37.29 | 37.29 | 32.59 | 18.02 | 44.19 | 45.35 | 15.38 | 17.90 | 37.58 | 35.85 | 25.38 | - | - | - | - | - | - |

DeepWalk | Graph | 46.74 | 31.75 | 38.06 | 36.15 | 9.66 | 26.70 | 61.86 | 16.71 | 47.06 | 38.46 | 32.38 | 25.74 | - | - | - | - | - | - |

M-NMF | Graph | 42.30 | 25.60 | 32.00 | 33.60 | 9.90 | 25.50 | 47.00 | 8.40 | 44.30 | - | - | - | - | - | - | - | - | - |

K-means | Feature | 34.65 | 16.73 | 25.42 | 38.49 | 17.02 | 30.47 | 57.32 | 29.12 | 57.35 | 33.37 | 30.20 | 24.51 | 33.09 | 9.36 | 11.31 | 52.79 | 19.84 | 29.01 |

Spectral-f | Feature | 36.26 | 15.09 | 25.64 | 46.23 | 21.19 | 33.70 | 59.91 | 32.55 | 58.61 | 41.28 | 43.99 | 25.20 | 29.71 | 11.65 | 17.76 | - | - | - |

ARGE | Both | 64.00 | 44.90 | 61.90 | 57.30 | 35.00 | 54.60 | 59.12 | 23.17 | 58.41 | 41.40 | 39.50 | 38.27 | - | - | - | - | - | - |

ARVGE | Both | 63.80 | 45.00 | 62.70 | 54.40 | 26.10 | 52.90 | 58.22 | 20.62 | 23.04 | 41.55 | 40.01 | 37.80 | - | - | - | - | - | - |

GAE | Both | 53.25 | 40.69 | 41.97 | 41.26 | 18.34 | 29.13 | 64.08 | 22.97 | 49.26 | 17.33 | 11.93 | 15.35 | - | - | - | - | - | - |

VGAE | Both | 55.95 | 38.45 | 41.50 | 44.38 | 22.71 | 31.88 | 65.48 | 25.09 | 50.95 | 28.67 | 30.28 | 20.49 | - | - | - | - | - | - |

MGAE | Both | 63.43 | 45.57 | 38.01 | 63.56 | 39.75 | 39.49 | 43.88 | 8.16 | 41.98 | 50.14 | 47.97 | 39.20 | 38.04 | 32.43 | 29.02 | - | - | - |

AGC | Both | 68.92 | 53.68 | 65.61 | 67.00 | 41.13 | 62.48 | 69.78 | 31.59 | 68.72 | 47.65 | 45.28 | 40.36 | 40.54 | 32.46 | 31.84 | 75.21 | 59.10 | 59.81 |

DAEGC | Both | 70.40 | 52.80 | 68.20 | 67.20 | 39.70 | 63.60 | 67.10 | 26.60 | 65.90 | 38.25 | 37.63 | 23.64 | 39.87 | 32.81 | 19.05 | - | - | - |

GMM-VGAE | Both | 71.50 | 54.43 | 67.76 | 67.44 | 42.30 | 63.22 | 71.03 | 30.28 | 69.74 | - | - | - | - | - | - | - | - | - |

DNENC-Att | Both | 70.40 | 52.80 | 68.20 | 67.20 | 39.70 | 63.60 | 67.10 | 26.60 | 65.90 | - | - | - | - | - | - | - | - | - |

DNENC-Con | Both | 68.30 | 51.20 | 65.90 | 69.20 | 42.60 | 63.90 | 67.70 | 27.50 | 67.50 | - | - | - | - | - | - | - | - | - |

FGC | Both | 72.90 | 56.12 | 63.27 | 69.01 | 44.02 | 64.43 | 70.01 | 31.56 | 69.10 | 51.10 | 44.12 | 34.79 | 48.25 | 35.24 | 35.52 | - | - | - |

GLGC (core) | Both | 71.62 | 52.26 | 65.27 | 68.67 | 40.97 | 60.10 | 68.69 | 31.25 | 67.78 | 47.00 | 41.66 | 35.33 | 52.91 | 32.06 | 34.51 | 78.01 | 53.62 | 67.97 |

GLGC (degree) | Both | 73.40 | 55.22 | 67.85 | 70.38 | 42.92 | 61.12 | 69.88 | 31.87 | 68.88 | 52.78 | 48.62 | 41.56 | 49.40 | 32.27 | 28.48 | 83.15 | 62.83 | 75.12 |

Method | Cora | Citeseer | Pubmed | Wiki | Large Cora | Coauthor Phy |
---|---|---|---|---|---|---|

AGC | 3.42 | 40.36 | 20.77 | 8.21 | 29.18 | 3172.58 |

DAEGC | 561.69 | 946.89 | 50,854.15 | 562.85 | 9339.67 | - |

FGC | 4.60 | 9.49 | 268.44 | 8.11 | 58.76 | - |

GLGC | 2.54 | 17.76 | 297.23 | 1.36 | 165.26 | 2441.03 |

Method | Cora | Citeseer | Pubmed | Wiki | Large Cora | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ACC% | NMI% | F1% | ACC% | NMI% | F1% | ACC% | NMI% | F1% | ACC% | NMI% | F1% | ACC% | NMI% | F1% | |

Baseline1 | 67.61 | 53.12 | 56.53 | 67.02 | 41.57 | 62.61 | 69.90 | 32.51 | 68.99 | 53.89 | 51.27 | 46.06 | 48.30 | 26.23 | 13.96 |

$f(A)=0$ | 69.46 | 56.10 | 63.25 | 66.93 | 41.73 | 61.09 | 64.45 | 24.63 | 64.40 | 56.30 | 52.44 | 44.85 | 43.46 | 27.48 | 31.73 |

$f(A)=A$ | 68.57 | 53.94 | 59.83 | 67.72 | 41.06 | 59.11 | 70.14 | 32.47 | 69.29 | 51.60 | 47.19 | 43.55 | 49.26 | 30.15 | 17.81 |

$f(A)=A+{A}^{2}$ | 72.90 | 56.12 | 63.27 | 69.01 | 44.02 | 64.43 | 70.01 | 31.56 | 69.10 | 51.10 | 44.12 | 34.79 | 48.25 | 35.24 | 35.52 |

$f(A)=A+{A}^{2}+{A}^{3}$ | 71.57 | 56.10 | 59.83 | 67.15 | 40.29 | 58.10 | 70.32 | 30.34 | 69.49 | 47.53 | 39.68 | 34.37 | 48.31 | 30.51 | 24.01 |

$f(A)=A+{A}^{2}+{A}^{3}+{A}^{4}$ | 71.49 | 52.55 | 63.96 | 67.33 | 40.03 | 58.23 | 68.49 | 27.32 | 68.21 | 43.49 | 35.83 | 27.17 | 46.87 | 26.26 | 30.91 |

Method | Cora | Citeseer | Pubmed | Wiki | Large Cora | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ACC% | NMI% | F1% | ACC% | NMI% | F1% | ACC% | NMI% | F1% | ACC% | NMI% | F1% | ACC% | NMI% | F1% | |

Baseline2 | 45.32 | 24.13 | 37.17 | 53.64 | 25.76 | 49.30 | 60.41 | 31.12 | 59.17 | 45.39 | 40.35 | 32.74 | 47.25 | 16.66 | 17.67 |

$f(A)=A$ | 73.40 | 55.22 | 67.85 | 70.38 | 42.92 | 61.12 | 69.88 | 31.87 | 68.88 | 52.78 | 48.62 | 41.56 | 49.40 | 32.27 | 28.48 |

$f(A)=A+{A}^{2}$ | 73.42 | 56.17 | 67.96 | 70.56 | 42.97 | 61.22 | 69.91 | 31.93 | 68.91 | 53.67 | 47.74 | 40.26 | 55.11 | 32.55 | 23.32 |

$f(A)=A+{A}^{2}+{A}^{3}$ | 71.12 | 52.66 | 63.93 | 69.56 | 41.61 | 60.34 | 68.96 | 31.57 | 68.03 | 53.05 | 47.14 | 39.81 | 54.32 | 29.27 | 26.14 |

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**MDPI and ACS Style**

Zhang, X.; Xie, X.; Kang, Z.
Graph Learning for Attributed Graph Clustering. *Mathematics* **2022**, *10*, 4834.
https://doi.org/10.3390/math10244834

**AMA Style**

Zhang X, Xie X, Kang Z.
Graph Learning for Attributed Graph Clustering. *Mathematics*. 2022; 10(24):4834.
https://doi.org/10.3390/math10244834

**Chicago/Turabian Style**

Zhang, Xiaoran, Xuanting Xie, and Zhao Kang.
2022. "Graph Learning for Attributed Graph Clustering" *Mathematics* 10, no. 24: 4834.
https://doi.org/10.3390/math10244834