- 概
- 符號說明
- Homophily on Feature Aspect
[1] Chen Y., Luo Y., Tang J., Yang L., Qiu S., Wang C. and Cao X. LSGNN: Towards general graph neural network in node classification by local similarity. 2023.
[2] Jin D., Wang R., Ge M., He D., Li X., Lin W. and Zhang W. RAW-GNN: Random walk aggregation based graph neural network. 2022.
[3] Lee S. Y., Kim S., Bu F., Yoo J., Tang J. and Shin K. Feature distribution on graph topology mediates the effect of graph convolution: Homophily perspective. ICML, 2023.
[4] Yang L., Li M., Liu L., Niu B., Wang C., Cao X. and Guo Y. Diverse message passing for attribute with heterophily. NeurIPS, 2021.
Zheng Y., Luan S. and Chen L. What is missing in homophily? Disentangling graph homophily for graph neural networks. 2024.
概
收集了一些基於 feature 而不是 label 的 homophily 的指標.
符號說明
- \(\mathcal{G} = (\mathcal{V}, \mathcal{E})\), 圖;
- \(\bm{X} \in \mathbb{R}^{|\mathcal{V}| \times M}\), node features;
- \(\mathcal{N}_u,\), 節點 \(u\) 的一階鄰居;
Homophily on Feature Aspect
-
Generalized edge homophily [2]:
\[h_{GE}(\mathcal{G}, \bm{X}) = \frac{1}{|\mathcal{E}|} \sum_{e_{uv} \in \mathcal{E}} \text{sim}(\bm{x}_u, \bm{x}_v), \]其中 \(\text{sim}(\cdot, \cdot)\) 可以是 cosine 相似度, 歐式距離等.
-
Local Similarity [1]:
\[h_{LS-cos} (\mathcal{G}, \bm{X}) = \frac{1}{|\mathcal{V}|} \sum_{u \in \mathcal{V}} \frac{1}{d_u} \sum_{v \in \mathcal{N}_u} \text{sim}(\bm{x}_u, \bm{x}_v). \] -
Attribute homophily:
\[h_{attr, m}(\mathcal{G}, \bm{X}_{:, m}) = \frac{1}{\sum_{u \in \mathcal{V}} X_{u, m}} \sum_{u \in \mathcal{V}} \bigg( X_{u, m} \frac{ \sum_{v \in \mathcal{N}_u} X_{v, m} }{ d_u } \bigg), \\ h_{attr}(\mathcal{G}, \bm{X}) = \sum_{m=1}^M h_{attr, m} (\mathcal{G}, \bm{X}_{:, m}). \] -
Class-controlled feature homophily [3]:
\[h_{CF}(\mathcal{G}, \bm{X}, \bm{Y}) = \frac{1}{|\mathcal{V}|} \sum_{v \in \mathcal{V}} \frac{1}{d_u} \sum_{v \in \mathcal{N}_u} \big( d(v, \mathcal{V} \setminus \{u\}), d(v, \{u\}) \big), \\ d(u, \mathcal{V}') = \frac{1}{|\mathcal{V}'|} \sum_{v \in \mathcal{V}'} \| (\bm{x}_u| \bm{Y}) - (\bm{x}_v | \bm{Y}) \|, \\ \bm{x}_u | \bm{Y} = \bm{x}_u - \bigg( \frac{ \sum_{Y_u = Y_v} \bm{x}_v }{ |\{ v| Y_u = Y_v, v \in \mathcal{V} \}| } \bigg). \]