Feature homophily metric
概
收集了一些基于 feature 而不是 label 的 homophily 的指标.
符号说明
- \(\mathcal{G} = (\mathcal{V}, \mathcal{E})\), 图;
- \(\bm{X} \in \mathbb{R}^{|\mathcal{V}| \times M}\), node features;
- \(\mathcal{N}_u,\), 节点 \(u\) 的一阶邻居;
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, 欧式距离等.
-
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]:
\[h_{CF}(\mathcal{G}, \bm{X}, \bm{Y}) = \frac{1}{|\mathcal{V}|} \sum_{v \in \mathcal{V}} \frac{1}{d_u} \sum_{v \in \mathcal{N}_u} \big( d(v, \mathcal{V} \setminus \{u\}), d(v, \{u\}) \big), \\ d(u, \mathcal{V}') = \frac{1}{|\mathcal{V}'|} \sum_{v \in \mathcal{V}'} \| (\bm{x}_u| \bm{Y}) - (\bm{x}_v | \bm{Y}) \|, \\ \bm{x}_u | \bm{Y} = \bm{x}_u - \bigg( \frac{ \sum_{Y_u = Y_v} \bm{x}_v }{ |\{ v| Y_u = Y_v, v \in \mathcal{V} \}| } \bigg). \]
