# GCARe: Mitigating Subgroup Unfairness in Graph Condensation through Adversarial Regularization
## 5 Experiments
### Experiment Settings
#### 5.1.2 Subgroup Division Evaluation Metrics  
In this paper, we consider fairness with respect to both node degrees and attributes. For Cora, the nodes were divided, according to their degrees, into three subgroups. The thresholds of the degrees used to make divisions were chosen to generate subgroups of comparable sizes. The subgroup sizes were (71, 48, 21) for the training nodes, and (400, 348, 252) for the testing nodes. For Ogbn-arxiv, the nodes were divided into five subgroups in a similar manner. The subgroup sizes were (25,134, 16,597, 18,607, 17,183, 13,420) for the training nodes, and (11,576, 10,557, 10,111, 9330, 7029) for the testing nodes. For Credit  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dataset** & **\#Nodes** & **\#Edges** & **\#Classes** & **\#Features** & **Train/Valid/Test** \\ \hline Cora & 2708 & 5429 & 7 & 1433 & 140/500/1000 \\ \hline Ogbn-arxiv & 169,343 & 1,166,243 & 40 & 128 & 90,941/29,799/48,603 \\ \hline Credit-defaulter & 30,000 & 2,873,716 & 2 & 13 & 6000/7500/7500 \\ \hline Recidivism & 18,876 & 642,616 & 2 & 18 & 1000/4719/4719 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dataset Statistics. The first two datasets were divided into subgroups according to node degrees, while the last two were divided according to their sensitive attributes.  
Figure 2: Framework of our method. At each iteration, both the original graph and condensed graph were fed to the condensation GNN, which is regularized by the proposed adversary term. Red dotted lines indicate a backward gradient.  
defaulter, we followed [34] in using the binary sensitive attribute Age to split the subgroups, whereby NoDefaultNextMonth was used as the label. The subgroups sizes were \((5442,578)\) for the training nodes, and \((6811,689)\) for the testing nodes. For Recidivism, we also followed the preprocessing method of [34]. The attribute _Race_ was used to split the nodes into two subgroups. The group sizes were \((477,523)\) for the training nodes, and \((2303,2416)\) for the testing nodes.  
For the degree bias, we evaluated the GNN's fairness with respect to two metrics: \(\Delta_{acc}\) was set as the accuracy gap between the most and least advantaged subgroups, and \(\sigma_{acc}\) was set as the standard deviation of the accuracies across all subgroups. By denoting the accuracy of group \(i\) as \(a_{i},\forall i\in\mathcal{G}\), the two metrics were computed as follows:  
\[\Delta_{acc} :=\max_{i\in\mathcal{G}}a_{i}-\min_{j\in\mathcal{G}}a_{j} \tag{3}\] \[\sigma_{acc} :=\sqrt{\mathbb{E}[(a_{i}-a)^{2}]} \tag{4}\]  
For attribute bias, we focused on the fairness metrics commonly adopted, namely statistical parity and equal opportunity [26], which evaluate the statistical dependencies between predictions and attributes. Suppose a binary attribute is denoted as \(s\), and prediction as \(\hat{y}\), then these two metrics are defined as follows:  
\[\Delta_{SP} :=\big{|}p(\hat{y}=1|s=0)-p(\hat{y}=1|s=1)\big{|} \tag{5}\] \[\Delta_{EO} :=\big{|}p(\hat{y}=1|y=1,s=0)-p(\hat{y}=1|y=1,s=1)\big{|} \tag{6}\]