# GCARe: Mitigating Subgroup Unfairness in Graph Condensation through Adversarial Regularization
## 1 Introduction
In this paper, we investigate the issue of the exacerbated subgroup unfairness that is present in current graph condensation techniques. Although a substantial body of literature exists on the topic of GNN fairness, existing algorithms of fair GNN training cannot be trivially applied in this case because they require that nodes in the training data are explicitly annotated with group IDs. However, current graph condensation methods do not preserve node attributes or group IDs during the compression process. Furthermore, current graph condensation methods are embedded within a complex bi-level optimization framework [10; 11; 14], making it even harder to integrate with modules like data augmentation [15], graph contrastive learning [16], or pseudo label prediction [17]. To address this problem, we propose **G**raph **C**ondensation with **A**dversarial **R**egularization (**GCARe**), which is a light-weight, efficient, and effective regularization term. GCARe renders condensed graphs that preserve the knowledge of different subgroups fairly to alleviate the unfairness of GNNs that are trained afterward. It achieves this by directly regularizing the condensation process with adversarial training. Specifically, we treat the condensation model as a generator, and append a discriminator to the output layer of the condensation model. During condensation, both the node features and group IDs are fed to the generator in order to produce hidden embeddings, from which the discriminator tries to predict the group IDs of the input nodes. Meanwhile, the generator is trained to fool the discriminator and to prevent it from making correct predictions. In this way, the group ID information is eliminated from the node representations, thus making the condensation model unbiased. Intuitively, GNNs trained on condensed graphs would inherit the unfairness of the GNNs that are utilized during the condensation procedure (i.e., the condensation model); this is because the latter one acts as the information bottleneck between the original graph and the former one. Therefore, it is necessary to impose a fair GNN to condense the graphs appropriately. Our contributions can be summarized as follows:  
1. We provide broad empirical evaluations, which reveal that performance disparities between advantaged and disadvantaged subgroups become more pronounced for GNNs that are trained on condensed graphs. 2. We introduce GCARe, an innovative paradigm that leverages adversarial learning to debias the condensation model, thereby achieving fair graph condensation. Notably, our proposed method maintains the same level of complexity as existing condensation techniques, and it can be implemented as a regularization term that is easy to plug and play. 3. Our comprehensive experiments demonstrate that GCARe not only mitigates fairness issues, but also enhances the overall performance of condensed graphs. For example, when applied to the Recidivism dataset, GCARe reduces the \(\Delta_{SP}\) value from 4.47% to 3.98%, as well as the \(\Delta_{EO}\) value from 3.40% to 2.52%, while simultaneously increasing the accuracy from 80.04% to 81.97%.