# GCARe: Mitigating Subgroup Unfairness in Graph Condensation through Adversarial Regularization
## 1 Introduction  
Graph neural networks (GNNs) have emerged as popular models for handling graph data in various fields, such as social networks, recommender systems, molecular topology, and chemistry, owing to their capacity to leverage graph structures and aggregate neighborhood information [1; 2; 3; 4; 5]. However, modern deep GNNs are data-hungry and demand extensive training data to realize their full potential. Training GNNs on large graphs introduces challenges, leading to increased computational complexity, memory requirements, training time, and carbon emissions [6]. Furthermore, hyperparameter tuning and architecture searches [7] on large graphs exponentially exacerbate these complexities. As a result, scalable GNN training has become increasingly important [8; 9]. Graph condensation [10; 11] is newly developed as a promising approach for compressing large graphs into smaller ones, significantly reducing both training time and memory consumption without sacrificing the performance of the GNNs trained on condensed graphs. It achieves this by feeding both the original and condensed graphs into the GNN, and by aligning the model gradients that correspond to each input at every iteration. The condensed graph is treated as if it were learnable parameters upon which convergence would induce GNNs, and this is such as if they were trained on the original graph. Once the condensation is complete, training GNNs on the condensed graph can be efficient. For instance, if the Ogbn-arxiv [12] dataset is condensed to a size of 0.05% for its nodes, training on a condensed graph leads to a four-time greater acceleration compared to training onthe original graph (measured on one NVIDIA RTX 2080 ti GPU ). Thus, the condensed graph can serve as a substitute for the original graph and can be utilized repeatedly. This is particularly helpful in applications like continual learning [13] and hyperparameter search, where one needs to train GNNs multiple times. Although current graph condensation methods are capable of distilling most of the knowledge from the original graphs, we find that this capability varies with respect to different node subgroups, leading to severe fairness issues. Specifically, the GNNs trained on condensed graphs exhibit much larger performance gaps between advantaged and disadvantaged node subgroups when compared to those trained on original graphs. Figure 1 demonstrates such phenomenon on two real-world datasets, namely Cora and Credit-defaulter. In the Cora dataset, the node degrees follow a long-tail distribution; in it, we split the nodes into 3 subgroups according to their degrees. In the Credit-defaulter dataset, nodes are annotated with the Age attribute; in it, we used this attribute to split all of the nodes into two subgroups. We evaluated two existing graph condensation methods, GCond [11] and DosCond [10], and compared them against their use with the original full graph. We report both the prediction accuracy and fairness metrics. For Cora, the subgroup fairness was measured by \(\Delta_{acc}\), which is the difference between the highest and lowest subgroup accuracy; for Credit, the fairness metric was \(\Delta_{EO}\), which is the difference in true positive rates between two subgroups. As shown in the figure, while GNNs trained on condensed graphs by GCond or DosCond can achieve similar overall accuracies to training on full data, they are more biased toward certain advantaged groups, and the performance on disadvantaged groups is sacrificed. Figure 1: Empirical evaluation on the Cora and Credit-defaulter datasets. For Cora, we used node degree to split the subgroups. For Credit, we used the node attribute Age. (**a**): The node degrees of Cora follow a long-tail distribution. (**b**): The Credit dataset was highly imbalanced regarding a specific attribute. (**c**): GCond and DosCond yielded comparable overall accuracy on both datasets. (**d**): GCond and DosCond resulted in a larger disparate impact for GNNs when compared to training on the original full data.