Let G=(V,E,X) as a graph, where V is the set of N nodes, E⊆V×V is the edge set, X∈ℝ^N×D is the node features with D dimensions. A∈{0, 1}^N×N is the adjacency matrix, where A_uv = 1 if there is an edge between nodes u and v. (u, v) denotes an edge between node u and node v. S∈ℝ^N×K is the vector containing sensitive attributes, K is the number of sensitive attributes can take on, (e.g., S_u∈{Female, Male, Unkown} for node u). g(·, ·):ℝ^H×ℝ^H → ℝ is the bivariate link predictor, and g(z_u, z_v) is the predicted probability of an edge (u,v)∈E in a given graph, where z_u and z_v are the node embedding vectors with dimension H for node u and v. The problem of fair link prediction aims to learn a synthetic graph Gf=(Vf,Ef,Xf), where a link predictor g(·, ·) trained on Gf will obtain comparable performance with it trained on the original graph G, and the link predictions are fair. In our experiemnts, |Vf|=|V| and Xf∈ℝN×D.

Previous research in fair machine learning has typically defined fairness in the context of binary classification as the condition where the predicted label is independent of the sensitive attribute. In the domain of link prediction, which involves estimating the probability of a link between pairs of nodes in a graph, fairness can be extended by ensuring that the estimated probability is independent of the sensitive attributes of the two nodes involved. In this subsection, we introduce two fairness concepts relevant to link prediction: demographic parity and equal opportunity.

In this subsection, we describe how to equip the learned graph with fairness-aware properties. This is achieved by incorporating a dyadic fairness regularizer, as specified in Equation 1, into the learning process of the fairness-enhanced graph. Further details on this process can be found in Section 2.2. The schematic diagram in Figure 2 illustrates the fairness objective of the fairness-enhanced graph learning within FairLink.

The concept of fairness constraint has been investigated in Zafar et al. (2015, 2017) by minimizing the disparity in fairness between users' sensitive attributes and the signed distance from the users' features to the decision boundary in fair linear classifiers. In this paper, we incorporate a fairness regularizer derived from Δ_DP (Chuang and Mroueh, 2021; Zemel et al., 2013), which quantifies the difference in the average predictive probability between various demographic groups. The fairness loss function Lfair at training epoch t is defined as follows:

where E is estimated Eu,v~V×V is generated from a link between any node pair in the graph. where Y^ represents the prediction probability of the downstream task. The variable N denotes the total number of instances, while N_{s = 0/1} refers to the total number of samples in the group associated with the sensitive attribute values of 0/1 respectively. The fundamental requirement for Δ_DP is that the average predictive probability Y^ within the same sensitive attribute group serves as a reliable approximation of the true conditional probability P(Y^=1|S=0) or P(Y^=1|S=1).

In this section, we address the first objective: determining how to learn a fairness-enhanced graph such that a link predictor trained on it exhibits comparable performance to one trained on the input graph. FairLink first computes the link prediction loss for the original graph, denoted as L(G), by calculating the cross-entropy loss between the predicted link distribution (based on the dot product scores of the node embeddings) and the actual link distribution. Similarly, the link prediction loss for the synthetic graph, L(Gf), is computed in the same manner. The gradients of both graphs with respect to the link predictors' weights, denoted as ∇θL(G) and ∇θL(Gf), are then obtained. We define the utility loss Lutil as the sum of the distances between these gradients across all training epochs.

Previous studies have utilized Cosine Distance to measure the distance between two gradients (Zhao et al., 2020; Jin et al., 2023; Liu and Shen, 2024b,a). While effective, Cosine Distance is scale-insensitive, meaning it ignores the magnitude of the vectors. Since the magnitude of the gradient is critical for optimization, incorporating it into the distance measurement is important. To address this limitation, we propose a combined approach that integrates Cosine Distance with Euclidean Distance, which accounts for vector magnitudes. Thus, the revised distance function D is defined as:

where D_cos denotes the Cosine Distance, D_euc denotes the Euclidean Distance, γ serves as a trade-off hyperparameter, and θ_t is the trainable parameters for link predictor at training epoch t. The definitions of these distances are as follows:

The utility loss at a specific epoch t, denoted as Lutil, is computed by summing the gradient distances between G and Gf across all training epochs. It is formally defined as follows:

Minimizing the utility loss ensures that the training trajectory of Gf closely follows that of G, leading to parameters learned on Gf closely approximating those learned on G. As a result, Gf preserves the essential information of the input graph G.

Optimizing a fairness-enhanced graph directly is computationally expensive due to the quadratic complexity involved in learning A_f. To address this challenge, previous work (Jin et al., 2023) proposed modeling A_f as a function of X_f. We initialize the node feature X_f by randomly selecting original features from each class. Note that learning a fairness-aware feature matrix for fair link prediction is important because this matrix will be used for node embedding when training a new link predictor on the fairness-enhanced graph. Therefore, it is necessary for the feature matrix itself to be fairness-aware. We further simplify this approach by using a multi-layer perceptron parameterized by ψ with a sigmoid activation function to model the relationship, thereby reducing the computational burden. Thus, the final loss function is as follows:

where T is the total training epochs, α and β are hyperparameters that govern the influence of two critical aspects: the gradient matching loss and the L₂ norm regularization, respectively.

Jointly optimizing X_f and ψ is often challenging due to the interdependence between them. To overcome this, we employ an alternating optimization strategy. We first update X_f for τ₁ epochs, then update ψ for τ₂ epochs. This process is repeated alternately until the stopping criterion is satisfied.

To achieve fair link prediction, we first use the fairness-enhanced graph Gf to train a link predictor. This link predictor can differ in architecture from the model that produced Gf and does not necessarily incorporate fairness considerations. In this paper, we define the link prediction function g(·, ·) as the inner product between the embeddings of two nodes u and v, for each node pair (u,v)∈V×V. Specifically, the function is defined as g(u, v) = u^⊤Σv, where Σ is a positive-definite matrix that scales the input vectors directionally. In our implementation, Σ is set to an identity matrix, simplifying g(·, ·) to the dot product, which is commonly used in link prediction research (Trouillon et al., 2016; Kipf and Welling, 2016b).

To evaluate the performance of our proposed method in both link prediction and fairness, we conducted a comprehensive comparison with the previously mentioned baselines using four real-world datasets. The results, which include the mean and standard deviations for all models across these datasets, are detailed in Table 1. From these results, we can draw the following observations:

In Figure 3, different colors are employed to distinguish between FairPR, Fairwalk, FairAdj, FLIP, FairEGM, and FairLink. Ideally, a debiasing method should be positioned in the upper-left corner of the plot to achieve the optimal balance between utility and unbiasedness. As depicted in the figures, methods based on FairLink generally provide the most favorable trade-offs between these two competing objectives. In contrast, while FairAdj usually offers superior debiasing with minimal utility loss, Fairwalk excels in maintaining high utility but is less effective in reducing bias. Although FairPR can achieve reasonable unbiasedness in embeddings, it significantly compromises utility compared to FairLink, as illustrated in the DBLP and Google+ datasets. In contrast, FairEGM does not show a notable debiasing effect.

The ability of FairLink to achieve a superior trade-off is attributed to its objective design, as outlined in Equation 8. Previous studies have demonstrated the effectiveness of gradient matching in generating synthetic data that maintains prediction accuracy during machine learning model training (Zhao et al., 2020; Jin et al., 2023, 2022a; Liu and Shen, 2024a). It is important to note that the synthetic graph derived from minimizing the gradient matching objective is not a singular solution; rather, it is quite flexible. Inspired by this insight, the proposed FairLink seeks to promote fairness in the learned data–the fairness-enhanced graph–by incorporating a fairness constraint into the gradient matching objective. In essence, FairLink aims to identify a solution that is close to the optimized graph space, where the gradient matching loss is zero, while simultaneously minimizing the fairness loss. A prior study also confirms the favorable fairness-utility trade-off in the learned graph when utilizing this design for the node classification task (Liu, 2023).

To validate the generalizability of the fairness-enhanced graph, we perform a cross-architecture analysis. Initially, we used GraphSAGE (SAGE) to generate synthetic graphs. These graphs are then evaluated across various architectures, including GCN, GAT, and VGAE, as well as on the original GraphSAGE model. Additionally, we apply FairLink with different structures to all datasets and assess the cross-architecture generalization performance of the fairness-enhanced graphs. The results of these experiments are documented in Figures 4A–D.

Compared to Table 1, FairLinkdemonstrates improved fairness performance over VGAEand Node2vecacross most model-dataset combinations. This indicates that FairLinkis versatile and consistently achieves gains across various architectures and datasets. Our fairness-enhanced graphs show generally superior performance in fairness metrics (e.g., Δ_DP and Δ_EO) and utility metrics (e.g., F1-score and AUC) across all datasets. Specifically, GraphSAGE excels in fairness across all datasets and achieves the best utility on Pubmed, DBLP, and Google+.

To evaluate the necessity of generating a synthetic graph for fair link prediction in architectures without built-in fairness considerations, we conducted an ablation study to compare the performance of the proposed FairLink with two of its variants: (1) FairLinkm, which is a model-centric variant of FairLink, excludes the synthetic graph and directly applies the dyadic fairness loss in Equation 5 to the training of a link predictor, and (2) FairLinkcos, which only uses the cosine distance function in the gradient matching process in Equation 6, by setting γ = 0. We evaluated both link prediction accuracy and fairness performance across various architectures that do not explicitly account for fairness.

For FairLink, we first trained GraphSAGE to obtain a fairness-enhanced graph. This graph was then used for training diverse architectures without any fairness design, including GraphSAGE, GAT, and VGAE, and we tested on the trained link predictor. In the variant without the synthetic graph generation, we incorporated the fairness loss directly into the training of GraphSAGE. However, similar to FairLink, we excluded fairness loss when training the other architectures, such as GAT and VGAE, to ensure a fair comparison.

(1) Data-centric vs. model-centric debiasing: From the results in the “(GAT)” and “(VGAE)” columns of Table 2, which correspond to architectures without fairness-aware designs, we observe that FairLinkcan generalize fairness when applied to different architectures, whereas FairLinkm cannot. Comparing the “(Self)” columns for FairLink and FairLinkm, it is evident that directly adding fairness loss during the training of a link predictor significantly degrades accuracy. This aligns with findings from prior studies, where applying fairness loss directly during the training of node classifiers led to a similar drop in performance (Qian et al., 2024; Dong et al., 2024). However, FairLink, by utilizing a gradient matching loss to preserve link prediction accuracy on the fairness-enhanced graph, successfully alleviates this trade-off. As a result, FairLink achieves both higher accuracy and better generalization of fairness across different architectures.

(2) Euclidean & cosine distance vs. cosine distance: From the results of FairLink and FairLinkcos, we observe a decline in link prediction performance when the Euclidean function is excluded from gradient matching. This finding highlights the importance of minimizing the gradient magnitude when optimizing the fairness-enhanced graph. In conclusion, FairLink, which utilizes both Euclidean and Cosine distance functions, is more effective at preserving the original graph's information in the learned graph.

Let L denote the number of MLP layers used for learning the adjacency matrix, and let d represent the number of hidden units. The complexity of FairLink is constituted by several calculations: (1) Calculation of A′: This step has a complexity of O(N²d²). (2) Forward Pass of GNN: Computing the forward pass on the full graph incurs a complexity of O(kLNd²), where k is the size of the sampled nodes per training instance. (3) Training on Fairness-Enhanced Graph: The complexity for training on the fairness-enhanced graph is O(LNd). (4) Gradient Matching Strategy: Including the calculation of additional matching metrics, the complexity of the gradient matching strategy is O(2|θ|+|A′|+|X′|). Considering T iterations and M different initializations, the total complexity is MT times the sum of the aforementioned complexities. (5) For link prediction tasks, calculating the dot product of node embeddings adds an extra cost of O(N²D). Therefore, the overall complexity of FairLink is the sum of all these components.

To evaluate whether it is feasible to learn a smaller fairness-enhanced graph, we implement method by initializing a synthetic graph with 75% of the nodes from the input training graph. In this experiment, we fine-tune all the hyperparameters in FairLink using the same settings as for the full graph on the DBLPdataset. We then compare the performance of a link predictor trained on both the full fairness-enhanced graph and the smaller graph. We assess the performance on two different architectures: the same architecture used for generating the graph (labeled as “Self”) and a different architecture (labeled as “GAT”).

From the results presented in Table 3, we find that FairLink is capable of learning a compact and generalizable fairness-enhanced graph for DBLPdataset. This demonstrates the scalability of FairLink for large-scale graphs and highlights its potential to be applied in the wild.

To identify the optimal architecture for FairLink, we conduct a cross-architecture experiment using different graph generation models. Specifically, we use one architecture to generate the fairness-enhanced graph and another architecture to train on the generated graph, followed by performance evaluation through testing.

From the results in Table 4A, we can find that the highest link prediction accuracy is achieved when the same architecture is used for both generation and testing. While GraphSAGE and VGAE exhibit similar levels of accuracy, a key distinction emerges when examining generalization performance. Specifically, fairness-enhanced graphs generated by GraphSAGE demonstrate better generalizability across different architectures, such as GCN, GAT, and VGAE. Furthermore, although both GCN and GraphSAGE show comparable fairness performance as shown in Table 4B, GraphSAGE exhibits a slight advantage in terms of generalization.

In recent years, numerous fairness definitions in machine learning have been proposed. These definitions generally fall into three categories. (1) Group fairness, which aims to ensure that certain statistical measures are approximately equal across protected groups (e.g., racial or gender groups) (Feldman et al., 2015; Hardt et al., 2016). (2) Individual fairness (Dwork et al., 2012; Kang et al., 2020; Dong et al., 2021, 2023) requires that similar individuals should be treated similarly. Compared with group fairness, individual fairness does not take sensitive features into account. Instead, it emphasizes fairness at the individual level, such as for each node in graph data. (3) Counterfactual fairness (Kusner et al., 2017; Ma et al., 2022; Zuo et al., 2022) seeks to ensure fairness by considering how decisions would hold under alternative scenarios. Counterfactual fairness is achieved when the prediction results for an individual remain consistent across their counterfactuals. In this context, “counterfactuals” refer to different versions of the same individual, where their sensitive features have been altered to various values. Thus, the prediction outcomes for the individual and their counterfactuals should be identical. In our experiments, we adopt two widely used definitions of group fairness: demographic parity and equal opportunity. Demographic parity (Feldman et al., 2015) requires that members of different protected classes are represented in the positive class at the same rate, meaning the distribution of protected attributes in the positive class should reflect the overall population distribution. In contrast, equal opportunity (Hardt et al., 2016) focuses on the model's performance rather than just the outcome; it requires that true positive rates are equal across different protected groups, ensuring that the model performs consistently for all groups. Methodologically, existing bias mitigation techniques in machine learning can be broadly categorized into three approaches: (1) Pre-processing, where bias is mitigated at the data level before training begins (Calders et al., 2009; Kamiran and Calders, 2009; Feldman et al., 2015); (2) In-processing, where the machine learning model itself is modified by incorporating fairness constraints during training (Zafar et al., 2017; Goh et al., 2016); and (3) Post-processing, where the outcomes of a trained model are adjusted to achieve fairness across different groups (Hardt et al., 2016).

Link prediction involves inferring new or previously unknown relationships within a network. It is a well-studied problem in network analysis, with various algorithms developed over the past two decades (Liben-Nowell and Kleinberg, 2003; Al Hasan et al., 2006; Hasan and Zaki, 2011). Specifically, heuristic methods define a score based on the graph structure to indicate the likelihood of a link's existence (Liben-Nowell and Kleinberg, 2003; Newman, 2001; Zhou et al., 2009). The primary advantage of heuristic methods is their simplicity, and most of these approaches do not require any training. Graph embedding methods learn low-dimensional node embeddings, which are then used to predict the likelihood of links between node pairs (Grover and Leskovec, 2016; Menon and Elkan, 2011). These embeddings are typically trained to capture the structural properties of the graph. Deep neural network methods have emerged as state-of-the-art for the link prediction task in recent years (Kipf and Welling, 2016a,b; Hamilton et al., 2017; Velickovic et al., 2018). This category includes GNNs, which leverage the multi-hop graph structure through the message-passing paradigm. Additionally, GNNs augmented with auxiliary information, such as pairwise information (Zhang M. et al., 2021), have been proposed to enhance link prediction. These advanced methods incorporate additional data to better capture the relationships between nodes (Zhang M. et al., 2021; Zhu et al., 2021; Wang et al., 2022).

With the success of GNNs, there has been increasing attention on fairness in graph representation learning (Dai et al., 2022). Some works have focused on creating fair node embeddings, which are subsequently used in link prediction (Bose and Hamilton, 2019; Buyl and De Bie, 2020; Cui et al., 2018). Others have directly targeted the task of fair link prediction (Masrour et al., 2020; Li et al., 2021). Specifically, dyadic fairness has been proposed for link prediction, which requires the prediction to be independent of whether the two vertices involved in a link share the same sensitive attribute (Li et al., 2021). To achieve dyadic fairness, the authors proposed FairAdj (Li et al., 2021), which leverages a variational graph auto-encoder (Kipf and Welling, 2016b) for learning the graph structure and incorporates a dyadic loss regularizer to enforce fairness. FairPageRank (FairPR) (Tsioutsiouliklis et al., 2021) is a fairness-sensitive variation of the PageRank algorithm. It modifies the jump vector to ensure fairness, both globally and locally. The locally fair PageRank variant specifically guarantees that each node behaves in a fair manner individually. DeBayes (Buyl and De Bie, 2020) adopts a Bayesian approach to model the structural properties of the graph, aiming to learn debiased embeddings using biased prior conditional network embeddings. Meanwhile, Fairwalk (Rahman et al., 2019) adapts Node2vec (Grover and Leskovec, 2016) to enhance fairness in node embeddings by adjusting the transition probabilities in random walks, weighing the neighborhood of each node based on their sensitive attributes. Finally, FLIP (Masrour et al., 2020) tackles graph structural debiasing by reducing homophily (the tendency of similar nodes to connect) in the graph. The fairness is assessed by the reduction in modularity, which measures the strength of the division of a graph into modules. FairEGM (Current et al., 2022), a collection of three methods that emulate the effects of a variety of graph modifications for the purpose of improving graph fairness.