Marginal Nodes Matter: Towards Structure Fairness
in Graphs

To account for the impact of graph structure toward fair decision-making in GNNs, we first need to identify the marginal nodes in the graph. To reflect node sociological origin [36] and the graph structure at the node level, we use two important centralities (i.e., closeness centrality and eigenvector centrality) to quantify the structure of each node. Closeness centrality [36] of a node measures its average nearness (inverse distance) to all other nodes in the graph. Eigenvector centrality [9] of a node measures the influence it has on all other nodes in the graph.

Closeness Centrality Closeness centrality [5, 49] of a node is the reciprocal of the sum of the shortest path distances from the node to all other nodes on the graph. The nodes with a high closeness centrality score have the shortest distances to all other nodes, which easily access other nodes and influence other nodes. For instance, in a social network, a person with a lower mean distance from others might find that their opinions reach others more quickly than the opinions of someone with a higher mean distance. Formally, the mathematical expression of closeness centrality of node $n_{i}$ is:

where $d(n_{i},n_{j})$ is the shortest-path distance between nodes $i$ and $j$, $N$ is the number of nodes in the graph.

Eigenvector Centrality Eigenvector centrality is a measure of the influence of each node in a graph [9]. A high eigenvector centrality score for a node indicates that it is connected to many other nodes with high scores. The eigenvector centrality score for node $i$ is calculated as the $i$-th element of the vector $\mathbf{x}$, which is defined by the equation $\mathbf{Ax}=\lambda\mathbf{x}$, where $\mathbf{A}$ is the adjacency matrix of the graph and $\lambda$ is the corresponding eigenvalue. According to the Perron-Frobenius theorem [47], if $\lambda$ is the largest eigenvalue of $\mathbf{A}$, then there exists a unique solution $\mathbf{x}$, all of whose entries are positive.[29] Throughout this paper, we use the indicator variable $s_{i}$ to represent the centrality metric of node $i$. Specifically, $s_{i}$ can take either closeness centrality or eigenvector centrality, depending on the context.

To explore the structure unfairness in GNNs, we perform node classification task with representative GNNs, i.e., graph convolutional networks (GCN) [35] and graph attention networks (GAT) [59]. Specially, we cluster nodes in the graph into multiple bins based on their indicators of structure (i.e., closeness centrality and eigenvector centrality). Then we calculate the average accuracy of each bin and compare the accuracy difference between bins.

Experimental Results. We present the experimental results on CoraFull and PubMed dataset with closeness and eigenvector centrality in Figure˜1. The X-axis represents different levels of closeness/eigenvector centrality, while Y-axis represents the prediction accuracy.

Results Analysis. The lower classification accuracy of marginal nodes shows that they are treated unfairly by GCN and GAT. Therefore, it is clear that structure unfairness in GCN and GAT can lead to discriminative results for marginal individuals, especially when life-changing decisions are at stake. Such unfairness can lead to discriminatory results for marginalized individuals, particularly in cases where significant decisions are at stake. As shown in the figures on the top row, GCN and GAT tend to achieve higher classification accuracy for nodes with higher closeness centrality and lower classification accuracy for nodes with lower closeness centrality. The experimental results presented in this section indicate that SFairGNN is capable of achieving structural fairness for downstream tasks.

Similarly, the experiment results on another indicator of structure (i.e., eigenvector centrality) have a similar trend. From the figure, GCN and GAT bias towards nodes with higher eigenvector centrality. In contrast, SFairGNN has comparable average node classification accuracy and has more even performance distribution and a lower correlation between performance and eigenvector centrality. The figures in the second row in Figure˜1 show that eigenvector centrality is indeed highly correlated to classification accuracy with GCN and GAT models. The observation here is quite similar to the experiment using closeness centrality as an indicator of structure.

Here, we analyze the root cause of the unfair decision-making of GNNs. As we have discussed, the neighborhood aggregation mechanism of GNNs treats nodes with different structures unfairly, especially for the marginal nodes. To further illustrate the unfair neighbor aggregation mechanism, we use Figure˜2a as a toy example. There are three groups of nodes with different egocentric structures, which are marked in yellow, blue, and green. The yellow indicates the central nodes, while the green indicates the marginal nodes. And the blue nodes situate between them. Since the green nodes and the blue nodes obtain less information than the yellow nodes, the GNNs will treat these three groups of nodes differently. Through the analysis of the toy example, we conclude that the aggregation mechanism of GNNs can lead to structure unfairness and put the marginal nodes into a disadvantaged situation since it is inherently difficult for marginal nodes to receive the global information of the graph. Although the aggregation mechanism in GNNs improves the performance of the graph analysis, it amplifies the structure fairness.

Since centralities and classification accuracy are both continuous variables, we propose the following two metrics to measure structure fairness. One is the correlation between the indicator of graph structure and the probability of correct classification. Another one is the variation of the average classification accuracy for nodes in different bins within different egocentric structure ranges.

Before delving into the neighborhood expansion based structure debiasing, we first investigate the distribution of closeness centrality over a synthetic graph to better understand the unfairness of the traditional information aggregation of GNNs. We plot the distribution of closeness centrality in Figure˜3a based on the graph shown in Figure˜2a. The synthetic graph in Figure˜2a contains central, middle, and marginal nodes colored yellow, blue, and green. It is observed from Figure˜3a that the closeness centrality varies significantly among the three colors, where the marginal nodes are much lower than the central nodes.

Motivated by the above observation, we propose the neighborhood expansion method to debias the structure unfairness by strengthening the marginal nodes’ connections to the center of the graph. Specifically, a heuristic threshold is adopted to determine whether a node is marginal or not. We name this threshold as margin line and denote it as $line$. The margin line serves as an important hyperparameter of the proposed method. The nodes with closeness centrality $s_{i}\leq line$ are regarded as marginal nodes; otherwise, they are central nodes. For each node $n_{i}$, we expand its original neighbor set $\mathcal{N}_{i}$ to a hyperset $\{\mathcal{N}^{(1)}_{i},\mathcal{N}^{(2)}_{i},\cdots,\mathcal{N}^{(h)}_{i}\}$ to indicate neighbors within $h$ hops. The $1$-hop neighbor set $\mathcal{N}_{i}^{(1)}$ is given by $\mathcal{N}_{i}$, which is generated based on the original adjacency matrix $\mathbf{A}$. We define a debiased adjacency matrix $\tilde{\mathbf{A}}$ with each element given as:

where $s_{i}$ is the indicator of structure for node $n_{i}$. In the debiased matrix $\tilde{\mathbf{A}}$, we only strengthen the connections of marginal nodes. Then we use the $h$-order power of debiased matrix $\tilde{\mathbf{A}}^{h}$ to generate neighbor set $\mathcal{N}_{i}^{(h)}$. We obtain a series of neighbor sets of node $n_{i}$ from different hops $\{\mathcal{N}^{(1)}_{i},\mathcal{N}^{(2)}_{i},\cdots,\mathcal{N}^{(h)}_{i}\}$, which will be used for hop-aware attentive information aggregation.

Figure˜2b, Figure˜2c, and Figure˜2d represent the graphs obtained by neighborhood expansions with $h=1,2$ and $3$, respectively. Note that the graph with $1$-hop expansion is the same as the original graph. To verify the effectiveness of neighborhood expansion, we illustrate the closeness centrality distributions of graphs with different hop expansions in Figure˜3. As we can see, increasing hop numbers for neighborhood expansion shrinks the closeness centrality gap between different node groups. The empirical study shows that neighborhood expansion can effectively alleviate the structure unfairness in terms of closeness centrality.

Since the neighbors of different hops contain different amounts of information, we propose hop-aware attentive information aggregation to provide customized information aggregation from neighbors of different hops. The overview is illustrated in Figure˜2. Specifically, at the $k$-th graph convolution layer, let $\mathbf{X}^{(k-1)}$ and $\mathbf{X}^{(k)}$ denote the input and output embedding matrices of all the nodes, respectively. The signal processing at each layer includes two steps: 1) hop-wise information aggregation and 2) hop fusion. With the series of neighbor sets $\{\mathcal{N}^{(1)}_{i},\mathcal{N}^{(2)}_{i},\cdots,\mathcal{N}^{(h)}_{i}\}$ obtained by the neighborhood expansion, we can compute the hop-wise embedding matrices $\{\mathbf{X}^{(1,k)},\mathbf{X}^{(2,k)},\cdots,\mathbf{X}^{(h,k)}\}$ at the $k$-th layer. Then we fuse these embedding matrices to obtain the final output node representation $\mathbf{X}^{(k)}$. We present the details of the two steps in the following:

Model Analysis. Here we analyze why the two key components of SFairGNN promote structure fairness: 1) neighborhood expansion strengthens the connection of marginal nodes in the graph, which positions them closer to the center of the graph and thus improves their accessibility to information from other nodes. 2) hop-aware attentive information aggregation enables SFairGNN to learn customized weights for information from neighbors of different hops away, which adaptively considers the varied strength of influence from neighboring nodes.

Avoid Constraining Loss over Fairness SFairGNN promotes the structure fairness of marginal nodes by neighbor expansion instead of explicitly optimizing fairness in loss function for two reasons. First, SFairGNN intends to improve fairness not for the measure’s own sake, but for a “healthier” graph structure. This minimizes the trade-off between the overall performance and fairness for SFairGNN, as lowering the performance of “privileged” nodes is the shortest path to achieve fairness for direct optimization. From Table˜2, we verify SFairGNN does not sacrifice the model performance for fairness. Second, compared to direct optimization, neighbor expansion is drastically more efficient. This is because computing centrality scores for every training iteration is prohibitively expensive.

Comparison to Related Works. Our proposed method focuses on structure fairness, while the previous works [48, 17, 57, 11] mainly focus on learning fair node representations regarding protected attributes. For the two methods which resemble SFairGNN the most, we discuss the difference between them and our proposed method below. Fairwalk [48] proposes a fairness-aware embedding method based on the random walk technique for attributed networks. FairGNN [17] uses adversarial debiasing and covariance constraint to regularize the GNN to obtain fair node representations and predictions. Both of them focus on the categorical sensitive attributes, while our proposed method does not require any sensitive features and aims to achieve structure-level debias.

Comparison to Cold-Start Our problem is different from the cold-start problem to a large extent, which does not aim to improve fairness across users but improve the recommendation performance for new users. In addition, while cold-start in recommender systems gradually warms up, unfairness in graphs, especially in the social context, can worsen over time and thus an algorithmic solution is needed.

In this section, we outlines our experimental setting for evaluating the effectiveness of SFairGNN in mitigating bias and achieving structural fairness. Specifically, we provide a comprehensive overview of the benchmark datasets, the baseline methods, and the implementation details both our proposed method and the baseline models.

Baselines In the experiments, we consider the widely used graph neural networks including graph convolutional networks (GCN) [35], graph attention networks (GAT) [59], and GraphSAGE [30] as baseline methods. The details of baseline methods are listed as follows:

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  GCN [35] is the pioneer work of graph neural networks. GCN is a scalable approach for semi-supervised learning on graph-structured data that is based on an efficient variant of convolutional neural networks.

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  GAT [59] incorporates the attention mechanism into the propagation step. This method computes the hidden states of each node by attending to its neighbors following a self-attention strategy.

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  GraphSAGE [30] efficiently generates node representations for previously unseen data. Instead of training individual representations for each node, GraphSAGE learns a function that generates representations by sampling and aggregating features from a node’s local neighborhood.

Datasets Following the previous works, we use six benchmark datasets to study the graph structure fairness on GNN models. They are Cora, CiteSeer, PubMed, CoraFull, Physics, and Photo [65]. The detailed statistics of these datasets are presented in Table˜1.

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  Cora, CiteSeer, and PubMed are citation networks [65], where nodes and edges denote papers and citations, respectively. Node features are bay-of-words for papers and node labels indicate the fields of papers.

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  CoraFull [8] is a well-known citation network that contains labels based on the paper topic. CoraFull contains a network extracted from the entire citation network of Cora, while the Cora dataset is its subset.

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  Physics [51] is co-authorship graph based on the Microsoft Academic Graph from the KDD Cup 2016 challenge. Nodes are authors and edges indicate co-authorship. Node features represent paper keywords for each author’s papers, and class labels indicate the most active fields of study for each author.

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  Photo [51] is part of the Amazon co-purchase graph, where nodes represent goods, and edges indicate the frequency of two goods being bought together. The node features are bag-of-words encoded product reviews and class labels are given by the product category.

Implementation Details We implement both SFairGNN and the baselines with PyTorch [1] and PyTorch Geometric [27]. We initialize model weights with xavier [28] and adopt the Adam optimizer to minimize the cross-entropy loss. We set the initial learning rate to $0.005$. We adopt the randomly initialized and trainable embedding vector for each node over a graph instead of node features provided by the dataset. We use $90/10$ as the training/test dataset split.

To address research question Q1, we conduct experiments to compare the proposed SFairGNN with baseline methods and evaluate their fairness performance using the proposed metrics: PCC (Pearson correlation coefficient) and STD (standard deviation). In this section, we utilize closeness centrality and eigenvector centrality as indicators of the graph structure in the experiments. Next, we present an analysis of the experimental results for both closeness centrality and eigenvector centrality.

To answer the research question Q2, we conduct a series of experiments to explore the effect of fusion methods in SFairGNN. We consider replacing the fusion method with several different strategies as follows:

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  SEQ represents the approach where the model aggregates information of different hop neighbors sequentially. This means the model first aggregates information from first hop neighbors and then moves on to aggregate information from other hop neighbors.

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  MAX represents the approach where the model selects the maximum value of each element from different hop embeddings for each node. This approach highlights the most important neighbor among different neighbors.

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  AVG represents the approach where the model takes the average of embeddings of different hop neighbors for each node. This approach treats all the neighbors equally during attentive information aggregation.

One can see from Figure˜4 that the fusion method MAX obtains the best performance among the three fusion methods because the MAX fusion method is capable of selecting the most important neighbor automatically to fuse the embeddings. Since the expanded neighbor may introduce noise to the graph structure, the MAX strategy can also reduce the influence of such adverse effects. On the other hand, the SEQ strategy obtains the worst performance on the metric PCC, which may be caused by the fact that SEQ treats different hop neighbors in the same way.

To answer the research question Q3, we conduct experiments to explore the sensitivity of SFairGNN to different hyperparameters, including the hop number $h$ and the margin line $line$. The hop number $h$ indicates the extent of the neighborhood expansion. The margin line $line$ determines whether the nodes are marginal nodes or not. For simplicity, we use closeness centrality as the indicator of structure in the experiments for hyperparameter sensitivity analysis.

2) when the hop number is too high, the performance drops because the original graph structure is damaged—being much different from the graph after neighborhood expansion.

Graph neural networks (GNNs) [64, 74, 72, 58, 71, 52, 31, 56, 69] have been the new state-of-the-art method to analyze graph-structured data, which are widely applied to social networks [32], and academic citation networks [35], knowledge graphs [73, 21], and to name a few. Starting with the success of graph convolutional network in the semi-supervised node classification task, a wide variety of GNN variants have enhanced and improved the node representation learning and downstream learning tasks [30, 59, 62]. Most of GNNs follow a message-passing strategy to learn node representations over a graph. GNNs apply a neighbor aggregator to update node representations iteratively via combining representations of neighbors and that of the node itself. GraphSAGE [30] learns representations by sampling and aggregating neighbor nodes, whereas GAT [59] uses the attention mechanism to aggregate representations from all neighbors. Besides, there is another branch of GNNs, spectral GCNs [12, 19, 35], which use spectral filters over graph laplacian to define the convolution operation.

Fairness [44, 46, 13] in machine learning algorithms attracts a lot of interest from both academic and industrial communities since life-changing decision-making needs to be fair in many sensitive environments, such as loan applications, health care [2], and hiring [7, 4]. One mainstream approach to achieve fairness is to use the in-processing approaches, which reduce the bias in the training data. In-processing approaches typically incorporate one or more fairness constraints into the model optimization process to maximize the performance of downstream tasks and improve fairness simultaneously [25, 67, 68]. Recently, a few works [11, 48, 17, 38, 41, 34, 22, 37, 33] focus on learning fair node representation regarding protected attributes (e.g., gender, age, race). [11] proposes an adversarial framework to enforce fairness constraints on graph learning, which can flexibly accommodate different combinations of fairness constraints during inference. [48] aims to study potential bias issues inherent within graph embedding and propose a fairness-aware embedding method named Fairwalk. FairGNN [17] uses adversarial debiasing and covariance constraint to regularize the GNN to yield fair predictions. There is also a line of works focusing on the fairness in graph structure [60, 40, 53, 15]. The position-aware graph neural network [66] also investigates the “position” of the node in graphs, which is related to our proposed method. A more comprehensive review of related works of fairness in graphs can be found in [23].