ComFairGNN: Community Fair Graph Neural Network

Graph Neural Networks (GNNs) utilize a layered architecture for neighborhood aggregation, where nodes progressively integrate information from their immediate neighbors across multiple iterations. For the $l$-th layer, the representation of a node $v$, denoted as $\mathbf{h}_{v}^{l}\in\mathbb{R}^{d_{l}}$, is formulated as follows:

where $d_{l}$ is the dimension of node representations in the $l$-th layer, $\text{AGGR}(\cdot)$ denotes an aggregation function such as mean-pooling (Kipf and Welling, 2016) or self-attention (Veličković et al., 2017), $\sigma$ is an activation function, $\mathcal{N}_{v}$ denotes the set of neighbors of $v$, and $\omega^{l}$ denotes the learnable parameters in layer $l$. The initial layer (layer 0) utilizes the raw input features as node representations, expressed mathematically as $\mathbf{h}_{v}^{0}=\mathbf{x}_{v}$, where $\mathbf{x}_{v}$ denotes the input attribute vector for node $v$.

By integrating node2vec (Grover and Leskovec, 2016) embeddings with $k$-means clustering (MacQueen et al., 1967), we can effectively partition the graph into communities based on the structural similarities captured by the embeddings.

Graph embedding using node2vec is a mapping that assigns each node in a graph to a vector in a $d$-dimensional space, denoted as $f:U\rightarrow\mathbb{R}^{d}$, where $d$ is a hyperparameter representing the number of dimensions in the vector space. For each node $u\in U$ in a graph $G$, multiple random walks are performed, as specified by the hyperparameter walk_num. This process generates a list of sequences, each comprising node IDs from individual random walks. A single random walk follows the standard approach in unweighted graphs, where at each step, the next node is selected uniformly at random from the current node’s neighbors. The hyperparameters walk_num and walk_len control the number of walks and the length of each walk, respectively. Next, node2vec uses the generated sequences to train a neural network and learn embedding vectors. Let us define a network neighborhood $N_{s}(u)$ as the set of nodes preceding and succeeding node $u$ in the generated walk sequences. The objective function is defined as:

$P(u^{\prime}\mid f(u))$ is the conditional probability represented as softmax.

Once the node embeddings are learned using node2vec, the next step involves clustering these embeddings to identify groups of similar nodes. Our primary focus for clustering is on $k$-means clustering, selected for its efficient linear runtime. $k$-means aims to minimize the total sum of squared errors, also known as inertia or within-cluster sum of squares (Shiao et al., 2023). This objective can be formally expressed as:

We specifically target group fairness scenarios where both the node label and the sensitive attribute are binary. Group fairness is the most prevalent and extensively studied form of fairness. In the context of group fairness, we consider a model or algorithm to be fair when its outputs exhibit no bias against any demographic group of sensitive attributes. In this work, we primarily focus on assessing the evaluation of existing graph fairness methods utilizing the two widely used metrics of group fairness, i.e., demographic parity (Dwork et al., 2012) and equal opportunity (Hardt et al., 2016).
Let $\hat{y}$ denote the prediction output of the classifier $f(\cdot)$. $y\in\{0,1\}$ and $s\in\{0,1\}$ represent the ground truth and the sensitive attribute, respectively.

Demographic Parity: Demographic parity, also referred to as statistical parity, necessitates that the prediction $\hat{y}$ is independent of the sensitive attribute $s$, symbolized as $\hat{y}\perp s$. Formally, demographic parity aims to have:

Equal Opportunity: Equal opportunity involves an identical true positive rate across all demographic groups. This concept can be expressed as:

During experiments, demographic parity and equal opportunity are utilized to assess fairness performance as below:

To debias nodes in a community with varying label neighborhood distribution, we propose a debiasing function based on the coreset sample nodes $c$, which are selected from different communities. Coreset selection methods aim to identify crucial subsets of training examples by employing specific heuristic criteria (Toneva et al., 2018). These subsets are considered essential for effectively training the model. Our study proposes a new method of graph coreset selection method to debias GNNs, addressing the local structural bias in their aggregation process. The coreset nodes are selected equally from each community for both groups to learn the debiasing function by contrasting between two groups of nodes, namely, subgroups $\mathcal{S}_{0}$ and $\mathcal{S}_{1}$. Note that GNNs neighborhood aggregation in Eq. 1, can only access its one-hop local contexts in each layer. The majority of real-world networks of nodes in a community tend to be from the same subgroup influenced by the sensitive attribute. Thus, it is a natural choice to select the coreset sample nodes of both groups from each community to balance the structural imbalance between different communities.

Here $c$ indicates the coreset sample nodes for all $k\in\{1,\ldots,K\}$. $K$ represents the total number of communities, and $C_{k_{S_{0}}}$ and $C_{k_{S_{1}}}$ represent selected sample nodes from subgroup $S_{0}$ and $S_{1}$ in community $k$ respectively.

To minimize the structural bias between the two groups in different communities, we contrast the output similarity of nodes with the same label in the coreset. This tends to maximize the similarity of nodes of the same label in different communities. This strategy enables GNNs to implicitly eliminate the structural bias between different communities.

To fundamentally eliminate the structural bias of GNNs neighborhood aggregation, which is the key operation in GNNs, we propose a neighborhood homophily distribution ratio to select the sample coreset from different communities.

A node in a graph is generally classified as highly homophilous when its edge homophily ratio $h(\cdot)$ is high, typically falling within the range $0.5\leq h(\cdot)\leq 1$, assuming an appropriate label context. Conversely, a node is considered heterophilous when its edge homophily ratio is low. Here, $h(\cdot)$ denotes the edge homophily ratio as previously defined.

As shown in figure 3, the neighborhood distribution ratio for identical label nodes in pokec-z dataset varies largely. If we consider that all nodes that share the same label have similar initial node features, then cleary, after GNN 1-hop neighborhood aggregation, the identical label nodes will have different output embedding. This inability to differentiate the embedding space of identical nodes in different communities of the graph can potentially lead the GNN to be biased towards certain subgroups. Considering this, the debiasing coreset nodes are sampled based on their neighborhood distribution from all communities of the graph.

Fairness-aware Graph Coreset: aims to identify a representative subset of training nodes $V_{w}\subset V_{\text{train}}=[n_{t}]$, where $|V_{w}|\leq c\ll n_{t}$, along with an associated set of sample weights. This selection is designed to approximate the training loss $\mathcal{L}(\theta)$ across all $\theta\in\Theta$, while maintaining fairness considerations. The graph coreset selection problem is formulated as:

which minimizes the worst-case error for all $\theta$. Both the graph adjacency matrix $A$ and node feature matrix $X$ are still needed to compute the full-graph embeddings $f_{\theta}(A,X)$ in the coreset loss. Since the goal of the fairness-aware graph, coreset is to debias the neighborhood aggregation of GNNs at the community level, we consider the coreset samples of both groups of $S_{0}$ and $S_{1}$ from all communities of the graph.

We focus on the task of node classification. We incorporate both the classification loss and the fairness loss associated with the classification to enhance the training process as illustrated in figure 2(e).

Here, $|Y|$ denotes the number of labeled examples (rows in $Y$), while $\hat{Y}_{i,j}$ represents the predicted probability of the $i^{th}$ example belonging to the $j^{th}$ class.

where $\text{sim}(\cdot,\cdot)$ is a cosine similarity function between two samples of same label. This fairness loss drives the coreset selection toward balancing the average intra-label similarities between the two label groups in $c$. It aims to constrain the similarity distribution of labels to be similar across the two groups, $S_{0}$ and $S_{1}$.

Fairness in Graph Neural Networks (GNNs) is an emerging area of research, researchers have performed signifi- cant work to measure the fairness of those systems(Wang et al., 2019; Fan et al., 2021; li2021user; Kleindessner et al., 2019; Horowitz and Kamvar, 2010). Generally, fairness can be categorized into group fairness and individual fairness. Group fairness focuses on ensuring that the model does not make decisions that favor or disadvantage certain groups. Group fairness is one of the mainly studied. EDITS (Dong et al., 2022) modifies the adjacency matrix and node attributes to produce unbiased graph-structured data by reducing the Wasserstein distance between different groups. Additionally, some studies (Dai and Wang, 2021; Wang et al., 2022b) achieve fairness in GNNs through adversarial training, resulting in a model that generates node embeddings indistinguishable with respect to the sensitive attribute. FairGNN (Dai and Wang, 2021) incorporates fairness regularization to address bias in graph-based tasks. It aims to ensure equitable treatment of different groups in both learned representations and final predictions. UGE (Wang et al., 2022a) addresses the unbiased graph embedding problem by learning node representations from a bias-free underlying graph. This approach eliminates the influence of sensitive node attributes on the embedding process. Another aspect of fairness, often referred to as counterfactual fairness, ensures that changes in sensitive attributes do not impact the prediction results (Agarwal et al., 2021a). NIFTY (Agarwal et al., 2021b) enhances counterfactual fairness and stability in node representations through a novel triplet-based objective function. It employs layer-wise weight normalization using the Lipschitz constant to improve performance and reliability. GEAR addresses graph unfairness through counterfactual graph augmentation and adversarial learning for attribute-insensitive embeddings. These combined approaches effectively reduce bias in graph-based models. GEAR (Ma et al., 2022) addresses graph unfairness through counterfactual graph augmentation and adversarial learning for attribute-insensitive embeddings. These combined approaches effectively reduce bias in graph-based models. Deg-FairGNN (Liu et al., 2023) aims to ensure fair outcomes for nodes with varying degrees within a group. Despite they introduce attribute and structural group fairness, overlooked the inherent local structural bias. In terms of individual fairness, nodes sharing similar attributes are treated equivalently (Dong et al., 2021). For example, REDRESS (Dong et al., 2021) proposes a ranking-based individual fairness optimization method to encourage each individual node relative similarity ranking list to the other nodes during the input and output. InFoRM (Kang et al., 2020) assesses GNN individual fairness using node similarity matrices. It incorporates a modified (d1, d2)-Lipschitz property to address fairness challenges in interconnected graph structures, moving beyond node features alone. However, none of these works are designed for community-level fairness on graphs.

There have been prior works that study the fairness paradoxes due to historical bias using statistical measures of fairness and counterfactual fairness. Counterfactual fairness (Kusner et al., 2017) introduces a path-specific causal effect on the prediction outcomes in discrimination analysis. The study demonstrates the paradox that arises when removing the sensitive attribute to achieve fairness; elements of the input feature can still contain discriminatory information that may not be immediately apparent. (Prost et al., 2022) demonstrates Simpson’s Paradox in the context of recommender system fairness, showing that the aggregate metric of fairness is influenced by differences in user distribution across the subgroups.