FairWire: Fair Graph Generation

Authors: o  deniz kose
yanning  shen
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 05:14:19

Abstract: Machine learning over graphs has recently attracted growing attention due to its ability to analyze and learn complex relations within critical interconnected systems. However, the disparate impact that is amplified by the use of biased graph structures in these algorithms has raised significant concerns for their deployment in real-world decision systems. In addition, while synthetic graph generation has become pivotal for privacy and scalability considerations, the impact of generative learning algorithms on structural bias has not yet been investigated. Motivated by this, this work focuses on the analysis and mitigation of structural bias for both real and synthetic graphs. Specifically, we first theoretically analyze the sources of structural bias that result in disparity for the predictions of dyadic relations. To alleviate the identified bias factors, we design a novel fairness regularizer that offers a versatile use. Faced with the bias amplification in graph generation models brought to light in this work, we further propose a fair graph generation framework, FairWire, by leveraging our fair regularizer design in a generative model. Experimental results on real-world networks validate that the proposed tools herein deliver effective structural bias mitigation for both real and synthetic graphs.

Main paper: 
 introduction: 
 The volume of graph-structured data has been explosively growing due to the advancement in interconnected systems. In this context, machine learning (ML) over graphs attracts increasing attention (1) , where specifically graph neural networks (GNNs) (2; 3; 4) have been proven to handle complex learning tasks over graphs, such as social recommendation (5) , traffic flow forecasting (6) .Despite the increasing research focus on graph ML, the deployment of these algorithms in real-world decision systems requires guarantees preventing disparate impacts. Here, algorithmic disparity refers to the performance gap incurred by ML algorithms with respect to certain sensitive attributes protected by anti-discrimination laws or social norms (e.g., ethnicity, religion). While algorithmic bias is a concern over tabular data (7; 8) , such bias becomes more critical for learning over graphs, as the use of graph structure in the algorithm design has been demonstrated to amplify the already existing bias (9) . Motivated by this, in this work, we specifically focus on structural bias and consequently the disparity in the predictions of dyadic relationships among nodes. Note that since the link predictions are informed by the proximity principle (nodes connect to other nodes that are similar to themselves), the bias in graph topology is directly reflected in link prediction. For example, in a social network, the denser connectivity within people from the same ethnic group leads to higher recommendation rates within these groups and may cause segregation in social relations (10) . Hence, the development of a fair link prediction algorithm is of crucial importance to prevent potential segregation.Fairness-aware algorithms typically require the knowledge of the sensitive attributes, the sharing of which can potentially create privacy concerns (11) . From a scalability perspective, sharing real graphs is also accompanied by difficulties due to the ever-increasing size of graphs. All these factors contribute to the value of synthetic graph generation for a number of applications, such as recommendation systems (12) , anomaly detection (13) . For graph generation, data-driven models are shown to achieve state-of-the-art results (14; 15; 16) , however, the fairness aspect of these models is under-explored. Recent works demonstrate that, in general, generative models tend to amplify the already existing bias in real data (17; 18) , which is a potential issue for graph generation as well.Faced with the aforementioned structural bias issues in graphs, in this work, we first carry out a theoretical analysis investigating the sources of such structural bias. Specifically, we deduce the factors that affect a commonly used bias metric, namely statistical parity (19) , for link prediction. Guided by the theoretical findings, a novel fairness regularizer, L FairWire is designed, which can be utilized for various graph-related problems, including link prediction and graph generation. In addition, an empirical analysis for a graph generation model is carried out, which reveals that the use of generative algorithms amplifies the already existing structural bias in real graph data. To resolve this issue, we design a new diffusion-based fair graph generation framework, FairWire, which leverages the proposed regularizer L FairWire . The training of diffusion model in FairWire is specifically designed to capture the correlations between the synthetic sensitive attributes and the graph connectivity, which enables fair model training with the existing techniques without revealing the real sensitive information. Overall, the contributions of this work can be summarized as follows: c1) A theoretical analysis that reveals the causes of disparity in the predictions of dyadic relations between nodes is derived. Differing from the existing analyses regarding the statistical parity in link prediction, our analysis considers a more general setting where sensitive attributes can be non-binary. c2) Based on the theoretical findings, we design a novel fairness regularizer, L FairWire , which can be directly utilized for link prediction, as well as for graph generation models to alleviate the structural bias in a task-agnostic way. c3) We conduct an empirical analysis for the effect of graph generation models on the structural bias, which reveals the possible bias amplification related to these models. c4) FairWire, a novel fair graph generation framework, is developed by leveraging L FairWire within a diffusion model. The diffusion model is trained to capture the relations between the sensitive attributes and the graph topology, facilitating fair model training without private information leakage. c5) Comprehensive experimental results over real-world networks show that the proposed framework can effectively mitigate structural bias and create fair synthetic graphs.Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?Answer: [Yes] Justification: For c1, please see Subsection 4.1. The contribution in c2 is presented in 4.2. For c3 and c4, please check Subsections 5.1 and 5.2, respoectively. Finally, the experimental results emphasized in c5 are presented in Section 6.Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 
related work: 
 Fairness-aware learning over graphs. Fairness-aware graph ML has attracted increasing attention in recent years (20; 21; 22) . Existing works mainly focus on: 1) Group fairness (9; 23; 24; 25), 2) Individual fairness (26; 27) , and 3) Counterfactual fairness (28; 29; 30) . To mitigate bias in graph ML, different strategies are leveraged, including but not limited to adversarial regularization (9; 24; 31; 32) , Bayesian debiasing (33) , and graph editing (28; 34; 35; 36; 37) . With a specific focus on link prediction, (19; 38) propose fairness-aware strategies to alter the adjacency matrix, while (39) designs a fairness-aware regularizer. Differing from the majority of existing strategies, our proposed design herein is guided and supported by theoretical results. Specifically, we rigorously analyze the factors in graph topology leading to disparity for link prediction considering non-binary sensitive attributes. Furthermore, the developed bias mitigation tool can be employed in a versatile manner for training the link prediction models, as well as for training the generative models to create synthetic, fair graphs. Synthetic Graph Generation. Generating synthetic graphs that simulate the existing ones has been a topic of interest for a long time (40; 41) , for which the success of deep neural networks has been demonstrated (12; 42; 43) . Recently, the use of diffusion-based graph generative models has been increasing, due to their success in reflecting several important statistics of real graphs in the synthetic ones (44; 45; 46; 16; 47; 48) .To the best of our knowledge, the only existing work that considers fair graph generation is (49) which only outputs a graph structure without nodal features, sensitive attributes and node labels, and also requires class labels as input. Furthermore, it focuses on the disparities in generation quality for different sensitive groups as the fairness metric, which may not be predictive for the fairness performance in downstream tasks. In contrast, our scheme herein does not require any class labels or training of a particular downstream task. In addition, differing from the existing diffusion models, our generation of graph topology and nodal features is guided by the sensitive attributes, which enables us to capture the correlations between the synthetic sensitive attributes and synthetic graph structure/nodal features. To the best of our knowledge, this work provides the first fairness-aware diffusion-based graph generation framework.Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes] Justification: Please see Limitations in Section 7.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
preliminaries: 
 Given an input graph G := (V, E), the focus of this study is investigating and mitigating the structural bias that may lead to unfair results for learning algorithms. Here,V := {v 1 , v 2 , • • • , v N }denotes the set of nodes and E ⊆ V × V stands for the set of edges. Nodal features and the adjacency matrix of the input graph G are represented by X ∈ R N ×F and A ∈ {0, 1} N ×N , respectively, where A ij = 1 if and only if (v i , v j ) ∈ E. This work considers a single, potentially non-binary sensitive attribute for each node denoted by s ∈ {1, • • • , K} N . In addition, S ∈ {0, 1} N ×K represents the one-hot encoding of the sensitive attributes. For the graphs with class information, y ∈ R N denotes the class labels. Node representations output by the lth GNN layer are H l+1 , with h l+1 i ∈ R F l+1 denoting the learned hidden representations for node v i . x i ∈ R F , and s i represent the feature vector, and the sensitive attribute of node v i , respectively. Furthermore, S k denotes the set of nodes whose sensitive attributes are equal to k. We define inter-edge setE χ := {e ij |v i ∈ S a , v j ∈ S b , a ̸ = b}, and intra-edge set E ω := {e ij |v i ∈ S a , v j ∈ S b , a = b}. Similarly, d χ i := vj ∈V-Sa A ij , ∀v i ∈ S a and d ω i := vj ∈Sa A ij , ∀v i ∈ S aare the inter-and intra-degrees of node v i , respectively. Finally, U A represents the discrete uniform distribution over the elements of set A.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [Yes] Justification: Our assumptions are explicitly presented in Subsection 4.1. Furthermore, all proofs are available in Appendices B and C. Guidelines:• The answer NA means that the paper does not include theoretical results.• All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced. • All assumptions should be clearly stated or referenced in the statement of any theorems.• The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. • Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. • Theorems and Lemmas that the proof relies upon should be properly referenced. 
inspection and mitigation of structural bias: 
 This section first derives the conditions for a graph topology that leads to optimal statistical parity for link prediction. Guided by the obtained conditions, a fairness regularizer will then be presented. Statistical parity for link prediction is defined as (19) , where g(v i , v j ) denotes the predicted probability for an edge between the nodes i and j. To the best of our knowledge, our analysis is the first theoretical investigation for the relation between ∆ SP and the graph topology considering multi-valued sensitive attributes, thus it generalizes previous findings with binary sensitive attributes (19) .∆ SP := |E (vi,vj )∼U V ×U V [g(v i , v j ) | s i = s j ] - E (vi,vj )∼U V ×U V [g(v i , v j ) | s i ̸ = s j ]|This subsection derives the conditions for a fair graph topology that achieves optimal statistical parity in the ensuing link prediction task. First, we will introduce the GNN model considered in this work.GNN model: Throughout the analysis, a stochastic graph view, Ã, is adopted, i.e., Ãij denotes the probability of an edge between the nodes v i and v j , and Ãij = Ãji . Let Z l+1 represent the aggregated representations by the lth GNN layer with ith row E Ã[z l+1 i ] := vj ∈V Ãij c l+1 j , where c l+1 i := W l h l i . Then, the hidden representation output by lth GNN layer for node v i can be written asE Ã[h l+1 i ] = σ( vj ∈V Ãij W l h l j ) = σ(E Ã[z l+1 i ]), where W l is the weight matrix and σ(•) is the non-linear activation employed in the lth GNN layer.The following assumptions are made for Theorem 1 that will be presented in this subsection:A1: ∥c i ∥ ∞ ≤ δ, ∀v i ∈ V. A2: E Ã[d ω i ] |S k | ≥ E Ã [d χ i ] N -|S k | , ∀v i ∈ S k , ∀k ∈ {1, • • • , K}. A3: vi,vj ∈V Ãij ≫ E Ã[d χ i ], ∀v i ∈ V.These assumptions are naturally satisfied by most of the real-world graphs. Assumption A1 implies that the representations, c i 's, are finite. For A2, note that most of the real-world social networks have considerably more intra-edges than inter-edges (50), i.e., E Ã[d ω i ] ≥ E Ã[d χ i ].Thus, unless |S i ∥ ≫ ∥S j |, i ̸ = j (extremely unbalanced sensitive group sizes), A2 holds. Finally, A3 holds with high probability as E Ã[d χ i ] = vi∈Ss i ,vj ∈V-Ss i Ãij . We also demonstrate that these assumptions are valid for the real-world networks we are using in Appendix A in order to further justify them.Building upon these assumptions, Theorem 1 reveals the factors leading to the disparity between the representations of different sensitive groups obtained at any GNN layer. Specifically, it upper bounds the term δ(l+1) k := ∥E Ã,vi∼U S k [h l+1 i | s i = k] -E Ã,vi∼U (V-S k ) [h l+1 i | s i ̸ = k]∥ 2 .The proof of the theorem is presented in Appendix B. Theorem 1. The disparity between the representations of nodes in a sensitive group S k and the representations of the remaining nodes output by the lth GNN layer, δ(l+1) k, can be upper bounded by:EQUATIONwhere L is the Lipschitz constant of the activation function σ(•),z l+1 i -mean(z l+1 j | v j ∈ V) ∞ ≤ ∆ z , ∀v i ∈ V, and p χ k := vi∈S k ,vj / ∈S k Ãi,j , p ω k := vi∈S k ,vj ∈S k Ãi,j .Representation disparity resulting from the aforementioned GNN-based aggregation is examined and explained by Theorem 1. The commonly used fairness measures, such as statistical parity (19) , are naturally a function of the representation disparity. Herein, we further investigate the said relation between the representation disparity and ∆ SP mathematically. Specifically, for a link prediction model described by a function g(v i , v j ) := h ⊤ i Σh j , Proposition 1 directly upper bounds ∆ SP . Here, h i denotes the representation for node v i that is employed for the link prediction task, i.e., the hidden representations in the final layer. The proof of Proposition 1 is presented in Appendix C. Proposition 1. For a link prediction model described by g(v i , v j ) := h ⊤ i Σh j , ∆ SP can be upper bounded by:EQUATIONwhere ∥h i ∥ 2 ≤ q, ∀v i ,δ max := max k (∥E Ã,vi∼U S k [h i | s i = k] -E Ã,vj ∼U (V-S k ) [h j | s j ̸ = k]∥ 2 ).Combining the findings of Theorem 1 and Proposition 1, Corollary 1 further demonstrates the factors (including the topological ones) that affect the resulting statistical parity in the link prediction task.Corollary 1. For a link prediction modelg(v i , v j ) := (h L+1 i ) ⊤ Σh L+1 j, where h L+1 j is the representation created by Lth (final) GNN layer, ∆ SP can be upper bounded by:∆ SP ≤ K k=1 |S k | N q∥Σ∥ 2 L δ F (L+1) α 1 + α 2 + 2 √ N ∆ z ,whereα 1 := p ω k |S k | - p χ k N -|S k | and α 2 := v i ,v j ∈V Ãij -p ω k -2p χ k N -|S k | - p χ k |S k | .The bias analysis in Subsection 4.1 brings to light the factors resulting in topological bias for a probabilistic graph connectivity. Corollary 1 shows that the topological bias can be minimized if α 1 = 0 and α 2 = 0. One can obtain α 1 = 0 by ensuringp ω k p χ k = |S k | N -|S k | , ∀k. Meanwhile, α 2 = 0 if p ω k = vi,vj ∈V ( Ã) -c|S k | -cN and p χ k = c|S k | for any constant c ∈ R.Overall, the optimal values of p ω k and p χ k that minimize both α 1 and α 2 follow as(p ω k ) * = v i ,v j ∈V Ãij |S k | 2 N 2and(p χ k ) * = ( v i ,v j ∈V Ãij )(N |S k |-|S k | 2 ) N 2. Therefore, in order to mitigate structural bias, we can design a regularizer that pushes the expected number of inter-edges and intra-edges towards (p χ k ) * and (p ω k ) * :L := K k=1 | vi,vj ∈V ( Ã⊙(Se k )(Se k ) ⊤ ) i,j -(p ω k ) * |+| vi,vj ∈V ( Ã⊙(Se k )(1-(Se k )) ⊤ ) i,j - (p χ k ) * |.Here, e k ∈ R K is the basis vector with only non-zero entry 1, at the kth element, and ⊙ denotes the Hadamard product. Note that such a regularizer is compatible with any learning algorithm that outputs probabilities of all possible edges in the graph, e.g., topology inference algorithms.Although L can be applied to several graph ML algorithms and its theory-guided design can promise effective topological bias mitigation, its design requires a single-batch learning setting due to the definitions of p χ k and p ω k (resulting in a complexity growing exponentially with N ). Specifically, p χ k and p ω k are calculated based on all edge probabilities related to the all nodes in S k . Therefore, regularizing the values of p χ k and p ω k will lead to scalability issues for large graphs. To tackle this challenge, we only focus on the optimal ratio between the expected number of intra-and inter-edges, i.e.,p ω k p χ k = |S k | N -|S k | , ∀k ∈ {1, • • • , K}, which is governed by α 1 .The idea is to manipulate the ratio between the expected number of intra-and inter-edges in each mini-batch of nodes for a better scalability. We call the corresponding batch-wise fairness regularizer L FairWire , which follows asEQUATIONwhere B denotes the set of nodes within the utilized minibatch. Note that the aforementioned versatile use of L also applies to L FairWire , which can directly be used in topology inference tasks. Specifically, for link prediction, the following loss function can be employed in training to combat bias:EQUATIONwhere Ãij denotes the predicted probability by the algorithm for an edge between v i and v j , and L CE is cross-entropy loss. The hyperparameter λ is used to adjust the weight of fairness in training.Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? Answer: [Yes] Justification: The proposed regularizer is presented with its derivation in Subsections 4.1 and 4.2. Furthermore, for reproducibility of the empirical evaluation, our codes are provided within the supplementary material of this submission. All experimental settings are also described in Subsection 6.1, and Appendix G. Guidelines:• The answer NA means that the paper does not include experiments.• If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. • If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. • Depending on the contribution, reproducibility can be accomplished in various ways.For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. • While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. , with an open-source dataset or instructions for how to construct the dataset). (d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility.In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. 
fair graph generation: 
 Generating synthetic graphs that capture the structural characteristics in real data attracts increasing attention as a promising remedy for scalability (ever-increasing size of real-world graphs) and privacy issues. Especially, sharing real sensitive attributes for fair model training exacerbates the privacy concerns due to the sensitive attribute leakage problem (11) . Thus, creating synthetic graphs with generative models becomes instrumental in applications over interconnected systems. In this work, we focus on diffusion models whose success in capturing the original data distribution has been shown for various types of networks (45; 46; 16; 47) . Despite the growing interest in these models, their effects on fairness have not yet been investigated, which limits their use in critical real-world decision systems. Motivated by this, in Subsection 5.1, we first empirically analyze the impact of diffusion models on the algorithmic bias by comparing the original and synthetic graphs in terms of different fairness metrics for link prediction. This empirical investigation reveals that the algorithmic bias is amplified while using generative models for graph creation. To resolve this critical issue, we develop FairWire in Subsection 5.2, a fair graph generation framework, which leverages our proposed regularizer L FairWire during the training of a diffusion model.To evaluate the effect of synthetic graph generation on bias, we first sample 10 different synthetic graphs for each of the 4 real-world networks (see Table 7 in Appendix E and Subsection 6.1 for more details on the datasets). Synthetic graphs are sampled using a diffusion model that is trained following the setup in (47) , which is a state-of-the-art algorithm for diffusion-based graph generation. Upon creating graphs, we evaluate them for the link prediction task on the same test set (generated from the real data) and report the corresponding utility (AUC) and fairness performance. Fairness performance is measured via two widely used bias metrics, statistical parity (∆ SP ) and equal opportunity (∆ EO ) (19) for which lower values indicate better fairness (see Subsection 6.1 for more details on the link prediction model and evaluation metrics). The obtained results are presented in Table 1 .In Table 1 , G denotes the original graphs, and the synthetic graphs are represented by G. Overall, Table 1 shows that graph generation via diffusion models indeed amplifies the already existing bias in the original graphs consistently for all the considered datasets. This brings the potential bias-related issues in synthetic graph creation to light and calls for robust bias mitigation solutions. 1 ) makes creating fair graphs via graph generation models of particular interest.The proposed fair graph generation framework, FairWire, is built upon structured denoising diffusion models for discrete data (51) . In the forward diffusion, FairWire employs a Markov process to create noisy graph data samples by independently adding or deleting edges. For denoising, a messagepassing neural network (MPNN) is trained to predict the clean graph based on noisy samples by using the guidance of sensitive attributes (and node labels if available). Finally, we sample synthetic graphs with the guidance of synthetic sensitive attributes that are initialized based on their distribution in the original data. If input graph has also node labels, during graph generation, these node labels are sampled based on their distribution conditioned on the sensitive attributes in the original graph.In the sequel, as our main novelty lies in the denoising process, we discuss the training process of FairWire (reverse diffusion process) in more detail, while the forward diffusion and sampling processes are explained in Appendix D. Note that the diffusion process is presented for attributed graphs, where synthetic nodal features X are also generated. However, the proposed approach can be readily adapted to graphs without nodal features.Reverse diffusion process: For denoising, we train an MPNN, ϕ θ parametrized by θ, which is shown to be a scalable solution for the generation of large, attributed graphs (47) . Specifically, ϕ θ inputs a noisy version of the input graph and the original sensitive attributes described by X t , A t , S and aims to recover the original nodal features X 0 and graph topology A 0 . Here, A 0 ∈ R N ×N ×2 denotes the one-hot representations for the edge labels. Note that the sensitive attributes are used to guide the MPNN to capture the relations between them and graph topology. Therefore, the sensitive attributes are initialized and kept the same during both training and sampling (the original distribution of sensitive attributes is used to initialize them during sampling). For a node v, the message passing at the lth layer can be described as:EQUATIONEQUATIONwhere ∥ stands for the concatenation operator. In this aggregation, WT →H , W S are all learnable parameters, while σ(•) consists of ReLU (52) and LayerNorm (53) layers. In addition, H (t,0) and h t are initialized as hidden representations created for X t and time step t via multi-layer perceptrons (MLP), respectively, and S (0) = S. After creating hidden representations for nodes and their sensitive attributes, final representation for a node v is generated viah v = h (t,0) v ∥h (t,1) v ∥ • • • ∥S (0) v ∥S (1) v ∥ • • • ∥h t .Note that when node labels are available, their one-hot representations Y, are also employed in this MPNN in the same way as S are utilized. Based on these final representations, node attributes, and edge labels are predicted. To create fair graph connections in the synthetic graphs, we regularize the predicted edge probabilities, Ã, via the designed fairness regularizer L FairWire . Overall, the training loss of the MPNN follows as:EQUATIONwhere λ adjusts the focus on the fairness regularizer.Remark (Applicability to general generative models): Although the designed regularizer in Subsection 4.2 is embodied in a diffusion-based graph generation framework in Subsection 5.2, L FairWire can be utilized in any generative model outputting synthetic graph topologies as a fairness regularizer on the connections, including but not limited to graph autoencoder-based or random walk-based graph generation models.Remark (Creation of synthetic sensitive attributes): We design a generative framework in Subsection 5.2 that outputs synthetic sensitive attributes whose effect on the connections is reflected by inputting them in the training of MPNN. We emphasize that the creation of these synthetic sensitive attributes also enables the use of existing fairness-aware schemes on the created graphs without leaking the real sensitive attributes.6 ExperimentsQuestion: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?Answer: [Yes] Justification: Codes to reproduce all the results in Section 6 are provided in the supplementary material to this submission.Guidelines:• The answer NA means that paper does not include experiments requiring code. • Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. 
datasets and experimental setup: 
 Datasets. In the experiments, four attributed networks are employed, namely Cora, Citeseer, Amazon Photo and Amazon Computer for link prediction. Cora and Citeseer are widely utilized citation networks, where the articles are nodes and the network topology depicts the citation relationships between these articles (54) . Amazon Photo and Amazon Computer are product co-purchase networks, where the nodes are the products and the links are created if two products are often bought together (55) . In addition to link prediction, we also evaluate the synthetic graphs on node classification, where the German credit (56) and Pokec-n ( 9) graphs are employed. For more details on the datasets and their statistics, please see Appendix E.Experimental Setup. In this section, we first report the performance of L FairWire for link prediction. For this task, the area under the curve (AUC) is employed as the utility metric. As fairness metrics, statistical parity and equal opportunity definitions in (19; 37) are used, where∆ SP := |E (vi,vj )∼U V ×U V [ Ãij = 1 | s i = s j ] -E (vi,vj )∼U V ×U V [ Ãij = 1 | s i ̸ = s j ]| and ∆ EO := |E (vi,vj )∼U V ×U V [ Ãij = 1 | A ij = 1, s i = s j ] -E (vi,vj )∼U V ×U V [ Ãij = 1 | A ij = 1s i , ̸ = s j ]|.Lower values for ∆ SP and ∆ EO indicate better fairness performance.To evaluate the generated synthetic graphs, we use both the link prediction and node classification tasks. Herein, we sample 10 synthetic graphs for each dataset with the trained diffusion models. Afterward, we train link prediction/node classification models (for more details on these models, please see Appendix G) on the sampled graphs, and test these models on the real graphs G (the test set is the same for all baselines and FairWire). Here, we consider the scenario where there is no access to the real graphs due to privacy concerns, and the models are trained on the synthetic graphs for downstream tasks. To evaluate these synthetic graphs on link prediction, the same utility and fairness metrics as in the link prediction task are used. For node classification, accuracy is employed as the utility measure withEQUATION∆ EO := |P (ŷ j = 1 | y j = 1, s j = 0) -P (ŷ j = 1 | y j = 1, s j = 1)| being the fairness metrics.For more details on the training of link prediction, node classification, diffusion models, and the hyperparameter selection for FairWire and baselines, see Appendix G. A sensitivity analysis is also provided in Appendix H for the effect of hyperparameter λ in (4) and in (7) . Note that the performance of the generative algorithms is generally reported in terms of the distances between the statistics of real data and the synthetic ones, instead of the fairness performance. For completeness, we report the distance metrics for node degree distribution and clustering coefficient distribution in Appendix F.Baselines. For link prediction, fairness-aware baselines include adversarial regularization (9) , FairDrop (37) , and FairAdj (19) . For graph generation, FairGen (49) , is the only existing fairnessaware baseline designed for node classification. For a comprehensive evaluation, we also employ adversarial regularization (9) and FairAdj (19) as in-processing and post-processing fairness-aware strategies within the generative model. For more details on the baselines, please see Appendix G. Comparative results for the link prediction task are presented in Table 2 , where we consider the setting L FairWire is employed as a fairness regularizer while training a GNN model for link prediction.The natural baseline here is to employ the same GNN model without any fairness interventions, which is denoted by GNN in Table 2 . Note that in Table 2 , c equals to 0.01 for Amazon Photo and the results are presented for c = 0.1 for Amazon Computer.The results in Table 2 demonstrate that employing L FairWire as a fairness regularizer leads to better fairness measures compared to the naive baseline, while also providing similar utility. Specifically, the proposed regularizer is observed to improve both fairness metrics, ∆ SP and ∆ EO , with improvements ranging from 20% to 80% for every evaluated dataset compared to the natural baseline, GNN. Furthermore, the obtained results show that L FairWire also outperforms the fairness-aware baselines Adversarial (9), FairDrop (37) , and FairAdj (19) in both fairness metrics. For certain datasets (e.g., Amazon Photo, Amazon Computer), we report the results of FairWire for different values of λ to illustrate the trade-off between fairness and utility and to show that FairWire leads to a better trade-off compared to the other fairness-aware baselines. Note that we could not include the results of FairAdj over the product networks (i.e., Amazon Photo, Amazon Computer) due to its substantial memory use during its optimization process, which led to out-of-memory errors for the infrastructure we use. In addition to the improved fairness performance, it can be observed that the employment of L FairWire generally results in the lowest standard deviation values for fairness metrics, which demonstrates the stability of the proposed strategy for bias mitigation. Overall, the results corroborate the effectiveness of L FairWire in enhancing fairness while also providing similar utility compared to the state-of-the-art fairness-aware baselines.Comparative results for graph generation are presented in Tables 3 and 4 , where the link prediction and node classification tasks are used to evaluate the synthetic graphs, respectively. In these tables, G represents the original data, and G stands for the synthetic graphs generated by the fairness-agnostic GraphMaker (47) . Overall, Tables 3 and 4 show that FairWire improves fairness metrics compared to G, fairness-agnostic synthetic graphs created via diffusion, without a significant utility loss for Note that, for link prediction, the fairness improvement reported for FairAdj in Table 3 is accompanied by a significant utility drop. Specifically, for a larger λ value (i.e., λ = 1), FairWire can provide better fairness measures (∆ SP = 7.01 ± 6.25 and ∆ EO = 3.06 ± 2.95) on Citeseer with a similar accuracy (83.47 ± 7.79) to FairAdj. Thus, the results in Table 3 demonstrate that FairWire provides a better utility/fairness trade-off compared to fairness-aware baselines on all evaluated datasets.In Table 4 , similar to the link prediction experiments (Table 2 ), the results of FairAdj for the Pokec-n network could not be obtained due to computational limitations. For FairGen (49) , we directly input the synthetic graph output by the algorithm to the node classification model we train, thus the results are obtained for a single synthetic graph. A possible explanation for the better accuracy of FairGen on the German dataset is that the algorithm is observed to output a denser synthetic network, which might be useful for the utility. It is observed that the synthetic graph output by FairGen for Pokec-n was not informative enough for the node classification task (we provide the codes for the FairGen algorithm in our supplementary material for the reproducibility of these results.) All in all, the results in Table 4 signify that the superior performance of FairWire in terms of fairness/utility trade-off can also be observed for node classification, which validates the efficacy of FairWire in creating fair synthetic graphs that also capture the real data distribution. Our analysis in Subsection 4.1 reveals that the ratio of intra-(edges connecting the same sensitive group) and interedges (edges between different sensitive groups) is a factor contributing to the structural bias. Specifically, the bias factor α 1 is minimized whenp ω k p χ k = |S k | N -|S k | , ∀k, where p ω k and p χ k are the expected number of intra-and inter-edges for the nodes in S k . This finding suggests that for a graph with multiple (> 2) sensitive groups, given the sizes of sensitive groups are not catastrophically unbalanced, the number of inter-edges (related to p χ k ) should be larger than the number of intra-edges (related to p ω k ) to alleviate structural bias (i.e., |S k | ≤ N -|S k |). However, for graphs encountered in several domains, the number of intra-edges is significantly larger than the number of inter-edges, due to the homophily principle (10) . Motivated by this, in Figure 1 , we visualize the distributions of intra-and inter-edges in synthetic graphs created by i) a fairness-agnostic strategy, GraphMaker (47) , and ii) FairWire, for Cora. In Figure 1 , intra-and inter-edges are colored with blue and red, respectively. Figure 1 reveals that the graph created by GraphMaker (47) predominantly consists of intra-edges, leading to the structural bias reflected in Table 3 . In contrast, FairWire exhibits a remarkable balancing effect, which provides a potential explanation for the improvement in fairness.Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?Answer: [Yes] Justification: Please see Subsection 6.1 and Appendix G.Guidelines:• The answer NA means that the paper does not include experiments.• The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 
conclusion: 
 This study focuses on the investigation and mitigation of structural bias for both real and synthetic graphs, where a novel fairness regularizer, L FairWire , is designed to alleviate the effects of bias factors identified in a developed theoretical bias analysis. Furthermore, the proposed fairness regularizer is leveraged in a fair graph generation framework, FairWire, which alleviates the bias amplification observed in graph generative models. Experimental results corroborate the effectiveness of the proposed tools in bias mitigation for both real and synthetic graphs.Limitations: This paper considers the setting where sensitive attributes are available during model training, which might limit its use for certain real-world applications. Thus, one future direction of this work would be to consider the partial availability of these sensitive attributes in the input graph data. Furthermore, although we showed that real-world graphs typically satisfy the assumptions in Subsection 4.1, another possible future work we consider is deriving a theoretical bias analysis without the dependency on these assumptions.Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?Answer: [Yes] Justification: All experimental results are obtained for multiple random seeds and their average together with standard deviations are reported.Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.• The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors). • It should be clear whether the error bar is the standard deviation or the standard error of the mean. Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?Answer: [NA] Justification:Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?Answer: [Yes]Justification: Citation and Amazon networks are provided by the DGL library (67) , which is explained and the corresponding license is explicitly provided in the README file in our supplementary material. Furthermore, the Pokec dataset, used in this study, was created and presented in (68) , which is again mentioned in the README file in our supplementary material.Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL.• The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. • For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. • If this information is not available online, the authors are encouraged to reach out to the asset's creators. 
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 Appendix: In order to show that the assumptions made in Subsection 4.1 are typically valid for real-world graphs, here we present the exact values of the terms within Assumptions 2 and 3 for the datasets we use. Specifically, we make the following assumptions:A2:E Ã[d ω i ] |S k | ≥ E Ã [d χ i ] N -|S k | , ∀v i ∈ S k , ∀k ∈ {1, • • • , K}, A3: vi,vj ∈V Ãij ≫ E Ã[d χ i ], ∀v i ∈ V.First, for Assumption 2, we obtained the real values of the terms l k :=E Ã,v i ∼U S k [d ω i ] |S k | and r k := E Ã,v i ∼U S k [d χ i ] N -|S k |, where we want l k ≥ r k ∀k ∈ {1, • • • , K}. For all different sensitive groups, these values are presented in Table 5 for all the used datasets. Overall, the results presented in both Tables 5 and 6 demonstrate that the assumptions made for the theoretical bias analysis in Section 4.1 are valid for the real-world graphs we are using. This supports that our analysis is applicable to most of the real-world data and settings.Here, without loss of generality, we will focus on the lth GNN layer, where the input representations are represented by H l and output representations are denoted H l+1 . The considered disparity measure follows as:EQUATIONLet's re-write the disparity measure δ (l+1) k by using definitions c l+1 i := W l h l i , andEQUATIONUsing Lemma A.1. in (57) , it can be shown that δ (l+1) k can be upper-bounded as:EQUATIONThen, we can divide the sums over all nodes into two: nodes in S k and the remaining ones.EQUATIONThen, by triangle inequality, it follows that:EQUATIONAssumption 2 in Subsection 4.1 ensures that1 |S k | vi∈S k Ãij - 1 N -|S k | vi /∈S k Ãij ≥ 0, ∀v j ∈ S k . Furthermore, third assumption presented in Subsection 4.1 guarantees that1 |S k | vi∈S k Ãij - 1 N -|S k | vi / ∈S k Ãij ≤ 0, ∀v j / ∈ S k . Utilizing Assumption 1 in Subsection 4.1, ∥c i ∥ ∞ ≤ δ, ∀v i ∈ V, the following upper bound can be derived.EQUATIONwhere δ F (l+1) stands for an F (l+1) dimensional vector with all elements being equal to δ. Then, by utilizing the definitions p χ k := vi∈S k ,vj / ∈S k Ãi,j , p ω k := vi∈S k ,vj ∈S k Ãi,j , the upper bound in (13) can be rewritten as:EQUATIONThe final result follows from the inequality, ∥δEQUATIONwhich concludes the proof.Statistical parity for link prediction is defined as (19) :EQUATIONBy considering each sensitive group explicitly, statistical parity can also be written as:EQUATION) Define E Ã,vi∼U S k [h i | s i = k] := p k and E Ã,vi∼U (V-S k ) [h i | s i ̸ = k] := q k . We further assume that ∥h i ∥ 2 ≤ q, ∀v i ∈ V and it holds that ∥E Ã,vi∼U S k [h i | s i = k] -E Ã,vj ∼U (V-S k ) [h j | s j ̸ = k]∥ 2 ≤ δ max k. Using the definitions for p k and q k and link prediction model g(v i , v j ) := h ⊤ i Σh j , ∆ SP can be reformulated as:EQUATIONBy triangle inequality, it follows thatEQUATIONFinally, by using Cauchy-Schwarz inequality and the assumption ∥h i ∥ 2 ≤ q, ∀v i ∈ V, we can conclude thatEQUATIONwhere the final inequality follows from the assumption that ∥E Ã,vi∼U S k[h i | s i = k] - E Ã,vj ∼U (V-S k ) [h j | s j ̸ = k]∥ 2 ≤ δ max k .Diffusion Models for Graph Generation. Briefly, diffusion models are composed of two main elements: a noise model q, and a denoising neural network ϕ θ . The noise model q progressively corrupts data to create a sequence of increasingly noisy data points. Inspired by the success of Gaussian noise for diffusion-based image generation (58), the earlier diffusion-based graph generation models employ Gaussian noise to create noisy graph data (45; 46) . However, such a noise model cannot properly capture the structural properties of discrete graph connections. Motivated by this, a discrete noise model is introduced in (51). The discrete noise model for graph structure is typically applied in the form of edge deletion and additions (16; 47) . After creating noisy data, a denoising network ϕ θ is trained to invert this process by predicting the original graph structure A from the noisy samples. While different neural network structures are examined as candidates for the denoising network, message-passing neural networks are shown to be a scalable solution for the creation of medium-to large-scale graphs (47) .Forward diffusion process: introduced in (51), we employ a Markov process herein to add noise to the input graph structure in the form of edge additions or deletions. These edge modifications can be executed by modeling the existence/non-existence of an edge as the edge class labels, where we have 2 classes, and applying a transition matrix that switches the labels with a certain probability. Then, given A 0 ∈ R N ×N ×2 denotes the one-hot representations for the edge labels, the noise model can be described by the transition matricesQ t A ∈ R 2×2 for t = 1, • • • , T , where q(A t | A t-1 ) = A t-1 Q t A , q(A t | A 0 ) = A 0 Qt A , and Qt A = Q 1 A Q 2 A • • • Q t A .For a uniform transition model, (59) proves that the empirical data distribution (the probability for the existence of an edge) is the optimal prior distribution. Following this finding, we specifically employ the transition matrix:EQUATIONwhere I ∈ R 2×2 is the identity matrix, 1 ∈ R 2 is a vector of ones, and m E ∈ R 2 describes the distribution of edge labels in the original graph. For the assignment of α t , cosine schedule is utilized, where ᾱt := cos(0.5π(t/T + s)/(1 + s)) 2 with a small s value for ᾱt = Π t τ =1 α τ . Note that for categorical nodal features, the forward diffusion process follows the same procedure.Sampling: Using the trained MPNN, synthetic graphs can be sampled iteratively. During sampling, we first sample the sensitive attributes of the nodes, S, based on its original distribution in the real graph. As the next step, we need to estimate the reverse diffusion iterationsp θ G t-1 = (A t-1 , X t-1 ) | G t = (A t , X t) , which is modeled as a product over nodes and edges (59) :EQUATIONTo compute each independent term, we marginalize over the predictions of the MPNN:EQUATIONwhere E c denotes all possible labels for edges, which are {0, 1} for an unweighted graph. Then, we can use our Markovian noise to modelp θ A t-1 ij | A ij = e, G t : p θ A t-1 ij | A ij = e, G t = q A t-1 ij | A ij = e, A t ij if q A t ij | A ij = e > 0 0otherwise.Note that posterior q A t-1 ij | A ij = e, A t ij can be computed in closed-form using Bayes rule. Leveraging this model, G t-1 can be sampled, which becomes the input of the MPNN at the next time step.For citation networks (Cora and Citeseer), the one-hot encoding representations of the words in the article descriptions constitute the binary nodal attributes. In these networks, similar to the setups in (19; 37) , the category of the articles is employed as the sensitive attribute. Furthermore, for product co-purchase networks, nodal attributes are again the one-hot encodings of the words in the product reviews and the product categories are utilized as the sensitive attributes. For the evaluation of node classification, we employ German (56) and Pokec-n (9) networks. Specifically, the German credit graph has 1,000 nodes representing the clients in a German bank, where the links are created based on the similarity of credit accounts. For this graph, the node labels classify clients into good vs. bad credit risks, where the clients' gender are employed as the sensitive attribute (28) . In addition, Pokec-n is sampled from an anonymized version of the Pokec network of 2012 (a social network from Slovakia), where nodes correspond to users who live in two major regions, and the region information is utilized as the sensitive attribute (9) . The working field of the users is binarized and utilized as the labels to be predicted in node classification. Statistical information for the utilized datasets are presented in Table 7 , where F is the total number of nodal features and K represents the number of sensitive groups.We evaluate the created synthetic graphs by FairWire via link prediction in Subsection 6.2. In order to provide a more traditional evaluation scheme, here we also report the 1-Wasserstein distance between the node degree distribution and clustering coefficient distribution of original graph and the synthetic ones. Table 8 presents the corresponding distance measures, where lower values for all metrics signify better performance. In Table 8 , ER and SBM stand for traditional baselines Erdos-Rényi model (60) anf SBM stochastic block model (61) , respectively. Furthermore, as deep learning based baselines, Feature-based MF represents feature-based matrix factorization (62) , GAE and VGAE stand for graph autoencoder and variational graph autoencoder (63) , respectively. Finally, GraphMaker in Table 8 corresponds to a diffusion-based graph generation baseline (47) . Note that all these baselines are fairness-agnostic. Overall, results in Table 8 signify that FairWire can create synthetic graphs that follow a similar distribution to the original data while also improving the fairness metrics.Link Prediction Model. For link prediction, we train a one-layer graph convolutional network (GCN), where the inner product between the output node representations signifies the corresponding edge probability. random data splits, and their average along with the standard deviations are reported. The weights of the GNN model for link prediction are initialized utilizing Glorot initialization (64) , where it is trained for 1000 epochs by employing Adam optimizer (65) . The learning rate, the dimension of hidden representations, and the dropout rate are selected via grid search for the proposed scheme and all baselines, where the value leading to the best validation set performance is selected. For learning rate the, the dimension of hidden representations, and the dropout rate, the corresponding hyperparameter spaces are {1e -1, 1e -2, 3e -3, 1e -3}, {32, 128, 512}, and {0.0, 0.1, 0.2}, respectively. Diffusion Model. Diffusion models are trained for 10000 epochs by employing Adam optimizer (65) , where the number of diffusion steps is 3. In the MPNN described in Subsection 5.2, hidden representation size for time step t is 32 for Cora and Citeseer and 16 for the Amazon Photo, German credit, and Pokec-n networks. In addition, hidden representation sizes for the nodal attributes and sensitive attributes are 512 and 64, respectively, for all datasets. The MPNN consists of two layers.Node Classification Model. For node classification, we employ one-layer and two-layers APPNP (66) networks for German credit and Pokec-n graphs, respectively. For training, 50% of the nodes are used, where the remaining nodes are split equally into two for the validation and test sets. The weights of the GNN model for node classification are initialized utilizing Glorot initialization (64) , where it is trained for 1000 epochs by employing Adam optimizer (65) . The learning rate, the dimension of hidden representations, and the dropout rate are selected via grid search for the proposed scheme and all baselines, where the value leading to the best validation set performance is selected. For learning rate the, the dimension of hidden representations, and the dropout rate, the corresponding hyperparameter spaces are {3e -2, 1e -2, 3e -3}, {32, 128, 512}, and {0.0, 0.1}, respectively.Hyperparameter Selection. For the link prediction task, we select the multiplier of L FairWire among the values {0.01, 0.05, 0.1, 0.5} via grid search (the multiplier of the cross-entropy loss is 1). The results for L FairWire in Table 2 are obtained for the λ values 0.05, 0.1, 0.01/0.05, 0.1 on Cora, Citeseer, Amazon Photo, and Amazon Computer, respectively. For adversarial regularization (9) , the multiplier of the regularizer is selected via a grid search among the values {0.1, 1, 10} (the multiplier of link prediction loss is again 1). The multiplers of the adversarial regularization for the results in Table 2 are {10, 1, 1, 1} on Cora, Citeseer, Amazon Photo, and Amazon Computer, respectively. Furthermore, the hyperparameter δ in FairDrop algorithm is tuned among the values {0.16, 0.25} (0.16 is the default value in their codes), where 0.16 led to the best fairness/utility trade-off on each dataset. For FairAdj (19) , we use the hyperparameter values suggested by (19) directly for the citation networks.For the generative models, we select the multiplier of L FairWire (λ in (7) ) among the values {0.05, 0.1, 1.0, 10.0} via grid search. The results for FairWire in Table 3 are reported for the λ values 10.0, 0.1 on Cora, and Citeseer, respectively (λ = 0.05 for Amazon photo in Table 11 ). For the results in Table 4 , λ equals to 10 and 1 for German credit and Pokec-n, respectively. For adversarial regularization (9) , the multiplier of the regularizer (again in the training loss of the MPNN) is selected via a grid search among the values {0.001, 0.01, 0.1}. The multiplers of the adversarial regularization for the results in Table 3 are {0.01, 0.01, 0.01} on Cora, Citeseer, and Amazon Photo, respectively. Furthermore, the hyperparameter η in FairAdj (19) algorithm is tuned among the values {0.001, 0.005, 0.01}, where 0.001 led to the best fairness/utility trade-off on each dataset.Baselines. Fairness-aware baselines in the experiments include adversarial regularization (9) , FairDrop (37) , FairAdj (19) , and FairGen (49) . Adversarial regularization refers to the bias mitigation technique where an adversary is trained to predict the sensitive attributes. In link prediction, the adversary is trained to predict the sensitive attributes of the nodes that are incident to the edges whose labels are of interest. Furthermore, FairDrop (37) is an edge dropout strategy where the dropout probabilities vary for intra-or inter-edges so as to create a more balanced graph topology. FairAdj (19) optimizes a fair adjacency with certain structural constraints for link prediction in an iterative manner considering both fairness and utility. Finally, FairGen (35) focuses on the disparities in generation quality (distances between different graph statistics) for different sensitive groups and employs fairness-aware regularizers during graph generation via a transformer-based model.For graph generation experiments, GraphMaker ( 47) is utilized to create synthetic graphs in a fairnessagnostic way. While an existing work that considers fair link prediction for synthetic graphs is not available to the best of our knowledge, we employ adversarial regularization (9) as an in-processing bias mitigation strategy during the training of the MPNN described in Subsection 5.2. Furthermore, we use FairAdj (19) as a post-processing bias mitigation strategy on the synthetic graphs generated via GraphMaker, and the processed synthetic graphs are then evaluated for the link prediction and node classification tasks.In order to examine the impact of hyperparameter selection on fairness improvements, the sensitivity analyses for the proposed tools are executed with respect to the hyperparameter λ.The results are obtained for changing λ values for both the link prediction (see ( 4)) and graph generation (see ( 7)) experiments and reported in Tables 9, 10 . Overall, the results signify that both L FairWire in the link prediction and FairWire, lead to better fairness measures compared to the natural baselines within a wide range of hyperparameter selection. Due to limited space, we present the comparative results for graph generation on Amazon Photo in Table 11 for link prediction. Similar to the link prediction experiments (Table 2 ), the results of FairAdj for the Amazon Photo network could not be obtained due to computational limitations. Overall, the results reported in Table 11 conclude again that FairWire can provide the best utility/fairness trade-off compared to other fairness-aware baselines.Experiments are carried over on 4 NVIDIA RTX A4000 GPUs.NeurIPS Paper Checklist 


Decision: This paper proposes a graph generation approach with fairness consideration. The authors offer theoretical analysis to bound the representation disparity between a specific group and the remaining nodes and the statistical parity. These theoretical results further lead to the proposed FairWire that aims to push the generated graph to have a desired number of intra-/inter-edges.


All reviewers appreciate the theoretical analysis of this work and acknowledge the empirical performance of FairWire. There are concerns about the clarity in terms of math notations and experiments (e.g., visualizing results in node classification, choice of base GNN, analysis of results). The authors' responses are quite comprehensive and address the concerns well. Another concern raised by the reviewers is that the paper adopted the fairness definition without much justification. I agree that whether a fairness consideration is the appropriate or not is often a much broader topic involving many disciplines. And a more comprehensive justification from many disciplines or regulations would be more ideal, though the justification we often provide is either hypothetical from empirical results or event reported in news articles.  Though being a valid concern, I think this is relatively a minor issue given the volume of works in fair graph learning, especially on group fairness and dyadic fairness.


Overall I think this paper offers a good theoretical understanding toward group/dyadic fairness with multi-valued sensitive attribute, and I recommend acceptance.

Reviews: review: paper_summary: paper_summary: The impact of generative learning algorithms on structural bias is investigated in this paper. The authors provide a theoretical analysis on the sources of structural bias which result in disparity. Then a novel fairness regularizer is designed to alleviate the structural bias for link prediction tasks over graphs. Furthermore, a fair generation framework called FairWire is proposed by leveraging the fairness regularizer in a generative model. Finally, extensive experiments on synthetic and real datasets are conducted to show the effectiveness of mitigating structural bias in machine learning algorithms on graphs.
strength_weakness: strength_weakness: 1. This paper aims to address the fairness in graph generation. But there are not sufficient experiments on different representative backbone generation models. The authors only provide one backbone generation model, I suggest more representative backbone generation model to solidify the effectiveness.
2. This paper provides the theory and algorithm based on the sensitive group which may limit the use to real world graphs.
questions: questions: 1. The author introduce GraphMaker as a backbone generative model and claimed that this method considers the bias mitigation when generating new graphs. Does it mean that FairWire can not been used on other generative models?
2. In the experiments, for the supervised tasks the author use link prediction tasks results, while for generation tasks, the author use node classification tasks results. What is the results of node classification tasks for supervised tasks? It would be better that the same tasks are used to evaluate both the supervised and generative tasks.
3. The authors claim the contribution in fairness over generative graph algorithm, but also provides a lot experimental results on supervised tasks. What is the difference in structural bias problem over generative tasks and supervised tasks?
4. From the experimental results, FairWire makes the performance of AUC worse. What is the influence of FairWire on the accuracy of the GNN model? Does it influence the model training or that because GNN models conducts bias on different samples? 
5. From the theoretical analysis of the paper, we have to know the sensitive group first, how would you define and select sensitive groups? What kind of influence does sensitive group have on GNN and the proposed FairWire method?
limitations: limitations: The authors adequately addressed the limitations of their methods. And It is hard to foresee any potential negative societal impact of this work.
paper_summary: paper_summary: This work considers the problem of fairness in learning over graphs. Namely, it adopts a criterion for fairness that quantifies the probability of a relationship existing between nodes whose sensitive attribute value matches (intra-edges) versus not (inter-edges). It first derives theoretical results that bound the discrepancy between the two, and uses these insights to design a "fairness regularizer" that is compatible with both link prediction and synthetic graph generation tasks. The authors perform an evaluation on these two tasks, outperforming existing baselines in fairness metrics while preserving the core utility metric (e.g., accuracy).
strength_weakness: strength_weakness: The paper is very well executed. In my opinion, its main weakness is that the considered fairness definition is adopted without a convincing justification. It is rather well-known in the network science literature [1,2] that individuals tend to form connections with those that are similar to them. Therefore, I do not understand why this characteristic is framed as inherently "bad" and something that should be corrected when training a model for link prediction or graph generation. I think this deserves a proper justification and discussion. However, this may be a point that is broader than the paper itself. 

[1] McPherson, M., Smith-Lovin, L., & Cook, J. M. (2001). Birds of a feather: Homophily in social networks. Annual review of sociology, 27(1), 415-444.

[2] Newman, M. E. (2003). Mixing patterns in networks. Physical review E, 67(2), 026126.
questions: questions: Please address the potential weakness discussed above. Additional questions and comments:

C1. Could you discuss what happens if there is >1 sensitive attribute? Do your results already apply via some transformation, or what would be required to generalise to this case?

C2. Line 131-32: I think this claim needs to be scoped to e.g. *social* networks, as this is definitely not always the case (see e.g. ref [2] above).

C3. Line 153: maximization is over $k$ presumably?

C4. Line 167; Eq 3: $\mathbf{Se}_{k}$ not defined as far as I can tell.

C5. Experiments in 6.2: I think it should be discussed what "base GNN" is being used. Appendix G mentions it is a GCN -- but what is the justification for using a single layer?

C6. I think it's worth summarizing the choice and justification for the considered sensitive attributes in the main text (currently in Appendix E).

C7. Table 5 is overflowing the margins, consider wrapping in `\resizebox`
limitations: limitations: Limitations and potential negative social impacts are discussed, although the latter only superficially, and I think it deserves more discussion given the primary area of the paper is fairness.
paper_summary: paper_summary: In this submission a graph diffusion model with fairness correction is introduced for graph generation. The model is also applied for link prediction. The authors introduce a graph regularizer for fairness based on a theoretical bound for subgroup distance and representation distance. This analysis is newly introduced by the authors. The authors apply the regularizer to link prediction, examining the accuracy-fairness tradeoff. The authors then test fair graph generation by examining link prediction and node classification tasks, but replace the training graphs with fairness-corrected generated graphs. Empirically it is shown that this greatly improves fairness while marginally reducing accuracy.
strength_weakness: strength_weakness: There is some missing/glossing-over existing work. The authors claim that the theoretical analysis is novel, however ref [24] in the paper also includes a theoretical analysis with >2 binary categories. It would be good for the authors to discuss this work and compare with their own. Also, the authors should consider citing "Debiasing Graph Representations via Metadata-Orthogonal Training" (doi/10.1109/ASONAM49781.2020.9381348), which seems relevant to the problem (also includes a fairness correction over potentially >2 attributes).

There are a non-trivial amount of clarity and presentation issues, which I discuss in my list of questions.
questions: questions: (1) In the second experiment type (S6.3), which link prediction model was trained on the generated graphs? And was the FairWire regularization also applied to the link prediction model during training?

(2) From Fig 1 we can see that FairWire effectively removes intra-edges from generated graphs. However, the structure of the generate graphs is not quite evident. Can the graphs be visualized in a more productive way? Maybe the adjacencies can be row-sorted into their communities.

(3) The authors derive two fairness criterion $\alpha_1$ and $\alpha_2$, though $\alpha_2$ is ignored at the mini-batch level for scalability. Can the authors give more intution about $\alpha_2$ and why it should ultimately not matter when applied in practice?
limitations: limitations: Yes

Comments: title: Thanks
comment: Thanks for the authors satisfactorily addressing my concerns. The clarification makes sense and I have raised the presentation score and the overall score.
Many thanks to the authors for responding to the points I raised, I think the additional clarifications make sense. The "sticking point" regarding the fairness definition is difficult to resolve in the absence of a practical study which shows that, indeed, graph ML systems do exhibit the type of bias the authors argue about, as has been shown for other types of ML techniques in a variety of deployment scenarios. Nevertheless, as I mentioned before, I do not see this as a reason to penalise this particular paper. I am retaining my original score as I think it accurately reflects my assessment of the work.
The authors have satisfactorily addressed my concerns. I have raised the presentation score and the overall score.