Unifying Adversarial Perturbation for Graph Neural Networks

TL;DR

This paper introduces PerturbEmbedding, a unified adversarial perturbation framework for GNNs that enhances robustness and generalization by applying perturbations directly to hidden embeddings, outperforming existing methods across datasets.

Contribution

The paper proposes a novel, unified perturbation method, PerturbEmbedding, that applies adversarial and random perturbations directly to GNN embeddings, improving robustness and generalization.

Findings

PerturbEmbedding significantly outperforms existing methods.

It enhances robustness against both random and adversarial attacks.

The method improves GNN performance across various datasets.

Abstract

This paper studies the vulnerability of Graph Neural Networks (GNNs) to adversarial attacks on node features and graph structure. Various methods have implemented adversarial training to augment graph data, aiming to bolster the robustness and generalization of GNNs. These methods typically involve applying perturbations to the node feature, weights, or graph structure and subsequently minimizing the loss by learning more robust graph model parameters under the adversarial perturbations. Despite the effectiveness of adversarial training in enhancing GNNs' robustness and generalization abilities, its application has been largely confined to specific datasets and GNN types. In this paper, we propose a novel method, PerturbEmbedding, that integrates adversarial perturbation and training, enhancing GNNs' resilience to such attacks and improving their generalization ability. PerturbEmbedding performs perturbation operations directly on every hidden embedding of GNNs and provides a unified framework for most existing perturbation strategies/methods. We also offer a unified perspective on the forms of perturbations, namely random and adversarial perturbations. Through experiments on various datasets using different backbone models, we demonstrate that PerturbEmbedding significantly improves both the robustness and generalization abilities of GNNs, outperforming existing methods. The rejection of both random (non-targeted) and adversarial (targeted) perturbations further enhances the backbone model's performance.

Figures

Tables

Table 1. Table 1 . Overview of different perturb strategies in a view of Bernoulli sampling process.

Strategy	Formula
PerturbEdge	$𝐀=𝐀+{△}_{𝐀}$
PerturbNode	$𝐗=𝐗+{△}_{𝐗}$
PerturbWeight	$𝐖=𝐖+{△}_{𝐖}$
PerturbEmbedding	$𝐇=𝐇+{△}_{𝐇}$

Table 2. Table 2 . Comparison results of different perturb strategies. The best results are in bold.

		Penn94	Chameleon	Squirrel	Film	Cora	CiteSeer	PubMed
	GCN	$82.10\pm 0.34$	$64.56\pm 2.85$	$50.72\pm 0.69$	$29.14\pm 0.62$	$87.28\pm 1.26$	$76.68\pm 1.64$	$87.38\pm 1.66$
Random (non-targeted)	GCN-PerturbNode	$82.80\pm 0.22$	$66.45\pm 2.85$	$51.59\pm 1.63$	$29.78\pm 1.16$	$88.09\pm 0.44$	$78.16\pm 1.59$	$\text{88.22}\pm \text{0.22}$
	GCN-PerturbEdge	$82.31\pm 0.35$	$65.83\pm 1.77$	$51.76\pm 1.77$	$29.50\pm 1.03$	$87.73\pm 1.49$	$\text{78.49}\pm \text{1.45}$	$87.64\pm 0.41$
	GCN-PerturbWeight	$82.65\pm 0.40$	$66.36\pm 2.03$	$\text{53.12}\pm \text{2.73}$	$\text{30.66}\pm \text{1.33}$	$87.57\pm 1.13$	$77.87\pm 1.26$	$87.47\pm 0.43$
	GCN-PerturbEmbedding	$\text{82.98}\pm \text{0.21}$	$\text{67.85}\pm \text{2.16}$	$51.12\pm 1.48$	$30.25\pm 1.48$	$\text{88.13}\pm \text{0.79}$	$78.02\pm 1.68$	$87.99\pm 0.33$
Adversarial (Targeted)	GCN-PerturbNode	$82.52\pm 0.28$	$68.16\pm 1.97$	$53.68\pm 1.63$	$29.61\pm 0.85$	$87.73\pm 0.38$	$78.11\pm 1.39$	$\text{88.35}\pm \text{0.56}$
	GCN-PerturbEdge	$82.18\pm 0.44$	$66.45\pm 1.97$	$53.20\pm 1.93$	$29.63\pm 1.20$	$87.44\pm 1.10$	$78.02\pm 1.42$	$87.74\pm 0.39$
	GCN-PerturbWeight	$82.47\pm 0.34$	$67.68\pm 1.52$	$51.91\pm 2.64$	$\text{29.87}\pm \text{1.76}$	$87.61\pm 1.17$	$78.27\pm 1.86$	$87.77\pm 0.23$
	GCN-PerturbEmbedding	$\text{82.72}\pm \text{0.37}$	$\text{68.51}\pm \text{1.15}$	$\text{56.52}\pm \text{1.46}$	$29.38\pm 1.91$	$\text{87.81}\pm \text{1.38}$	$\text{78.63}\pm \text{1.27}$	$87.90\pm 0.27$
	GAT	$81.32\pm 0.76$	$68.99\pm 1.15$	$61.92\pm 2.61$	$28.88\pm 1.67$	$87.57\pm 1.19$	$75.46\pm 1.72$	$87.04\pm 0.43$
Random (non-targeted)	GAT-PerturbNode	$82.10\pm 0.33$	$71.32\pm 1.85$	$62.32\pm 0.79$	$29.88\pm 1.22$	$87.73\pm 0.78$	$76.48\pm 0.75$	$87.54\pm 0.24$
	GAT-PerturbEdge	$81.50\pm 0.57$	$71.36\pm 1.81$	$62.25\pm 1.59$	$29.59\pm 2.00$	$87.65\pm 1.12$	$76.04\pm 1.93$	$87.22\pm 0.44$
	GAT-PerturbWeight	$81.57\pm 0.81$	$71.01\pm 1.53$	$62.69\pm 1.62$	$29.84\pm 1.93$	$87.69\pm 1.67$	$76.15\pm 1.37$	$87.16\pm 0.16$
	GAT-PerturbEmbedding	$\text{82.28}\pm \text{0.42}$	$\text{71.49}\pm \text{1.57}$	$\text{62.86}\pm \text{0.68}$	$\text{29.99}\pm \text{0.36}$	$\text{88.01}\pm \text{0.39}$	$\text{77.05}\pm \text{0.64}$	$\text{87.57}\pm \text{0.41}$
Adversarial (Targeted)	GAT-PerturbNode	$81.33\pm 0.47$	$71.23\pm 1.29$	$63.11\pm 1.39$	$29.21\pm 1.16$	$87.77\pm 1.11$	$75.86\pm 1.99$	$\text{87.38}\pm \text{0.72}$
	GAT-PerturbEdge	$81.45\pm 0.30$	$69.82\pm 1.16$	$62.07\pm 0.91$	$29.11\pm 0.70$	$87.73\pm 1.59$	$76.28\pm 1.72$	$87.15\pm 0.43$
	GAT-PerturbWeight	$81.83\pm 0.64$	$70.92\pm 0.81$	$62.11\pm 1.28$	$29.08\pm 1.30$	$87.81\pm 1.13$	$75.82\pm 1.63$	$87.08\pm 0.31$
	GAT-PerturbEmbedding	$\text{81.85}\pm \text{0.80}$	$\text{71.54}\pm \text{2.49}$	$\text{63.34}\pm \text{1.24}$	$\text{29.34}\pm \text{1.86}$	$\text{87.93}\pm \text{1.21}$	$\text{76.33}\pm \text{2.17}$	$87.31\pm 0.32$
	LINKX	$84.71\pm 0.52$	$68.42\pm 1.38$	$61.81\pm 1.80$	$36.10\pm 1.55$	$85.35\pm 0.97$	$74.08\pm 1.52$	$87.44\pm 0.76$
Random (non-targeted)	LINKX-PerturbNode	$84.75\pm 0.24$	$70.13\pm 0.48$	$63.00\pm 0.48$	$37.04\pm 0.55$	$85.79\pm 1.30$	$\text{74.76}\pm \text{1.54}$	$87.73\pm 0.27$
	LINKX-PerturbEdge	$84.65\pm 0.42$	$70.04\pm 0.86$	$62.21\pm 0.20$	$36.96\pm 0.42$	$85.47\pm 1.04$	$74.49\pm 0.82$	$87.72\pm 0.26$
	LINKX-PerturbWeight	$84.54\pm 0.29$	$70.83\pm 0.74$	$63.07\pm 0.79$	$36.86\pm 0.82$	$85.75\pm 1.31$	$73.83\pm 1.21$	$87.47\pm 0.43$
	LINKX-PerturbEmbedding	$\text{85.42}\pm \text{0.47}$	$\text{71.18}\pm \text{0.84}$	$\text{63.73}\pm \text{0.49}$	$\text{37.47}\pm \text{0.82}$	$\text{86.00}\pm \text{1.97}$	$74.37\pm 1.58$	$\text{88.22}\pm \text{0.27}$
Adversarial (Targeted)	LINKX-PerturbNode	$84.61\pm 0.30$	$70.57\pm 0.61$	$62.94\pm 0.63$	$37.51\pm 0.72$	$85.71\pm 1.14$	$74.24\pm 1.66$	$87.50\pm 0.48$
	LINKX-PerturbEdge	$84.74\pm 0.13$	$70.57\pm 0.36$	$61.96\pm 0.41$	$37.57\pm 0.88$	$84.91\pm 1.29$	$73.74\pm 1.17$	$87.47\pm 0.60$
	LINKX-PerturbWeight	$84.84\pm 0.19$	$70.92\pm 0.67$	$63.17\pm 0.35$	$37.38\pm 0.91$	$85.43\pm 1.81$	$74.61\pm 0.50$	$87.64\pm 0.11$
	LINKX-PerturbEmbedding	$\text{84.93}\pm \text{0.20}$	$\text{71.32}\pm \text{0.42}$	$\text{63.32}\pm \text{0.78}$	$\text{37.75}\pm \text{0.53}$	$\text{86.36}\pm \text{2.45}$	$\text{74.70}\pm \text{1.81}$	$\text{88.42}\pm \text{0.38}$

Table 3. Table 3 . Comparison with other adversarial training GNNs. The best results per dataset is highlighted. (M) denotes out of memory.

	Penn94	Chameleon	Squirrel	Film	Cora	Citeseer	Pubmed
GCN	$82.10\pm 0.34$	$64.56\pm 2.85$	$50.72\pm 0.69$	$29.14\pm 0.62$	$87.28\pm 1.26$	$76.68\pm 1.64$	$87.38\pm 0.66$
GCN + VAT	$55.98\pm 1.44$	$59.17\pm 3.78$	$27.91\pm 0.89$	$29.14\pm 0.62$	$\text{88.37}\pm \text{1.60}$	$78.22\pm 1.69$	$86.83\pm 0.98$
GCN + ADV_Train	$72.38\pm 0.81$	$59.69\pm 1.66$	$39.02\pm 1.54$	$\text{30.65}\pm \text{1.16}$	$87.81\pm 1.67$	$78.01\pm 1.85$	$87.70\pm 0.53$
GCN + TW-AWP	(M)	$47.15\pm 6.53$	$31.58\pm 1.35$	$29.45\pm 1.33$	$87.97\pm 0.91$	$76.07\pm 0.95$	$86.28\pm 0.98$
GCN + Flag	(M)	$66.49\pm 2.48$	$50.10\pm 1.02$	$29.23\pm 0.84$	$87.48\pm 0.95$	$75.48\pm 1.31$	$87.88\pm 0.53$
GCN + PerturbEmbedding	$\text{82.98}\pm \text{0.21}$	$\text{68.51}\pm \text{1.15}$	$\text{56.52}\pm \text{1.46}$	$30.25\pm 1.48$	$88.13\pm 0.79$	$\text{78.63}\pm \text{1.27}$	$\text{87.99}\pm \text{0.33}$

Table 4. Table 5 . Statistics of the graph datasets. #C is the number of distinct node classes, h is the homophily ratio.

Dataset	Nodes	Edges	Features	#C	Edge hom	h	#Training Nodes	#Validation Nodes	#Testing Nodes
Penn94	41,554	1,362,229	5	2	.470	.046	50% of nodes per class	25% of nodes per class	Rest nodes
chameleon	2,277	3,6101	2,325	5	.23	.062	48% of nodes per class	32% of nodes per class	Rest nodes
film	7,600	29,926	931	5	.22	.011	48% of nodes per class	32% of nodes per class	Rest nodes
squirrel	5,201	216,933	2,089	5	.22	.025	48% of nodes per class	32% of nodes per class	Rest nodes
Cora	2,708	5,278	1,433	7	.81	.766	48% of nodes per class	32% of nodes per class	Rest nodes
Citeseer	3,327	4,676	3,703	6	.74	.627	48% of nodes per class	32% of nodes per class	Rest nodes
Pubmed	19,717	44,327	500	3	.80	.664	48% of nodes per class	32% of nodes per class	Rest nodes

Equations

$${\mathbf{H}}^{(0)}=(\mathbf{A}+{△}_{\mathbf{A}})\mathbf{X}{\mathbf{W}}^{(0)}=\mathbf{AX}{\mathbf{W}}^{(0)}+{△}_{\mathbf{A}}\mathbf{X}{\mathbf{W}}^{(0)}={\mathbf{H}}^{(0)}+{△}_{\mathbf{H}}$$

$${\mathbf{H}}^{(0)}=\mathbf{A}(\mathbf{X}+{△}_{\mathbf{X}}){\mathbf{W}}^{(0)}=\mathbf{AX}{\mathbf{W}}^{(0)}+\mathbf{A}{△}_{\mathbf{X}}{\mathbf{W}}^{(0)}={\mathbf{H}}^{(0)}+{△}_{\mathbf{H}}$$

$${\mathbf{H}}^{(l+1)}=\mathbf{A}{\mathbf{H}}^{(l)}({\mathbf{W}}^{(l)}+{△}_{\mathbf{W}})$$

$$={\mathbf{H}}^{(l+1)}+{△}_{\mathbf{H}}$$

$${\mathbf{H}}^{(l)}={\mathbf{H}}^{(l)}+{△}_{\mathbf{H}}$$

$$\underset{\theta }{\min }L\left(\mathbf{X}+{△}_{\mathbf{X}},\mathbf{A},\mathbf{W},\mathbf{H}\right)s.t.\Vert {△}_{\mathbf{X}}{\Vert }_p\le {\delta }_{\mathbf{X}}$$

$$\underset{\theta }{\min }L\left(\mathbf{X},\mathbf{A}+{△}_{\mathbf{A}},\mathbf{W},\mathbf{H}\right)$$

$$\underset{\theta }{\min }L\left(\mathbf{X},\mathbf{A},\mathbf{W}+{△}_{\mathbf{W}},\mathbf{H}\right)s.t.\Vert {△}_{\mathbf{W}}{\Vert }_p\le {\delta }_{\mathbf{W}}$$

$$\underset{\theta }{\min }L\left(\mathbf{X},\mathbf{A},\mathbf{W},\mathbf{H}+{△}_{\mathbf{H}}\right)s.t.\Vert {△}_{\mathbf{H}}{\Vert }_p\le {\delta }_{\mathbf{H}}$$

$$\underset{\theta }{\min }\underset{\beta }{\max }$$

$$\text{where}{△}_{\mathbf{X}}$$

$$\underset{\theta }{\min }\underset{\beta }{\max }$$

$$where{△}_{\mathbf{A}}=Generat$$

$${\mathbf{M}}_{i,j}={{\mathbf{Z}}_i^{(m)}}^T\cdot {\mathbf{Z}}_j^{(m)},{\mathbf{Z}}^{(m)}={\mathbf{MLP}}_{\beta }(A)$$

$${△}_{\mathbf{A}}=-{\mathbf{Top}}_t(\mathbf{M}\odot \mathbf{A})$$

$$\underset{\theta }{\min }\underset{\beta }{\max }$$

$$\text{where}{\Delta }_{\mathbf{W}}$$

$$\underset{\theta }{\min }\underset{\beta }{\max }$$

$$\text{where}{△}_{\mathbf{H}}$$

$${\mathbf{h}}_v^{(k)}={\mathrm{COMBINE}}^{(k)}\left({\mathbf{h}}_v^{(k-1)},\mathrm{AGGR}\left\{{\mathbf{h}}_u^{(k-1)}:u\in N(v)\right\}\right)$$

$${\mathbf{h}}_v^{(k)}=\sigma \left(\sum _{u\in N(v)}\left({\mathbf{W}}^{(k-1)}{h}_v^{(k-1)}{\tilde{A}}_{uv}\right)\right)$$

$${\mathbf{H}}^{(k)}=\sigma \left(\tilde{\mathbf{A}}{\mathbf{H}}^{(k-1)}{({\mathbf{W}}^{(k-1)})}^T\right)$$

$$Y={\mathrm{MLP}}_f\left(\sigma \left(W[h_A;h_X]+h_A+h_X\right)\right)$$

$$\underset{\theta }{\min }\underset{\Vert \delta {\Vert }_p\le \mathbf{ϵ}}{\max }L\left(f_{\theta }\left(x+\delta \right),y\right)$$

$${\delta }_{t+1}=\begin{array}{c}{\prod }_{\Vert \delta {\Vert }_{\infty }\le \mathbf{ϵ}}\end{array}\left({\delta }_t+\alpha \cdot sign\left({\nabla }_{\delta }L\left(f_{\theta }\left(x+{\delta }_t\right),y\right)\right)\right)$$

Full text

Graph Neural Networks; Graph Adversarial Training;

††copyright: none††ccs: Computing methodologies Machine learning

1. Introduction

Graph structured data is ubiquitous in many real-world applications, such as social networks (Tang et al., 2009; Peng et al., 2022), knowledge bases (Vashishth et al., 2019) and recommendation systems (Chen et al., 2021, 2019). As powerful tools for representation learning on graphs, graph neural networks (GNNs) have shown promising results in many graph-related tasks such as node classification (Chen et al., 2022) and graph classification (Defferrard et al., 2016). However, despite the great success of Graph neural networks, recent studies have shown that GNNs are vulnerable to adversarial attacks on node features and graph structure (Deng et al., 2019; Xu et al., 2019; Athalye et al., 2018). Moreover, Due to the additional consideration of connections between examples, GNNs could be more sensitive to the perturbations since the perturbations from neighbor nodes exacerbate the impact on a target node (Feng et al., 2019).

The sensitivity of GNNs to perturbations, particularly those that propagate through the graph’s structure, underscores the need for robust training strategies. Adversarial training is one strategy that has been proposed to mitigate this sensitivity (Madry et al., 2017; Goodfellow et al., 2014; Shafahi et al., 2019; Athalye et al., 2018; Miyato et al., 2016). The key to adversarial training is augmenting data with the worst-case adversarial examples, such that the trained model can correctly classify future adversarial examples. Recently methods (Feng et al., 2019; Xu et al., 2019; Zhu et al., 2019; Zügner and Günnemann, 2019; Jin et al., 2020; Kong et al., 2020; Wu et al., 2020) have implemented adversarial training to augment graph data, aiming to bolster the robustness and generalization of GNNs. These methods typically involve applying perturbations to the node feature, weights, or graph structure and subsequently minimizing the loss by learning more robust graph model parameters under the perturbations. Specifically, perturbations can be further categorized into targeted and non-targeted based on whether they are generated using specific label information. Random perturbations are non-targeted and do not rely on specific label information, while adversarial perturbations are targeted, intentional, and take the target label into account when calculating the loss. Despite the effectiveness of adversarial training in enhancing GNNs’ robustness and generalization abilities, its application has been largely confined to specific datasets and GNN types.

This paper studies the vulnerability of Graph Neural Networks (GNNs) to adversarial attacks on node features and graph structure. Several open problems related to adversarial training on GNNs persist. Firstly, finding a universal perturbation strategy/method suitable for all cases, considering the divergence of different datasets and models is a daunting task. Secondly, the generation of adversarial perturbations in previous methods primarily relied on gradient information, which is a time-consuming process. Thus, discovering a simple and efficient method still remains a challenge.

We propose a novel method, PerturbEmbedding, that integrates adversarial perturbation and training, enhancing GNNs’ resilience to such attacks and improving their generalization ability. PerturbEmbedding performs perturbation operations directly on every hidden embedding of GNNs and provides a unified framework for most existing perturbation strategies/methods. We also offer a unified perspective on the forms of perturbations, namely random (non-targeted) and adversarial (targeted) perturbations. Through experiments on various datasets using different backbone models, we demonstrate that PerturbEmbedding significantly improves both the robustness and generalization abilities of GNNs. The rejection of both random (non-targeted) and adversarial (targeted) perturbations further enhances the backbone model’s performance. Our contributions can be concluded as follows:

• •

We propose a novel method, PerturbEmbedding, which performs perturbation operations directly on every hidden embedding of GNNs and provides a unified framework for most existing perturbation strategies/methods. In other words, these perturbation strategies/methods can be regarded as one special form of PerturbEmbedding.

• •

We offer a unified perspective on the forms of perturbations, namely random (non-targeted) and adversarial perturbations (targeted), to understand the mechanism of perturbation.

• •

We conduct experiments on various datasets using different backbone models to demonstrate that PerturbEmbedding significantly improves both the robustness and generalization abilities of GNNs, outperforming existing methods.

2. Related work

2.1. Graph Neural Networks

Graph Neural Networks (GNNs) have shown promising results in various graph-based applications (Chen et al., 2024; Lv et al., 2025c, a, b). Based on the strong homophily assumption that nodes with similar properties are more likely to be linked together, (Bruna et al., 2013) firstly propose a Convolutional Neural Network in graph domain based on the spectrum of the graph Laplacian. GCN (Kipf and Welling, 2016) introduces a simple and well-behaved layer-wise propagation rule for neural network models that operate directly on graphs GAT (Velickovic et al., 2017) introduces an attention-based architecture, the idea is to compute the hidden representations of each node leveraging masked self-attentional layers. However, such an assumption of homophily is not always true for heterophilic graphs (Zhang et al., 2024; Yang et al., 2025), for example, in online transaction networks, fraudsters are more likely to build connections with customers instead of other fraudsters. To make GNNs generalize to heterophilic graphs, MixHop (Abu-El-Haija et al., 2019) Proposes a mixed feature model that aggregates messages from multi-hop neighbors by mixing powers of the adjacency matrix. LINKX (Lim et al., 2021) proposes a simple method that combines two simple baselines MLP and LINK, which overcomes the scalable issues on large graphs.

Despite these advancements, recent studies (Bojchevski and Günnemann, 2019; Dai et al., 2018) have revealed that GNNs are susceptible to adversarial attacks, which involve small but intentional perturbations to node features or graph structure, such as adding or deleting edges. This vulnerability underscores the critical need for developing robust GNNs capable of withstanding such targeted manipulations.

2.2. Adversarial Training

Adversarial training (Madry et al., 2017; Goodfellow et al., 2014; Shafahi et al., 2019; Athalye et al., 2018; Miyato et al., 2016; Yang et al., 2024; Liu et al., 2024, 2025) is a widely used countermeasure to tackle the vulnerability to intentional perturbations of deep neural networks on image and text data, which involves an alternative min-max process. The key of adversarial training is generating adversarial examples from clean examples with perturbations maximally attacking the training objective such that the trained model can correctly classify the future adversarial examples.

Specifically, the perturbations can be further categorized into targeted and non-targeted based on whether they are generated using specific label information. Random perturbations are used to generate adversarial samples, which aim to slightly modify the sample in such a way that the target model misclassifies it without any particular preference towards a specific output. These perturbations are small, random changes added to the input data with the intent to deceive the model into making an incorrect classification decision. Since random perturbations do not rely on specific label information, they are not designed to misclassify the input data into any particular category but rather to cause the model’s prediction to be incorrect. Hence, this type of attack is considered non-targeted. Targeted attacks are designed to generate adversarial samples that can change the prediction of a model to a specific target label. In these attacks, the perturbations are purposeful and consider the target label when calculating the loss. Attackers optimize a loss function that quantifies the discrepancy between the model’s output and the desired target label to generate adversarial perturbations. Therefore, these perturbations are intentional and meant to cause the model to misclassify the input data into a specific, attacker-chosen category. Since the loss calculation involves a specific label, this type of attack is considered targeted. In mathematical terms, adversarial training is posed as a min-max problem, aiming to find the optimal solution for the worst-case scenario. (Madry et al., 2017) showed that this min-max problem could be reliably tackled by Stochastic Gradient Descent (SGD) for the outer minimization and Projected Gradient Descent (PGD) for the inner maximization. (Miyato et al., 2016) extends adversarial and virtual adversarial training to the text domain by applying perturbations to the word embeddings in a recurrent neural network rather than to the original input itself. (Shafahi et al., 2019) presents an algorithm that eliminates the overhead cost of generating adversarial examples by recycling the gradient information computed when updating model parameters.

However, adversarial training for GNNs presents challenges. GNNs inherently treat examples as interconnected due to the graph structure, unlike the independent assumption in traditional adversarial training. This interconnectedness means that GNNs are particularly sensitive to perturbations, as changes in one node can propagate through the graph and significantly affect other nodes. The adjacency matrix, which captures these connections, can amplify the impact of adversarial perturbations, making GNNs more vulnerable to attacks that exploit the graph’s topology.

2.3. Adversarial Training for Graphs

Several studies (Deng et al., 2019; Xu et al., 2019; Athalye et al., 2018) have shown that GNNs are vulnerable to adversarial attacks on the node features and/or graph structure. Recently many methods have applied adversarial training to augment graph data to to improve the robustness and generalization of GNNs. Specifically, these methods mainly apply perturbation to the node feature, weights or graph structure. (Dai et al., 2018) suggest dropping edges randomly in adversarial training to generate perturbations on the adjacency matrix A. (Feng et al., 2019) designs a dynamic regularizer forcing GNN models to learn to prevent the propagation of perturbations on graphs, enhancing the model’s robustness against perturbations on node input features. (Xu et al., 2019) introduce projection gradient descent (PGD) to optimize edge perturbation. (Dai et al., 2019) defines adversarial perturbations in the embedding space with an adaptive $L_{2}$ norm constraint that depends on the connectivity pattern of node pairs. (Kong et al., 2020) regard adversarial training as a method of data augmentation and propose FLAG which iteratively augments node features with gradient-based adversarial perturbations during training, and boosts performance at test time. (Jin and Zhang, 2019) raise a latent adversarial training method that injects perturbations on the hidden layer.

In contrast to these specialized methods, our work aims to develop a unified adversarial training framework that encapsulates a diverse set of adversarial strategies. This framework is designed to comprehensively alleviate the vulnerabilities of GNNs by targeting different aspects of the graph data, namely the adjacency matrix (A), node features (X), and hidden representations (H). By integrating these strategies, our framework aims to provide a more holistic defense against adversarial attacks, ensuring that GNNs can robustly learn and generalize across a wide range of graph-based tasks.

3. Notations

Notations. Let $G=(V,E)$ be a graph, where $V$ is the set of nodes, $E$ is the set of edges. $X$ $\in$ $R^{N\times F}$ denotes feature matrix, where the vector $x_{v}$ corresponds to the feature of node $v$, and c is the dimensionality of features. $A\in\{0,1\}^{N\times N}$ is the adjacency matrix, where $A_{uv}=1$ if there exists an edge between node $v_{i}$ and $v_{j}$, otherwise $A_{ij}=0$. For semi-supervised node classification tasks, only part of nodes have known labels $Y^{o}=\{y_{1},y_{2},...y_{n}\}$, where $y_{j}\in\{0,1,...,c-1\}$ denotes the label of node $j$, $c$ is the number of classes. The goal of GNNs is to predict the class of unlabeled nodes.

4. Method

In this section, we introduce our proposed unified framework PerturbEmbedding, which is universally applicable to all Graph Neural Networks (GNNs). We begin by delineating the intricacies of our approach and subsequently demonstrate that the most common existing perturbing methods, namely PerturbNode, PerturbEdge, and PerturbWeight, can be unified into our framework PerturbEmbedding. After that, we offer a unified perspective on the forms of perturbations, namely random (non-targeted) and adversarial perturbations (targeted).

4.1. A Unified Specific framework

As Figure 1 suggests, most existing adversarial GNNs mainly apply perturbations to the node features, weights or graph structure, and then minimize the loss by learning the more robust graph model parameters $\theta$ under the perturbed samples. For simplicity, we use PerturbEdge, PerturbNode and PerturbWeight to refer to these perturbation strategies in the following text. We prove that the effects of the most common existing perturbation strategies, i.e., PerturbEdge, PerturebNode and PerturbWeight can be unified into our proposed framework–directly perturbing the embedding process, which is referred to PerturbEmbedding.

Unifying perturbation Strategies. As shown in Figure 1, in intuition, applying perturbation to features, edges or weights will all ultimately act on the embedding $\mathbf{H}$. This inspires us to explore the specific connection between different perturbation strategies. As a starting point, we demonstrate that PerturbEdge, PerturbNode, PerturbWeight, and PerturbEmbedding can all be formulated in Table 1. More importantly, we find that PerturbEdge, PerturbNode, and PerturbWeight are actually special cases of PerturbEmbedding, and thus can be expressed in a uniform framework. We provide the equivalent operation of PerturbEmbedding for each strategy below.

PerturbEdge. The earliest and most primitive way of perturbing the graph is to randomly drop edges (Dai et al., 2018). The joint training of such cheap adversarial perturbations is shown to slightly improve the robustness of standard GNN models towards both graph and node classification tasks. Taking Message-Passing GNNs as an Example, applying perturbations $\triangle_{\mathbf{A}}$ on Edge $\mathbf{A}$ is equivalent to directly applying perturbations $\triangle_{\mathbf{H}}$ on embedding $\mathbf{H}^{(0)}$ at the 1-th layer, which blends the information of both node features and the graph:

[TABLE]

where $\triangle_{\mathbf{A}}=\{{\triangle_{\mathbf{A}}}_{ij}\in\{0,-1\}\land{\triangle_{\mathbf{A}}}_{ij}\in\mathbf{A}\}$ indicates the edges that need to be dropped.

PerturbNode. In addition to edges, among various defensive strategies, adversarial training based on feature perturbation generally exhibits greater robustness compared to other methods (Xue et al., 2021). The intentionally perturbed features can be considered as augmented data, which aids the model in extrapolating to out-of-distribution testing data and mitigates overfitting (Xue et al., 2021). Taking Message-Passing GNNs as an Example, applying perturbations $\triangle_{\mathbf{X}}\in R^{N\times F}$ on node feature $\mathbf{X}$ is equivalent to directly applying perturbations $\triangle_{\mathbf{H}}$ on embedding $\mathbf{H}^{(0)}$ at the $1$-th layer, which blends the information of both node features and the graph:

[TABLE]

PerturbWeight. Adversarial weight perturbation (Wu et al., 2020) (AWP) aims to minimize the worst-case loss within a small region centered at the model weights, seeking parameters whose neighborhood regions exhibit low training loss. In this way, the weight loss landscape has smaller curvature at the final learned weights, which in turn shrinks the generalization gap. Applying perturbations $\triangle_{\mathbf{W}}$ on Weight $\mathbf{W}^{(l)}$ is equivalent to directly applying perturbations $\triangle_{\mathbf{H}}$ on embedding $\mathbf{H}^{(l+1)}$ at the $(l+1)$-th layer:

[TABLE]

PerturbEmbedding. LAT-GCN (Jin and Zhang, 2019) proposes injecting adversarial perturbations only into the first layer $\mathbf{H}^{(0)}$ of GCN, leading to indirect perturbations of the graph, which implicitly enforces robustness against structural attacks. However, unlike LAT-GCN, which only injects adversarial perturbations into the first layer of GCN, PerturbEmbedding applies perturbations $\triangle_{\mathbf{H}}$ on embedding $\mathbf{H}^{(l)}$ at every layer for various types of GNNs:

[TABLE]

Based on the above descriptions, we find that PerturbEdge, PerturbNode, and PerturbWeight are equivalent to PerturbEmbedding, and they can be regarded as special cases of PerturbEmbedding, which makes PerturbEmbedding the most flexible strategy.

4.2. Unified random perturbation (non-targeted) for GNNs

Motivated by (Yu et al., 2022) which adds directed random noises to the embedding for different data, it can smoothly adjust the uniformity of learned embeddings and has been proven to be a very effective data augmentation method for recommendation. Here we also add random noise directly to the graph to generate random perturbations, then try to minimize the loss by learning the more robust graph model parameters $\theta$ under the random perturbations, which is different from previous adversarial training methods which tries to maximize task-oriented loss by generating adversarial perturbations. In this way, the learned GNNs are expected to be resistant to future adversarial attacks. We consider adding random noise/perturbation to the node features $\mathbf{X}$, edges $\mathbf{A}$, weight $\mathbf{w}$ and embedding $\mathbf{H}$ to generate perturbations.

Random node perturbation. Here we directly add random noises to node feature to generate perturbations during training, then train neural networks by incorporating the random feature perturbations on the inputs as follows:

[TABLE]

where $\theta$ is the parameters of GNNs, $\mathbf{H}$ is the hidden representation, $\triangle_{\mathbf{X}}$ denotes the random perturbation bounded by the $\delta_{\mathbf{X}}$ norm ball. The optimization of Eq. 5 seeks to find the optimal weights having the smaller losses at the neighborhood regions centered at the input features, the maliciously perturbed features could also be treated as the augmentation data, which helps the model extrapolate the out-of-distribution testing data and alleviate the overfitting.

Random edge perturbation. We apply the earliest and most primitive way to perturb the graph structure – randomly drop edges. To be specific, we inject random perturbations to the original graph while training GNNs:

[TABLE]

where $\triangle_{\mathbf{A}}=\{{\triangle_{\mathbf{A}}}_{ij}\in\{0,-1\}\land{\triangle_{\mathbf{A}}}_{ij}\in A\}$, the random perturbations are expected to be small, unnoticeable and should not corrupt the majority of graph structures. The joint training of such cheap perturbations is shown to slightly improve the robustness of standard GNN models towards graph and node classification tasks(Sun et al., 2022).

Random weight perturbation. We directly inject random perturbations to weight $\mathbf{W}$ while training GNNs:

[TABLE]

where $\triangle_{\mathbf{W}}$ denotes the random perturbation bounded by the $\delta_{\mathbf{W}}$ norm ball, Eq. 7 seeks to obtain parameter $\mathbf{W}$ whose neighborhood regions have low training loss. Therefore, the weight loss landscape is expected to be smoothed to reduce the generalization gap.

Random Embedding perturbation. We inject random perturbations to hidden embedding $\mathbf{H}$ of GNNs, specifically, we inject random perturbations to every hidden layer while training GNNs and minimize the loss under the random embedding perturbations:

[TABLE]

where $\triangle_{\mathbf{H}}$ denotes the random embedding perturbation bounded by the $\delta_{\mathbf{H}}$ norm ball. Hidden layer blends the information of both node features, the graph and weight, injecting adversarial perturbations to hidden embedding $\mathbf{H}$ indirectly represent the perturbations in $\mathbf{X}$, $\mathbf{G}$ and $\mathbf{W}$, which is more flexible compared with other methods and implicitly enforces robustness to structural attacks.

4.3. Unified adversarial perturbation (targeted) for GNNs

Previous adversarial GNNs perturb graphs with gradient-based adversarial perturbations during training, which is a time-consuming process. we introduce a novel generator for graph structure data, which tries to maximize task-oriented loss by generating targeted adversarial perturbations, and GNNs (discriminator) tries to minimize the loss by learning the more robust graph model parameters $\theta$ under the targeted adversarial perturbations. Specifically, We apply different forms of targeted perturbations to graph, namely features, edges, weight, and representation.

Adversarial node perturbation. We inject perturbations to the node features $\mathbf{X}$ while training GNNs. The adversarial learning objective following the min-max formulation:

[TABLE]

where the generator, with parameters $\beta$, aims to generate worst-case adversarial perturbations $\triangle_{\mathbf{X}}$ to maximize task-oriented loss, while the GNNs minimize the loss under these perturbations.

Adversarial edge perturbation. GNNs are sensitive to small perturbations of graph structure (Dai et al., 2018), to defends against perturbations on graph topology, we inject perturbations to the origin graph while training GNNs, The adversarial learning objective following the min-max formulation:

[TABLE]

where the generator creates targeted adversarial perturbations $\triangle_{\mathbf{A}}$ based on the adjacency matrix $\mathbf{A}$, using a learnable MLP to generate a soft mask matrix $\mathbf{M}$, which is then used to obtain $\triangle_{\mathbf{A}}$.

[TABLE]

where $\mathbf{Z}^{(m)}$ is the node representation. Then, we obtain the targeted adversarial perturbations $\triangle_{\mathbf{A}}$:

[TABLE]

where $\odot$ means the element-wise matrix multiplication, and $\mathbf{Top}_{t}$ selects the top $t$-percentage of elements with the largest values. In this way, we can efficiently generate adversarial perturbations $\triangle_{\mathbf{A}}$.

Adversarial Weight perturbation. The adversarial learning objective of following min-max formulation:

[TABLE]

where the targeted adversarial perturbations $\triangle_{\mathbf{W}}$ are generated based node embedding $\mathbf{W}$, $\beta$ is the model parameters of Generator.

Adversarial Embedding perturbation. For GNNs, the features of nodes are aggregated by the neighborhoods. Perturbation on the hidden representation output to affect the nodes with their neighborhoods would produce a more challenging view. Specifically, we inject random perturbations to every hidden embedding while training GNNs, we optimize the following min-max formulation:

[TABLE]

where the targeted adversarial perturbations $\triangle_{\mathbf{H}}$ are generated based node embedding $\mathbf{H}$, $\beta$ is the model parameters of Generator.

The unified adversarial perturbation (targeted) algorithm is detailed in Appendix B, which alternates between updating the generator and GNN parameters using stochastic gradient descent.

5. Experiments

In this section, we conduct extensive experiments on various datasets to empirically demonstrate our method’s effectiveness by answering the following questions:

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(RQ 1) How does PerturbEmbedding enhance the GNNs in node classification tasks across various datasets and backbone models?

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(RQ 2) How does PerturbEmbedding perform under different types of perturbations, including random and adversarial, in maintaining the robustness of GNNs?

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(RQ 3) What are the individual contributions of the components within PerturbEmbedding to its overall performance, and how do they compare to specialized perturbation methods?

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(RQ 4) How efficient is PerturbEmbedding, and what are the implications for practical applications?

5.1. Experimental Setup.

Datasets. We employ 7 graph datasets in our experiments, including 3 widely used homophilous datasets Cora, Citeseer and PubMed (Yang et al., 2016), and 4 heterophilous datasets Penn94, Chameleon, Squirrel and Chameleon. Dataset statistics are in Appendix Table 5.

Training and Evaluation. We conduct experiments on Cora, Citeseer and PubMed with 60/20/20 train/val/test splits, for Penn94, we follow the setting in (Lim et al., 2021) with 50/25/25 train/val/test splits, for datasets of Film, Squirrel, Chameleon, we follow the widely used semi-supervised setting in (Pei et al., 2020) with the standard 48/32/20 train/val/test splits. We use classification accuracy as the evaluation metric. For a fair comparison, we conduct five independent runs and report the mean result with the standard deviation over it.

Baseline methods. We compare proposed unified framework PerturbEmbedding with other perturbation strategies, including PerturbEdge, PerturbNode, and PerturbWeight. According to the generation form of perturbations, they can be further divided into random perturbations (non-targeted) and adversarial perturbations (targeted). We combine different perturbation strategies and forms of perturbations, and adopt them on various GNNs as the backbone model, and compare their performances on different datasets.

Backbone models. To illustrate that our method can be applied in different different scenarios and graph neural networks with different structures. we mainly consider three mainstream GNNs as our backbone models: GCN (Kipf and Welling, 2016), GAT(Velickovic et al., 2017) and LINKX(Lim et al., 2021). GCN and GAT are the most representative homophilous methods and adopt the message-passing framework. Different from message-passing GNNs that aggregate messages from neighbor nodes for the target node, LINKX(Lim et al., 2021) separately embeds node features and adjacency information with MLPs, combines the embeddings by concatenation, which achieves superior performance on heterophilous graphs. For GCN and GAT, we apply PerturbEmbedding to the first layer output $H^{(1)}$ after graph convolution. For LINKX, we apply PerturbEmbedding to every hidden layer.

5.2. Overall results

To answer RQ 1 adn RQ 2, we apply our method in different methods, and Table 2 summarizes the overall results.

Effect of random/adversarial perturbations. It is observed that random/adversarial perturbation methods consistently outperform GNNs without random/adversarial perturbations in node classification tasks for all 7 datasets. Besides, we see that the effects of random/adversarial perturbation methods vary over different datasets and backbone models.

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For example, random perturbation methods on LINKX obtain an average accuracy improvement of 2.1% on Chameleon, while 0.35% on Pubmed. Meanwhile, random perturbation methods achieve 1.46% accuracy improvement for GCN on Citeseer, while only 0.97% for GAT.

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Adversarial perturbation methods on GCN obtain an average accuracy improvement of 3.1% on the chameleon, while 0.48% on Film. Meanwhile, adversarial perturbation methods achieve 3.1% accuracy improvement for GCN on squirrel, while only 0.85% for GAT.

Comparison of random perturbations and adversarial perturbations. We find that rejecting random perturbations and adversarial perturbations could both improve the performance of the backbone model, and the performance of random perturbations and adversarial perturbations is comparable. Taking PerturbEmbedding as the example, we have 21 settings under the node classification task, each of which is a combination of different backbone models and datasets. (e.g., GCN-Cora), it is shown that PerturbEmbedding with random perturbations achieves the optimal results in 11 settings while adversarial perturbed get 10 optimal results. which indicates that simply injecting random noise into the embedding during the training process can achieve comparable results.

Comparison with other adversarial GNNs. We compare our method with the following representative adversarial GNNs: 1) Attribute Perturbations methods: Graph Adversarial Training (Feng et al., 2019) (GraphAT), Free Large-scale Adversarial Augmentation on Graphs (Kong et al., 2020) (Flag). 2) Structure Perturbations methods: Adversarial Training (Xu et al., 2019) (ADV_Train). 3) Weight Perturbations methods: Weighted Truncated Adversarial Weight Perturbation (Wu et al., 2020) (WT-AWP) . Table 3 shows GCN’s performance with different adversarial augmentations and PerturbEmbedding outperforms all other methods in most datasets, which demonstrates its effectiveness.

5.3. Robustness analysis

To answer RQ 3, We study the robustness of perturbation methods by measuring their ability to handle perturbed graphs. Specifically, we conduct experiments on two settings: 1) Random perturbations: we randomly add a certain ratio of edges into the graph and perform the node classification. 2) Adversarial attacks: We also conduct experiments to evaluate the effectiveness of our method against adversarial attacks.

Random perturbations. We conduct experiments on citeseer and Chameleon, using GCN and GAT as the backbone model. We randomly add a certain ratio of edges into these datasets and perform the node classification, Figure 2 and 3 show the results. We find that all the random perturbation methods have positive effects when the perturbation rate increases from 0% to %30, which indicates that the random perturbation methods strengthen the robustness of GNN models. Besides, our proposed PerturbEmbedding shows its versatility and outperforms other perturbation methods in noisy situations.

Adversarial Attacks. We also conduct experiments to evaluate the effectiveness of perturbation methods against adversarial attacks. We apply Meta attacks and PGD attacks to perturb the graph structures on Cora, CiteSeer and Chameleon, using GCN, GAT and LINKX as the backbone model. Figure 4 and 5 presents the results.

5.4. Component analysis

Comparison of different perturbation methods. Our proposed method works well in all settings, exhibiting its strong adaptability to various scenarios. Overall, we have 42 settings under the node classification task, each of which is a combination of different backbone models, different forms of perturbation and datasets. (e.g., GCN-Random-Cora). It is shown that PerturbEmbedding achieves optimal results in 34 settings. The reason why PerturbEmbedding outperforms other methods is that other methods can be regarded as a special form of PerturbEmbedding, directly applying perturbations to embedding could achieve finer granularity of perturbation, thus it could be adapted to various scenarios, making PerturbEmbedding the most flexible and useful method.

Uniformity analysis. We plot the node Embedding Distribution with Gaussian kernel density estimation (KDE) in R2 and von Mises-Fisher (vMF) KDE on angles. Figure 6 and 7 show the node Embedding Distribution of GCN and our method trained on cora and chameleon. We observe that the features learned from PerturbEmbedding (right in Figure 6 and 7) are more uniform compared with GCN (left in Figure 6 and 7), which explains PerturbEmbedding could preserve more information of the data than GCN (Wang and Isola, 2020), thus improves the robustness and generalization abilities of GNNs.

5.5. Efficiency analysis

Training process analysis. We conduct experiments to analyze the loss during the training process when employing different adversarial methods. Figure 8 shows the change of loss in GCN training processes when employing different adversarial methods on Cora. Furthermore, similar training loss curves can be drawn under other experimental settings. The experimental results suggest that PerturbEmbedding presents the smallest sample variance among all methods, thus achieving the fastest convergence and the most stable performance.

Analysis on Computational Efficiency. We compare the training time of different perturbation methods on the Cora dataset, using GCN as the backbone model. In Table 4, we observe that PerturbEmbedding is faster than other perturbation methods, and has the closest time to the GCN.

[TABLE]

6. Conclusion

In this paper, we propose a novel method, PerturbEmbedding, that integrates adversarial perturbation and training, enhancing GNNs’ resilience to such attacks and improving their generalization ability. PerturbEmbedding performs perturbation operations directly on every hidden embedding of GNNs and provides a unified framework for most existing perturbation strategies/methods. We also offer a unified perspective on the forms of perturbations, namely random and adversarial perturbations. Through experiments on various datasets using different backbone models, we demonstrate that PerturbEmbedding significantly improves both the robustness and generalization abilities of GNNs, outperforming existing methods. The rejection of both random (non-targeted) and adversarial (targeted) perturbations further enhances the backbone model’s performance.

Appendix A Preliminary

Message-passing GNNs. Based on the strong homophily assumption that nodes with similar properties are more likely to be linked together, most of the existing GNN models (GCN, GAT) adopt the message-passing framework, where each node sends messages to its neighbors and simultaneously receives messages from its neighbors. Formally, the $k$-th iteration of message passing, or the $k$-th layer of GNN forward computation is:

[TABLE]

where $\mathbf{h}^{(k)}_{v}$ is the embedding of node $v$ at the $k$-th layer, and $N(v)$ is a set of nodes adjacent to node $v$, and $\mathbf{h}^{(0)}_{v}=x_{v}$. $\operatorname{COMBINE}^{(k)}$ is differentiable function and $\operatorname{AGGR}$ represents the aggregation operation.

A special form of Message-passing GNNs is graph convolutional networks (GCN) (Kipf and Welling, 2016), the forward inference at the $k$-th layer of GCN is formally defined as:

[TABLE]

where $\sigma(\cdot)$ is the ReLU function, ${\mathbf{W}^{(k-1)}}$ is the trainable weight matrix at layer k - 1. Let $\tilde{A}_{i\mathrel{\mathop{\ordinarycolon}}}$ denote the $i$th row of $\tilde{A}$ and $\mathbf{H}^{(k)}=\left[{(\mathbf{h}^{(k)}_{1})}^{T};...;{(\mathbf{h}^{(k)}_{N})}^{T}\right]$, we than have the standard form of GCN:

[TABLE]

where $\mathbf{\hat{A}}=\mathbf{A}+\mathbf{I},\tilde{\mathbf{A}}={\mathbf{\hat{D}}}^{-\frac{1}{2}}\mathbf{\hat{A}}{\mathbf{\hat{D}}}^{-\frac{1}{2}}$.

LINKX. However, such an assumption of homophily is not always true for heterophilic graphs, Message-passing GNNs fail to generalize to this setting. (Lim et al., 2021) proposes a simple method LINKX that combines two simple baselines MLP and LINK, which overcomes the scalable issues and achieves superior performance on large graphs. LINKX separately embeds node features and adjacency information with MLPs, combines the embeddings by concatenation, then uses a final MLP to generate predictions:

[TABLE]

where $h_{A}=\operatorname{MLP}_{A}(A)$, $h_{X}=\operatorname{MLP}_{X}(X)$.

Adversarial Training. Standard adversarial training seeks to solve the min-max problem as:

[TABLE]

where $y$ is the label, $\lVert\cdot\rVert$ is some $l_{p}$-norm distance metric, $\boldsymbol{\epsilon}$ is the perturbation budget, and $L$ is the objective function. The typical approximation of the inner maximization under an $l_{\infty}$-norm constraint is as follows :

[TABLE]

where perturbation $\delta$ is updated iteratively, and $\begin{matrix}\prod_{\lVert\delta\rVert_{\infty}\leq\boldsymbol{\epsilon}}\end{matrix}$ performs projection onto the $\boldsymbol{\epsilon}$-ball in the $l_{\infty}$-norm. For maximum robustness, this iterative updating procedure usually loops M times, which makes PGD computationally expensive. While there are M forward and backward steps within the process, $\theta$ gets updated just once using the final $\delta_{M}$.

Appendix B Algorithm details

Appendix C Dataset Details

In this paper, we conduct experiments on 7 graph datasets, including three homophilous datasets and 4 heterophilous datasets, here are the details of these 7 graph datasets in our experiments, and dataset statistics are in Appendix Table 5.

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Cora, Citeseer and PubMed: These 3 different datasets are most widely used homophilous datasets where majority of edges connect nodes within the same class. Specifically, these 3 datasets are citation graphs, where each node represents a scientific paper. These graphs use bag-of-words representations as the feature vectors of nodes. Each node is assigned a label indicating the research field.

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Penn94 (Traud et al., 2012) is a friendship network from the 2005 Facebook 100 university student networks, where nodes represent students. Each node is labeled with the user’s reported gender. Major, second major/minor, dorm/house, year, and high school are the features of the node.

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Film/Actor is an actor co-occurrence network in which nodes represent actors and edges represent co-occurrence on the same Wikipedia page. It is the actor-only induced subgraph of the film-director-actor-writer network.

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Chameleon and Squirrel are Wikipedia networks (Rozemberczki et al., 2021), in which nodes represent Wikipedia web pages and edges are mutual links between pages. And node features correspond to several informative nouns on the Wikipedia pages. Each node is assigned one of five classes based on the average monthly traffic of the web page.