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AdvSynGNN: Structure-Adaptive Graph Neural Nets via Adversarial Synthesis and Self-Corrective Propagation

Rong Fu, Muge Qi, Chunlei Meng, Shuo Yin, Kun Liu, Zhaolu Kang, Simon Fong

Year: 2026Venue: arXiv preprintArea: cs.LGType: PreprintEmbeddings: 122

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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 91%

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Summary

The paper introduces AdvSynGNN, a graph neural network architecture designed to handle structural noise and non-homophilous topologies. It combines multi-resolution structural synthesis, contrastive pretraining, a heterophily-adaptive transformer backbone, and an adversarial propagation engine. The system uses a generative component to identify connectivity alterations and a discriminator to enforce coherence, while label refinement is achieved through residual correction guided by per-node confidence metrics.

Entities (6)

AdvSynGNN · model · 100%Adversarial Propagation · technique · 95%Graph Neural Networks · method · 90%Heterophily · concept · 90%Residual Correction · technique · 90%Transformer Backbone · component · 90%

Relation Signals (5)

AdvSynGNN → uses → Adversarial Propagation

confidence 95% · Central to our contribution is an integrated adversarial propagation engine... Central to our contribution is an integrated adversarial propagation engine

AdvSynGNN → addresses → Heterophily

confidence 90% · adaptively accommodates heterophily by modulating attention mechanisms through learned topological signals

AdvSynGNN → contains → Transformer Backbone

confidence 90% · We develop a transformer backbone that adaptively accommodates heterophily

AdvSynGNN → employs → Residual Correction

confidence 90% · label refinement is achieved through a residual correction scheme guided by per-node confidence metrics

Graph Neural Networks → suffersfrom → Heterophily

confidence 85% · Graph neural networks frequently encounter significant performance degradation when confronted with structural noise or non-homophilous topologies.

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Abstract

Abstract:Graph neural networks frequently encounter significant performance degradation when confronted with structural noise or non-homophilous topologies. To address these systemic vulnerabilities, we present AdvSynGNN, a comprehensive architecture designed for resilient node-level representation learning. The proposed framework orchestrates multi-resolution structural synthesis alongside contrastive objectives to establish geometry-sensitive initializations. We develop a transformer backbone that adaptively accommodates heterophily by modulating attention mechanisms through learned topological signals. Central to our contribution is an integrated adversarial propagation engine, where a generative component identifies potential connectivity alterations while a discriminator enforces global coherence. Furthermore, label refinement is achieved through a residual correction scheme guided by per-node confidence metrics, which facilitates precise control over iterative stability. Empirical evaluations demonstrate that this synergistic approach effectively optimizes predictive accuracy across diverse graph distributions while maintaining computational efficiency. The study concludes with practical implementation protocols to ensure the robust deployment of the AdvSynGNN system in large-scale environments.

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AdvSynGNN: Structure-Adaptive Graph Neural Nets via Adversarial Synthesis and Self-Corrective Propagation Rong Fu University of Macau mc46603@um.edu.mo &Muge Qi Peking University 2301210659@stu.pku.edu.cn &Chunlei Meng Fudan University clmeng23@m.fudan.edu.cn &Shuo Yin Tsinghua University yins25@mails.tsinghua.edu.cn &Kun Liu University of Southampton kundy9909@gmail.com &Zhaolu Kang Peking University kangzl9966@gmail.com &Simon Fong University of Macau ccfong@um.edu.mo Corresponding author: mc46603@um.edu.mo Abstract Graph neural networks frequently encounter significant performance degradation when confronted with structural noise or non-homophilous topologies. To address these systemic vulnerabilities, we present AdvSynGNN, a comprehensive architecture designed for resilient node-level representation learning. The proposed framework orchestrates multi-resolution structural synthesis alongside contrastive objectives to establish geometry-sensitive initializations. We develop a transformer backbone that adaptively accommodates heterophily by modulating attention mechanisms through learned topological signals. Central to our contribution is an integrated adversarial propagation engine, where a generative component identifies potential connectivity alterations while a discriminator enforces global coherence. Furthermore, label refinement is achieved through a residual correction scheme guided by per-node confidence metrics, which facilitates precise control over iterative stability. Empirical evaluations demonstrate that this synergistic approach effectively optimizes predictive accuracy across diverse graph distributions while maintaining computational efficiency. The study concludes with practical implementation protocols to ensure the robust deployment of the AdvSynGNN system in large-scale environments. Keywords Adversarial Graph Learning, Transformer Architectures, Multi-scale Embeddings, Generative Pretraining, Adaptive Signal Calibration, Vertex Classification, Computational Efficiency 1 Introduction Graph-based semi-supervised learning plays a central role in applications where labeled data are scarce yet relational structure is abundant. Classical message-passing models and propagation frameworks remain foundational, and recent work increasingly emphasizes scalable pretraining and engineering practices required for production graphs. Foundational propagation paradigms and p-Laplacian message transmission established key algorithmic primitives [13, 21], while adaptive smoothing and corrected-smoothing lines of work demonstrated that shallow, well-calibrated pipelines can rival deeper GNNs in many settings [85, 26]. Contemporary research advances both methodology and systems: scalable and adaptive pretraining improves applicability on industrial graphs [61], and engineering toolkits make large-scale GNN recommendations practical [58]. Empirical studies that disentangle feature, structural, and label homophily show that simple homophily metrics do not fully predict model behavior and expose failure modes for attention-style learners [90]. Complementary work explores adversarial robustness and data-centric defenses, including augmentations guided by external models [87], and dynamic multi-relational modeling for forecasting tasks in finance and other domains [53]. Despite this progress, three concrete and verifiable pain points limit the adoption of expressive transformer-like architectures on real-world graphs. First, transformer attention can be brittle on low-homophily graphs: when labels are not well aligned with local connectivity, attention mechanisms that implicitly assume locality misallocate weight and harm downstream accuracy. Second, models are fragile to structural noise: representations learned without training-time defenses often change drastically under modest edge perturbations, and robustness is typically measured post hoc rather than enforced during training. Third, scaling expressive architectures to million-node graphs incurs prohibitive memory and runtime cost, which impedes deployment in production systems. AdvSynGNN addresses these gaps through three components emphasizing robustness, adaptability, and efficiency. A multi-scale structural encoding stage with contrastive pretraining yields geometry-aware embeddings stable under topology changes. A structure-aware transformer injects learned structural bias and feature-difference cues into attention, avoiding implicit homophily assumptions. An adversarial propagation module jointly trains a topology generator, discriminator, and representation learner so that structural perturbations act as training-time regularizers. These modules are coupled by a per-node confidence estimator that gates residual correction and ensures contraction under mild spectral controls, with remedies such as spectral clipping and confidence capping when bounds are violated. The integrated design matters because adaptive per-node propagation prevents uniform amplification of errors from unreliable nodes, structure-aware attention reduces mismatch between attention allocation and label distributions on heterophilous graphs, and training-time adversarial perturbations embed robustness into representations instead of relying on post-hoc defenses. Together these elements produce a balanced pipeline that reconciles expressivity, robustness, and computational tractability. Our contributions are as follows. We introduce AdvSynGNN, a modular end-to-end architecture that couples adversarial topology synthesis, heterophily-aware transformer attention, contrastive multi-scale structural pretraining, and node-confidence-weighted residual correction. We provide a practical theoretical analysis that supplies sufficient contractivity conditions for the iterative residual correction and propose engineering strategies for settings where spectral bounds are challenged. We present comprehensive empirical evaluations on homophilous and heterophilous benchmarks that measure accuracy, robustness, and embedding stability, and we report ablation diagnostics that disentangle the roles of adversarial regularization and confidence-weighted propagation. Finally, we release implementation notes and hyperparameter recipes to facilitate implementation and community follow-up. 2 Related Work 2.1 Architectural development for graph representation Graph representation learning evolved from spectral and spatial formulations to architectures capturing multi-scale and long-range interactions. Early spectral filters and message passing established convolutional patterns [14, 28], later extended by attention-based variants [65]. Methods reconciling spectral and spatial views introduced precomputation and simplified baselines such as SIGN [19, 47, 49], while unified analyses clarified design trade-offs [8]. For heterophilous graphs, decoupled pipelines and structural encodings (degree, feature differences) improved robustness [40, 18, 33, 71, 31]. These advances underscore the importance of multi-scale context and topology-aware design. 2.2 Semi-supervised propagation, contrastive pretraining and theory Propagation-based and residual-correction methods remain central for semi-supervised graph learning, where shallow predictors with principled correction can match deeper GNNs, inspiring nonlinear and adaptive variants for label efficiency [26, 57]. Contrastive and diffusion-based pretraining further enhance transferability under distribution shifts [36, 44, 86]. Theoretical analyses clarify convergence regimes and noise amplification in propagation [59], while benchmark taxonomies characterize method behavior across structural settings [41]. These insights guide pretraining objectives and propagation regularization. 2.3 Robustness to structural noise and adversarial augmentation Robustness to noisy or manipulated graphs has driven strategies such as denoising, adversarial edge modification, and generator-based augmentation during training [22, 7, 4, 78]. Methods prune spurious substructures or enforce invariant features for out-of-distribution generalization [78, 6], while diffusion-based and structural augmentations expose models to diverse topologies for improved resilience [69, 67]. Recent work emphasizes principled augmentation and explores its interaction with calibration and confidence mechanisms [44, 6]. 2.4 Transformer-style architectures and structural encodings Graph transformers provide global receptive fields and flexible attention beyond local neighborhoods. Early adaptations introduced degree-aware normalization and positional encodings [79, 29, 17], while recent designs add neighborhood- and label-enhanced signals, feature-difference encodings, and heterophily-aware attention biases [75, 33, 82]. Evidence that plain transformers can be strong learners with structural priors motivates hybrids combining attention and propagation for expressivity and stability [47, 71]. 2.5 Scalability, efficiency and pretraining at scale Scaling to large graphs relies on algorithmic and system-level optimizations. Mixed precision and checkpointing enable deeper models under resource limits [51, 5], while noise masking, tensor decompositions, and randomized sparse computations reduce per-iteration cost [38, 54, 43]. Linear-time architectures and system-aware designs further cut overheads [83, 81]. Lightweight pretraining via self-supervised clustering improves downstream accuracy without heavy supervision [30, 86], and alternatives to backpropagation offer hardware-friendly training [88]. 2.6 Domain applications, evaluation and relation to prior work Specialized frameworks target biomolecular forecasting, financial anomaly detection, and neuroimaging analysis [37, 70, 10]. Standardized benchmarks like OGB reveal performance variation across structural regimes [24, 41], while comparative studies contextualize gains and identify where architecture, pretraining, or augmentation drive improvements [12, 60, 34, 25, 15]. 2.7 Positioning and relation to prior work Our method integrates multi-scale structural encodings, contrastive alignment, adversarial augmentation, and heterophily-aware transformers. These components have been shown to enhance robustness and generalization [78, 44, 69, 6, 33, 7]. Unlike prior work that treats these elements separately, we unify them in a pipeline that adaptively modulates propagation and enforces consistency under topology perturbations. Empirical evaluation spans benchmarks with varying homophily and scale, comparing against lightweight baselines and recent transformer-based graph learners [40, 26, 47, 38]. Figure 1: Overview of the AdvSynGNN framework for structure-adaptive graph learning. The pipeline begins with Multi-scale Feature Synthesis, which generates node embeddings XMSX_MS by aggregating local and multi-hop contextual signals. In the core processing stage, we employ Contrastive Representation Alignment to stabilize embeddings via a self-supervised loss ℒsslL_ssl. Simultaneously, an Adversarial Synthesis module, consisting of a GAN-based Generative Adversary and a Structural Discriminator, proposes heterophily-oriented edge flips to produce a perturbed adjacency A~′ A . These signals feed into the Adaptive Residual Correction engine, where label estimates are refined through confidence-weighted propagation using per-node calibration cic_i (or αi _i) to mitigate structural noise. The refined representations are then processed by a Heterophily-Adaptive Graph Transformer that incorporates a learned structural attention bias ϕij _ij to differentiate between compatible and noisy neighbors. Finally, the Robust Diffusion module computes a steady-state prediction Z(∞)Z^(∞), which is integrated via Prediction Fusion and a lightweight Ensemble to produce the final resilient node labels YfinalY_final. Shaded blocks indicate modules that are jointly optimized during the end-to-end training phase. 3 Methodology In the methodology section, we outline the training and inference procedure for AdvSynGNN, which is summarized in Algorithm 1. This process is followed by a detailed theoretical analysis of the convergence properties and robustness of our approach, with key insights from the analysis of spectral norm bounds discussed in Appendix D and the justification for adversarial perturbations as a sensitivity control and regularizer, which is elaborated in Appendix E. 3.1 Problem formalization and objectives We consider the learning task over an attributed graph structure where structural noise and class heterophily may coexist. Let the topological domain be represented as =(,ℰ)G=(V,E), where V denotes the set of N discrete nodes and ℰE represents the observed connectivity. The global state of the graph is characterized by a feature arrangement X∈ℝN×dfX ^N× d_f and an associated binary adjacency matrix A∈0,1N×NA∈\0,1\^N× N. In our setting, the observed adjacency A is treated as a potentially perturbed instantiation of the underlying latent manifold. To facilitate stable message passing, we derive the symmetric normalized Laplacian proxy as A~=D−1/2(A+I)D−1/2 A=D^-1/2(A+I)D^-1/2 (1) where D signifies the degree diagonal matrix such that Dii=∑j(Aij+Iij)D_i= _j(A_ij+I_ij), and I denotes the identity matrix representing self-loops. The node set is partitioned into a labeled subset ℒV_L and an unlabeled subset V_U. The supervision signal is provided as a label matrix Y∈0,1N×CY∈\0,1\^N× C for C distinct categories. Our primary objective is to optimize a robust mapping function ℱ:X,→Y^F:\X,G\→ Y that minimizes the empirical risk over ℒV_L while maintaining structural resilience. This is achieved by generating class-probability estimates Z=σ(Φ(X,A~′;Θ))Z=σ ( (X, A ; ) ) (2) where Φ represents the integrated AdvSynGNN encoder parameterized by Θ , A~′ A denotes the adversarially rectified adjacency, and σ corresponds to the softmax activation for categorical distribution. The framework specifically targets the recovery of the true posterior P(Y|,X)P(Y|G,X) under conditions where the homophily ratio h=1||∑i∈|j∈i:yi=yj||i|h= 1|V| _i |\j _i:y_i=y_j\||N_i| (3) is significantly low, necessitating a mechanism that can adaptively transition between smoothing and filtering operations. Here, iN_i represents the local neighborhood of node i as defined by the graph topology. 3.2 Integrated graph-learning architecture We design a single, end-to-end framework composed of four tightly coupled modules that jointly produce robust node representations and resilient label estimates under structural noise: multi-resolution feature synthesis, contrastive representation alignment, confidence-driven residual correction, and topology-adaptive transformation. The input graph and primitive data are written as =(,ℰ),N=||,X∈ℝN×df,A∈0,1N×N,A~=D−1/2AD−1/2,Y∈0,1N×C. splitG&=(V,E), N=|V|,\\ X& ^N× d_f, A∈\0,1\^N× N,\\ A&=D^-1/2AD^-1/2, Y∈\0,1\^N× C. split (4) where G denotes the input graph with node set V and edge set ℰE, N is the number of nodes, X is the node-feature matrix with feature dimension dfd_f, A is the binary adjacency, D=diag(A)D=diag(A1) is the degree diagonal matrix, A~ A is the symmetric degree-normalized adjacency used throughout the propagation modules, and Y is the one-hot (or multi-hot) label matrix with C classes. 3.3 Feature synthesis pipeline We first form compact node descriptors by absorbing edge-level signals and assembling multi-hop contextual embeddings. Edge-to-node aggregation is implemented as i _i =1|i|∑j∈i(Weij+be), = 1|N_i| _j _i (W_ee_ij+b_e ), (5) i _i =GeLU(ℳ(i)). =GeLU (M(v_i) ). (6) where iN_i denotes the neighbourhood of node i, |i||N_i| its cardinality, ij∈ℝdee_ij ^d_e are optional edge features, We∈ℝdh×deW_e ^d_h× d_e and be∈ℝdhb_e ^d_h are learnable parameters that map edge descriptors into a hidden space of dimension dhd_h, ℳ(⋅)M(·) is a missing-value handling / masking operator, and GeLU(⋅)GeLU(·) denotes the Gaussian Error Linear Unit activation. To capture local and longer-range topology we construct multi-scale structural embeddings by repeated normalized propagation and concatenation: X(k) X^(k) =A~kX,k∈0,1,…,K, = A^\,kX, k∈\0,1,…,K\, (7) XMS X_MS =[X(0)‖X(1)‖⋯∥X(K)]. = [\,X^(0)\,\|\,X^(1)\,\|\,·s\|\,X^(K)\, ]. (8) where A~k A^k denotes k-step propagation under the symmetric normalized adjacency, K is the maximal propagation depth, and ∥\| denotes column-wise concatenation that yields the multi-resolution representation XMSX_MS used by downstream modules. 3.4 Contrastive representation alignment We regularize encoder outputs via a normalized contrastive objective that encourages stability across randomized augmentations: ℒssl=−1N∑i=1Nlog⁡exp⁡(s(i,i′)/τ)∑j=1Nexp⁡(s(i,j′)/τ).L_ssl=- 1N _i=1^N (s(h_i,h_i )/τ ) _j=1^N (s(h_i,h_j )/τ ). (9) where ih_i denotes the encoded representation for node i and i′h_i is an independently sampled augmentation of the same node, s(⋅,⋅)s(·,·) is cosine similarity, and τ>0τ>0 is a temperature hyperparameter that controls the sharpness of the induced distribution; in practice we use a modest number of non-correlated negatives per anchor to stabilize optimization. 3.5 Adaptive residual correction We refine label estimates by propagating label residuals in a node-adaptive manner and then re-integrating scaled corrections. The initial residual and the confidence-weighted propagation rule are R(0) R^(0) =Z(0)−Yobs, =Z^(0)-Y_obs, (10) Ri(t+1) R_i^(t+1) =(1−ci)Ri(0)+ci(A~R(t))i, =(1-c_i)\,R_i^(0)+c_i\, ( AR^(t) )_i, (11) ci c_i =σ(c⊤[i∥1|i|∑j∈ij]+bc). =σ\! (w_c [x_i\| 1|N_i| _j _ix_j ]+b_c ). (12) where Z(0)Z^(0) denotes initial soft predictions with observed labels filled and unlabeled entries zero-padded, YobsY_obs contains available labels and zero for missing entries, R(t)∈ℝN×CR^(t) ^N× C is the residual matrix at iteration t and Ri(t)R_i^(t) denotes its i-th row, ci∈(0,1)c_i∈(0,1) is a learnable per-node confidence produced by a sigmoid σ(⋅)σ(·), and c,bcw_c,b_c parameterize the confidence estimator. After T propagation steps we normalize residual magnitudes using the labeled set and re-integrate the scaled corrections: snorm s_norm =1|ℒ|∑j∈ℒ‖Rj(0)‖1, = 1|L| _j \|R_j^(0)\|_1, (13) Zi(r) Z_i^(r) ←Zi(0)+snormRi(T)max⁡(ε,‖Ri(T)‖1). ← Z_i^(0)+s_norm R_i^(T) ( ,\|R_i^(T)\|_1). where ℒL indexes labeled nodes, ∥⋅∥1\|·\|_1 denotes the element-wise ℓ1 _1 norm, and ε>0 >0 is a small regularizer to avoid division by zero; this normalization preserves directionality of residual corrections while aligning magnitudes to a stable labeled-set reference. In practice ‖A~‖2>1\| A\|_2>1 frequently arises on heterophilous graphs; we enforce κ<1κ<1 via spectral-clipping and confidence-ceiling with negligible accuracy loss. 3.6 Heterophily-adaptive attention To accommodate dissimilar neighbors we augment multi-head attention with an explicit learned structural bias. Head-specific projections and attention logits are computed as i(k) _i^(k) =Linearq(k)(i),j(k)=Lineark(k)(j), =Linear_q^(k)(x_i), _j^(k)=Linear_k^(k)(x_j), (14) ψij(k) _ij^(k) =i(k)⊤j(k)dh+⊤MLP([i∥j]), = q_i^(k) k_j^(k) d_h+w MLP ([x_i\|x_j] ), (15) ωij(k) _ij^(k) =exp⁡(ψij(k))∑l∈iexp⁡(ψil(k)), = ( _ij^(k)) _l _i ( _il^(k)), (16) i′ _i =∥k=1H(∑j∈iωij(k)v(k)j). = \|_k=1^H ( _j _i _ij^(k)W_v^(k)x_j ). (17) where Linear(⋅)(k)Linear_(·)^(k) are head-specific linear maps, dhd_h is the per-head dimension, w parameterizes an MLP-based structural bias acting on concatenated features [i∥j][x_i\|x_j], H denotes the number of heads, v(k)W_v^(k) are value projection matrices, and ∥\| denotes concatenation over heads. 3.7 Adversarial propagation with generative networks An adversarial generator synthesizes plausible edge flips while a discriminator penalizes unrealistic global modifications. The generator outputs edge flip probabilities and the perturbed soft-adjacency is formed as Pij=σ(MLP([i‖j‖ij])),P_ij=σ (MLP([x_i\|x_j\|e_ij]) ), (18) where Pij∈[0,1]P_ij∈[0,1] denotes the flip probability for the candidate pair (i,j)(i,j) and ije_ij are optional edge features. The discriminator is a degree-normalized message-passing network with layerwise updates i(ℓ+1)=ReLU(∑j∈(i)Dii−1/2Djj−1/2|(i)|(ℓ)j(ℓ)),h_i^( +1)=ReLU ( _j (i) D_i^-1/2D_j^-1/2 |N(i)|W^( )h_j^( ) ), (19) where (ℓ)W^( ) are learnable layer weights and a permutation-invariant readout maps node embeddings to a scalar authenticity score. Given generator probabilities, the soft perturbed adjacency used for downstream attention and diffusion is A~ij′=Aij⋅(1−Pij)+(1−Aij)⋅Pij, A_ij =A_ij·(1-P_ij)+(1-A_ij)· P_ij, (20) where A~′ A denotes the perturbed soft adjacency that is optionally re-normalized to preserve degree-normalization properties. Z(t+1)=clip[0,1]((1−γ)Z(r)+γA~′Z(t))Z^(t+1)=clip_[0,1] ((1-γ)\,Z^(r)+γ\, A \,Z^(t) ) (21) where Z(t)∈ℝN×CZ^(t) ^N× C denotes class-probability predictions at diffusion iteration t, Z(r)Z^(r) is the residual-reintegrated prediction matrix produced by the adaptive residual correction module, A~′ A is the (possibly adversarially perturbed) soft normalized adjacency used for diffusion, γ∈[0,1]γ∈[0,1] controls diffusion strength, and clip[0,1](⋅)clip_[0,1](·) enforces valid probability outputs elementwise. In practice we run this iteration for a fixed number of steps or until the change ‖Z(t+1)−Z(t)‖F\|Z^(t+1)-Z^(t)\|_F falls below a small tolerance, producing the diffusion steady-state Z(∞)Z^(∞) used in fusion. Adversarial training uses a Wasserstein objective with gradient penalty to stabilise optimization and includes engineering constraints to prevent excessive perturbation and to preserve contractivity where required. The perturbed adjacency replaces A~ A in residual propagation, attention computations, and diffusion steps during training so that the model learns to be robust to plausible structural changes. 3.8 Prediction fusion and ensemble We fuse the heterophily-adaptive attention outputs with diffusion-corrected predictions and allow a lightweight ensemble over complementary predictors: Y Y =ρ⋅σ(Y¯)+(1−ρ)Z(∞), =ρ·σ( Y)+(1-ρ)\,Z^(∞), (22) Yfinal Y_final =∑k=13κkℱk(X,A~′),∑k=13κk=1. = _k=1^3 _k\,F_k(X, A ), _k=1^3 _k=1. (23) where Y¯ Y is the structure-aware output from the attention module, Z(∞)Z^(∞) denotes the diffusion steady-state obtained from iterative application of the robust diffusion operator, ρ∈[0,1]ρ∈[0,1] balances the two streams, ℱk\F_k\ are complementary predictors, κk≥0 _k≥ 0 are mixing coefficients summing to unity, and A~′ A denotes the (possibly adversarially perturbed) adjacency used at inference time. 3.9 Temporal dynamic adaptation For evolving graphs we let temporal signals modulate confidence and attention. A snapshot-aware confidence scalar is defined by ci(τ)=σ(c⊤[i‖AGG(jj∈i)‖Δτi]+bc),c_i^(τ)=σ\! (w_c [x_i\|AGG(\x_j\_j _i)\| _i ]+b_c ), (24) where τ indexes the snapshot, Δτi _i denotes a compact temporal descriptor for node i (for example the time since last update), and AGG(⋅)AGG(·) denotes a neighbourhood aggregator. Temporal proximity is incorporated into attention logits by adding a learned temporal kernel term: ψij(k)←ψij(k)+⊤tanh⁡([i‖j‖gθ(|τi−τj|)]), _ij^(k)← _ij^(k)+v \! (W[x_i\|x_j\|g_θ(| _i- _j|)] ), (25) where gθ(⋅)g_θ(·) parameterizes temporal decay, and ,W,v are learned projections that allow the attention mechanism to prefer temporally proximate interactions when appropriate. 4 Experimental Evaluation Table 1: Benchmark evaluation of forecasting performance (Mean Absolute Error). Datasets are described in the text (ECG: physiological time series; Traffic: traffic flow; Motor: industrial sensor series). Dataset Length TimeGAN SigCWGAN GMMN RCGAN GAT-GAN AdvSynGNN ECG 16 0.061 0.053 0.058 0.058 0.060 0.055 64 0.121 0.148 0.149 0.151 0.049 0.044 128 0.152 0.147 0.148 0.154 0.048 0.042 256 0.154 0.167 0.156 0.168 0.047 0.040 Traffic 16 0.027 0.034 0.020 0.027 0.030 0.025 64 0.141 0.107 0.130 0.136 0.017 0.014 128 0.140 0.118 0.124 0.149 0.016 0.013 256 0.134 0.109 0.180 0.129 0.004 0.003 Motor 16 0.354 0.385 0.339 0.347 0.161 0.148 64 0.157 0.497 0.140 0.147 0.127 0.118 128 0.686 0.741 0.536 0.510 0.135 0.124 256 0.492 0.712 0.473 0.493 0.133 0.122 4.1 Experimental framework We evaluate AdvSynGNN on diverse benchmarks including citation networks, e-commerce, protein interactions, co-authorship graphs, and molecular collections: OGBN-ArXiv (169169K nodes, homophily 0.650.65) [24], OGBN-Products (2.42.4M nodes) [24], OGBN-Proteins (132132K nodes, multi-label) [24], DBLP (≈105≈ 10^5–10610^6 edges), and PCQM4Mv2 (millions of molecular graphs) [23]. These datasets span homophilous and heterophilous regimes, enabling comprehensive robustness assessment. For comparison, we include state-of-the-art baselines: GraphGAN-style generative models [66], GCN [28], transformer-based architectures (GraphGPS, Graphormer) [55, 76], and hybrid GAN–GNN variants. All methods use identical splits and comparable hyperparameter budgets for fairness. In our experiments, we perform sensitivity analysis on key hyperparameters, as shown in Appendix J. We also evaluate the computational efficiency of AdvSynGNN, with results summarized in Appendix L, where we discuss parameter counts and runtime performance on large-scale benchmarks. Further, we present the analysis of negative sampling strategies in Appendix I, highlighting the impact of various sampling choices on node classification accuracy and robustness. 4.1.1 Forecasting datasets We also include a set of time-series forecasting benchmarks used in the GAN-based comparisons (ECG, Traffic and Motor). ECG comprises physiological heartbeat sequences sampled at multiple lengths [52]; Traffic refers to traffic-flow time-series commonly used in transport forecasting [11]; and Motor is an industrial sensor suite studied in prior forecasting evaluations [64]. These dataset descriptions are provided here for clarity; the MAE table below reports our measured errors for each sequence length without repeating dataset citations in the table body. 4.2 Quantitative assessment 4.2.1 Forecasting (Mean Absolute Error) Table 1 reports mean absolute error (MAE) on three forecasting datasets at multiple sequence lengths. The datasets are described in the preceding paragraph and the table presents raw MAE values for each evaluated method and horizon. The AdvSynGNN variant consistently achieves the lowest MAE across lengths, indicating that adversarial topology synthesis and confidence-driven refinement provide benefits that extend to temporally-structured prediction tasks. The evaluated baselines include TimeGAN [80], SigCWGAN [39], GMMN [35], RCGAN [1], and GAT-GAN [27]. 4.2.2 Node-level Classification and Graph-level Regression. We benchmark AdvSynGNN against twelve baselines on five datasets: four node-level classification tasks (accuracy) and PCQM4Mv2 for quantum-chemistry regression (MAE). Using identical splits and early-stop protocols, AdvSynGNN consistently achieves the best results, showing that adversarial confidence propagation mitigates label noise and spurious edges. On PCQM4Mv2, it surpasses transformer-based competitors, confirming the benefits of multi-scale embeddings and heterophily-aware attention. Table 2: Performance summary across five random seeds (mean ± std). Node classification accuracy (%, higher is better) and PCQM4Mv2 MAE (lower is better). Bold indicates the best result. Method ArXiv Products Proteins DBLP PCQM4Mv2 (MAE↓ ) GCN[28] 71.74±0.2171.74± 0.21 83.90±0.1883.90± 0.18 72.51±0.3172.51± 0.31 86.01±0.2286.01± 0.22 0.148±0.0030.148± 0.003 GraphGAN[66] 68.50±0.2868.50± 0.28 80.25±0.2480.25± 0.24 70.12±0.3570.12± 0.35 82.30±0.1982.30± 0.19 0.144±0.0040.144± 0.004 DnnGAN[89] 70.85±0.1970.85± 0.19 82.67±0.2082.67± 0.20 71.25±0.2771.25± 0.27 87.45±0.2387.45± 0.23 0.139±0.0030.139± 0.003 GMP-GL[77] 71.20±0.2271.20± 0.22 83.10±0.1783.10± 0.17 71.80±0.2971.80± 0.29 87.90±0.2187.90± 0.21 0.137±0.0050.137± 0.005 Att-GAN[63] 71.55±0.2571.55± 0.25 83.45±0.1983.45± 0.19 72.05±0.2672.05± 0.26 88.25±0.2088.25± 0.20 0.135±0.0040.135± 0.004 TenGAN[32] 71.90±0.2371.90± 0.23 83.80±0.1883.80± 0.18 72.30±0.2872.30± 0.28 89.50±0.2289.50± 0.22 0.133±0.0030.133± 0.003 GTGAN[62] 72.05±0.2072.05± 0.20 83.95±0.1683.95± 0.16 72.45±0.2472.45± 0.24 90.25±0.1990.25± 0.19 0.131±0.0040.131± 0.004 Graphormer[76] 72.27±0.1872.27± 0.18 84.18±0.1584.18± 0.15 72.17±0.2372.17± 0.23 92.60±0.1792.60± 0.17 0.136±0.0030.136± 0.003 LargeGT[16] 72.35±0.2172.35± 0.21 79.81±0.2679.81± 0.26 72.25±0.2572.25± 0.25 91.85±0.1891.85± 0.18 0.134±0.0040.134± 0.004 SGFormer[73] 72.63±0.1772.63± 0.17 84.75±0.1484.75± 0.14 79.53±0.2079.53± 0.20 92.20±0.1692.20± 0.16 0.129±0.0030.129± 0.003 AdvSynGNN 75.48±0.1575.48± 0.15 89.31±0.1389.31± 0.13 86.40±0.1886.40± 0.18 94.86±0.1294.86± 0.12 0.108±0.0020.108± 0.002 4.2.3 Link prediction We evaluate AdvSynGNN on link prediction across four networks: arXiv-AstroPh, arXiv-GrQc, Wikipedia [50], and Amazon2M [9], using AUC as the metric. All methods share identical edge splits and early-stopping, with results averaged over five seeds. AdvSynGNN achieves the highest AUC on all tasks, confirming that adversarial perturbations enhance link recovery while preserving global structure. Table 3: Link prediction AUC (%, higher is better). Bold indicates the best result. All results are averaged over five independent runs with different random seeds. Method arXiv-AstroPh arXiv-GrQc Wikipedia Amazon2M GraphGAN [66] 85.5±0.3185.5± 0.31 84.9±0.2984.9± 0.29 81.3±0.3381.3± 0.33 78.50±0.4278.50± 0.42 DnnGAN [89] 96.0±0.1896.0± 0.18 95.0±0.2095.0± 0.20 99.0±0.0999.0± 0.09 80.25±0.3880.25± 0.38 GFformer [84] 94.2±0.2294.2± 0.22 93.5±0.2493.5± 0.24 98.1±0.1598.1± 0.15 85.12±0.3585.12± 0.35 VCR-GRAPHORMER [20] 94.8±0.1994.8± 0.19 94.0±0.2194.0± 0.21 98.5±0.1298.5± 0.12 76.09±0.4576.09± 0.45 SGFormer [73] 95.0±0.1795.0± 0.17 94.5±0.1994.5± 0.19 98.7±0.1198.7± 0.11 89.09±0.2889.09± 0.28 Proformer [42] 95.1±0.1695.1± 0.16 94.6±0.1894.6± 0.18 98.8±0.1098.8± 0.10 89.48±0.3089.48± 0.30 NodeFormer [72] 94.9±0.2094.9± 0.20 94.4±0.2294.4± 0.22 98.6±0.1398.6± 0.13 87.85±0.3287.85± 0.32 Graphformers [76] 94.7±0.2194.7± 0.21 94.2±0.2394.2± 0.23 98.4±0.1498.4± 0.14 85.90±0.3485.90± 0.34 STAR [2] 94.0±0.2494.0± 0.24 93.8±0.2593.8± 0.25 98.0±0.1698.0± 0.16 84.75±0.3784.75± 0.37 Ada-SAGN [46] 94.8±0.1994.8± 0.19 94.3±0.2094.3± 0.20 98.5±0.1298.5± 0.12 87.84±0.3187.84± 0.31 NTFormer [3] 93.0±0.2793.0± 0.27 92.5±0.2992.5± 0.29 97.5±0.1897.5± 0.18 78.03±0.4378.03± 0.43 AdvSynGNN 98.8±0.0998.8± 0.09 98.1±0.1198.1± 0.11 99.2±0.0799.2± 0.07 90.86±0.2590.86± 0.25 4.3 Unified Component and Robustness Ablation We quantify each module’s contribution and their synergy on three datasets. All results are averaged over 5 random seeds; p-values (paired t-test vs. Full) are reported in parentheses. Table 4: Ablation study on node classification accuracy (%, mean ± std) and robustness (ROC-AUC drop) under 5% hybrid perturbation on OGBN-Proteins. “w/o A+B” denotes simultaneous removal of modules A and B; Shapley values approximate marginal contribution on Proteins. Configuration Node Accuracy (%) ± std ΔAUC (p) Shapley ϕφ (%) ArXiv Proteins WikiCS Full AdvSynGNN 75.48 ± 0.15 86.40 ± 0.18 81.22 ± 0.21 0 – w/o GAN-only 73.65 ± 0.20 84.25 ± 0.22 78.30 ± 0.25 -2.07 24.7 w/o confidence-only 74.20 ± 0.19 84.91 ± 0.20 78.95 ± 0.23 -1.26 15.1 w/o multi-scale + bias 73.65 ± 0.20 84.12 ± 0.22 78.40 ± 0.24 -1.83 22.0 w/o GAN + w/o confidence 72.11 ± 0.23 82.93 ± 0.26 76.95 ± 0.28 -5.90 57.8 w/o GAN + multi-scale 72.90 ± 0.24 83.50 ± 0.25 77.60 ± 0.27 -4.15 19.8 w/o confidence + multi-scale 73.10 ± 0.22 83.70 ± 0.23 77.85 ± 0.26 -3.85 18.3 w/o GAN + confidence 72.11 ± 0.23 82.93 ± 0.26 76.95 ± 0.28 -5.90 38.7 Only GAN + confidence 74.12 ± 0.21 84.98 ± 0.20 78.85 ± 0.24 -1.42 – Only multi-scale + GAN 73.95 ± 0.22 84.75 ± 0.23 78.60 ± 0.25 -1.65 – Only confidence + multi-scale 74.50 ± 0.19 85.10 ± 0.21 79.00 ± 0.23 -1.30 – Only GAN 72.80 ± 0.24 83.40 ± 0.25 77.50 ± 0.27 -2.95 – Only confidence 73.20 ± 0.23 83.85 ± 0.24 77.90 ± 0.26 -2.55 – 4.4 Robustness analysis under structural perturbations To assess resilience, we subject OGBN-Proteins to systematic structural noise and measure relative performance loss. Specifically, we perform random edge deletions at rates of 5%5\% and 10%10\%, random edge additions at 5%5\% and 10%10\%, and a hybrid perturbation that simultaneously deletes and adds 5%5\% of edges. Table 5 reports relative ROC–AUC degradation for each perturbation type. We quantify perturbation impact using the relative change in AUC: ΔAUC=AUCperturbed−AUCcleanAUCclean×100%. = AUC_perturbed-AUC_cleanAUC_clean× 100\%. (26) where AUCperturbedAUC_perturbed is the area under the ROC curve after applying the structural modification and AUCcleanAUC_clean is the baseline value on the original graph. AdvSynGNN exhibits markedly smaller degradation than competing methods, with a maximum observed drop of approximately 2.05%2.05\% under hybrid perturbation, indicating strong robustness brought by adversarial propagation and confidence-weighted residuals. Table 5: Relative ROC–AUC degradation under structural perturbations on OGBN-Proteins Method 5% Del 10% Del 5% Add Hybrid GCN[28] -3.21 -6.74 -4.83 -7.95 GraphGAN[66] -5.47 -9.82 -7.16 -11.03 Graphormer[76] -2.78 -5.63 -3.95 -6.41 SGFormer[73] -1.95 -4.27 -2.86 -5.12 AdvSynGNN -0.82 -1.93 -1.14 -2.05 4.5 Temporal dynamics analysis We further evaluate incremental learning on temporal benchmarks drawn from the Temporal Graph Benchmark (TGB) Wikipedia revision history, processing monthly snapshots and measuring both final accuracy and the degree of catastrophic forgetting. We quantify knowledge retention by the metric =1||−1∑k=1||−1(Acc(||)−Acc(k)),K= 1|T|-1 _k=1^|T|-1 (Acc(A_|T|)-Acc(A_k) ), (27) where kA_k denotes the model performance evaluated at snapshot k and T is the set of snapshots; smaller (less negative) values of K indicate better retention. Table 6 presents final accuracy, K and a simple parameter-stability measure computed as the expected parameter change across adjacent snapshots. AdvSynGNN attains the highest final accuracy and the smallest forgetting measure, demonstrating that chronological attention modulation and recency-sensitive confidence weighting effectively capture and preserve evolving relationships. Table 6: Incremental learning performance on TGB–Wikipedia Method Final Accuracy (%) K (%) Parameter Stability TGAT[74] 88.14 -12.67 0.318 TGN [56] 89.51 -9.24 0.285 APAN[68] 90.74 -7.85 0.241 AdvSynGNN 95.42 -2.31 0.127 Finally, theoretical stability for temporal propagation is enforced by maintaining a contraction bound across snapshots: supτmaxi⁡ci(τ)⋅‖A~τ‖2<1, _τ _ic_i^(τ)·\| A_τ\|_2<1, (28) where τ indexes temporal snapshots, ci(τ)c_i^(τ) is the snapshot-wise confidence scalar for node i, and ‖A~τ‖2\| A_τ\|_2 denotes the spectral norm of the normalized adjacency at time τ. Satisfying this inequality ensures that per-snapshot residual operators remain contractive and iterative refinement converges. 5 Conclusion We presents AdvSynGNN, an integrated architectural paradigm that harmonizes multi-resolution structural synthesis and contrastive learning objectives with a heterophily-adaptive transformer and adversarial propagation. By incorporating a node-specific confidence-weighted residual correction mechanism, the proposed framework successfully addresses the inherent trade-offs between expressive capacity, structural resilience, and computational overhead. Comprehensive empirical evaluations confirm that our approach yields superior predictive performance and embedding invariance across a diverse spectrum of graph topologies. Systematic ablation analyses further elucidate how adversarial regularization and adaptive gating collectively suppress error propagation while enhancing global stability. 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Wang, C. Lang, Y. Jin, Y. Li, and H. Ling (2024) DFA-gnn: forward learning of graph neural networks by direct feedback alignment. Advances in Neural Information Processing Systems 37, p. 59289–59313. Cited by: §2.5. [89] M. Zhao and Y. Zhang (2022) GAN-based deep neural networks for graph representation learning. Engineering Reports 4 (11), p. e12517. Cited by: Table 2, Table 3. [90] Y. Zheng, S. Luan, and L. Chen (2024) What is missing for graph homophily? disentangling graph homophily for graph neural networks. Advances in Neural Information Processing Systems 37, p. 68406–68452. Cited by: §1. Appendix A AdvSynGNN algorithm The full training and inference procedure of AdvSynGNN is summarized in Algorithm 1. Algorithm 1 AdvSynGNN: Unified Training, Spectral Clipping, and Inference 1:Graph =(,ℰ)G=(V,E), features X, observed labels YℒY_L, epochs E, residual steps T, GAN critic steps ncn_c, spectral tolerance ϵ>0ε>0, confidence ceiling c¯∈(0,1) c∈(0,1) 2:Predictions YfinalY_final 3:Initialize encoder, generator, and discriminator networks. 4:Set Z(0)←PadLabels(Yℒ)Z^(0) (Y_L). 5:Compute multi-scale features XMSX_MS via Eq. (7)–(8). 6:for epoch←1epoch← 1 to E do 7: Forward encoder for node embeddings h and compute ℒsslL_ssl via Eq. (9). 8: Compute per-node confidences c via Eq. (12) and initial residual R(0)R^(0) via Eq. (10). 9: ⊳ Spectral clipping and confidence ceiling 10: ν←PowerIter(A~)ν ( A) 11: A~←A~⋅min⁡(1,(ν+ϵ)−1) A← A· (1,(ν+ε)^-1) 12: c←min⁡(c,c¯)c← (c, c) ⊳ Element-wise clipping 13: for t←0t← 0 to T−1T-1 do 14: Update residuals R(t+1)R^(t+1) via Eq. (11). 15: end for 16: Re-integrate residuals to obtain Z(r)Z^(r) via Eq. (13). 17: ⊳ Adversarial perturbation learning 18: for k←1k← 1 to ncn_c do 19: Update discriminator using WGAN-GP objective. 20: end for 21: Update generator using adversarial and regularization losses. 22: Build perturbed adjacency A~′ A via Eq. (20) and renormalize. 23: Compute attention outputs Y¯ Y via Eq. (17) and diffusion Z(∞)Z^(∞) via Eq. (21). 24: Fuse predictions via Eq. (22)–(23) and compute total loss. 25: Update all modules via gradient-based optimization. 26:end for 27:Inference: Run encoder and ensemble outputs via A~ A or Monte-Carlo perturbations. 28:return YfinalY_final Subroutine: Power iteration (spectral norm estimate). Use a few iterations (e.g., 10) of power iteration to estimate ν≈‖A~‖2ν≈\| A\|_2. This estimate is used only for light-weight preprocessing and diagnostics; it need not be exact. Appendix B Theoretical analysis and proofs This appendix establishes a convergence guarantee for the confidence-weighted residual propagation and describes sufficient spectral conditions that ensure stable temporal updates. The derivations are presented to be directly usable by implementers and to clarify the assumptions underlying the contraction arguments. B.1 Residual propagation: fixed point and contraction Theorem B.1 (Residual convergence). Let A~∈ℝN×N A ^N× N denote the symmetric degree-normalized adjacency matrix and let c∈(0,1)Nc∈(0,1)^N be the vector of per-node confidence scalars. Consider the affine iteration R(t+1)=(I−diag(c))R(0)+diag(c)A~R(t).R^(t+1)= (I-diag(c) )\,R^(0)+diag(c)\, A\,R^(t). (29) where R(t)∈ℝN×CR^(t) ^N× C denotes the residual matrix after t steps. If the spectral quantity κ=(maxi⁡ci)⋅‖A~‖2κ\;=\; ( _ic_i )· A _2 (30) satisfies κ<1κ<1, then the mapping induced by (29) is a contraction in the spectral norm and the iterates converge linearly to a unique fixed point R⋆R . where in (29) the matrix R(t)R^(t) has rows Ri(t)R_i^(t), the scalar cic_i denotes the i-th component of c, diag(c)diag(c) denotes the diagonal matrix with c on the diagonal, and ∥⋅∥2 · _2 denotes the spectral operator norm. Proof. Define the affine operator ℱ:ℝN×C→ℝN×CF:R^N× C ^N× C by ℱ(R)=(I−diag(c))R(0)+diag(c)A~R.F(R)= (I-diag(c) )R^(0)+diag(c)\, A\,R. (31) For any two matrices R and R′R we have ‖ℱ(R)−ℱ(R′)‖2 (R)-F(R ) _2 =‖diag(c)A~(R−R′)‖2 = (c)\, A\,(R-R ) _2 ≤‖diag(c)A~‖2⋅∥R−R′∥2. ≤ (c)\, A _2· R-R _2. (32) Using the submultiplicative property of the operator norm and the identity ∥diag(c)∥2=maxi⁡ci (c) _2= _ic_i we obtain ‖diag(c)A~‖2≤(maxi⁡ci)⋅‖A~‖2=κ. (c)\, A _2≤ ( _ic_i )· A _2=κ. (33) Since κ<1κ<1, the operator ℱF is a contraction in the spectral norm and Banach’s fixed point theorem implies the existence of a unique fixed point R⋆R and linear convergence of the iteration to R⋆R . The linear error bound follows directly from repeated application of (32). ∎ B.2 Temporal contraction condition A snapshot-wise contraction argument extends to time-varying graphs by applying the previous argument at each timestamp. For a snapshot at time τ let A~τ A_τ denote the symmetric degree-normalized adjacency and let c(τ)c^(τ) denote the corresponding confidence vector. If for every snapshot τ the scalar κτ=(maxi⁡ci(τ))⋅‖A~τ‖2 _τ\;=\; ( _ic_i^(τ) )· A_τ _2 (34) satisfies κτ<1 _τ<1, then the propagation operator for that snapshot is contractive and the snapshot iterates converge to a unique fixed point. where ci(τ)c_i^(τ) denotes the confidence for node i at snapshot τ and ∥A~τ∥2 A_τ _2 denotes the spectral norm of the snapshot adjacency. B.3 Practical remarks on spectral bounds Empirical graphs, and particularly those with heterophilous structure, can exhibit spectral radii greater than one after naive normalization. Practically useful safeguards include estimating an empirical upper bound on ∥A~∥2 A _2 prior to training, applying light spectral scaling to enforce ∥A~∥2≤1 A _2≤ 1 when necessary, and capping learned confidences via ci←min⁡(ci,c¯)c_i← (c_i, c) with a chosen ceiling c¯<1 c<1 to preserve contractivity. The spectral scaling can be implemented by a small number of power iterations to estimate the largest singular value followed by rescaling using its reciprocal plus a small tolerance. These operations are designed to be minimally invasive to the original topology while restoring the sufficient condition used in the convergence argument. B.4 Practical enforcement of the convergence condition Theorem B.1 ensures linear convergence when the scalar κ defined in (30) is less than one. Real-world graphs may violate this inequality. To guarantee contractivity while altering the operator minimally, we adopt a two-step procedure that we call spectral clipping and confidence ceiling. First, estimate the largest singular value ν≈∥A~∥2ν≈ A _2 by applying a small number of power iterations. Then rescale the normalized adjacency as A~←A~⋅min⁡(1,1ν+ϵ), A← A· \! (1,\, 1ν+ε ), (35) where ϵ>0ε>0 is a small tolerance such as ϵ=10−4ε=10^-4. After rescaling, apply a confidence ceiling by replacing each cic_i with ci′=min⁡(ci,c¯)c_i = (c_i, c) for a chosen c¯∈(0,1) c∈(0,1). where ν denotes the power-iteration estimate of the spectral norm of A~ A and ϵε is a numerical tolerance. Lemma B.2 (Spectral clipping with confidence ceiling). Let ϵ>0ε>0 be a positive tolerance and let c¯∈(0,1) c∈(0,1) be a chosen ceiling for node confidences. Define A~′=min⁡(1,1∥A~∥2+ϵ)A~,ci′=min⁡(ci,c¯). A \;=\; \! (1,\, 1 A _2+ε )\, A, c_i \;=\; (c_i, c). (36) Then the spectral norm satisfies ∥diag(c′)A~′∥2≤c¯ (c )\, A _2≤ c, and the propagation iteration using A~′ A and c′c is contractive with contraction factor at most c¯ c. where c′c denotes the vector of clipped confidences and diag(c′)diag(c ) the corresponding diagonal matrix. Proof. By construction ∥A~′∥2≤1 A _2≤ 1. The operator norm of the product satisfies ∥diag(c′)A~′∥2≤∥diag(c′)∥2⋅∥A~′∥2. (c )\, A _2≤ (c ) _2· A _2. Since ∥diag(c′)∥2=maxi⁡ci′≤c¯ (c ) _2= _ic_i ≤ c and ∥A~′∥2≤1 A _2≤ 1, the right-hand side is at most c¯ c. Choosing c¯<1 c<1 yields the desired contractivity bound. ∎ Empirical validation. We evaluate this safeguard on three heterophilous benchmarks: wikiCS, Chameleon, and Squirrel. Table 7 reports the estimated spectral norm ∥A~∥2 A _2, test accuracy, and the number of residual iterations required to reach a spectral-norm precision threshold. Without clipping some datasets exhibit κ>1κ>1 and the iteration diverges; after clipping the contraction factor satisfies κ≤c¯κ≤ c and convergence is obtained within a modest number of steps with negligible change in accuracy. Table 7: Impact of spectral clipping on heterophilous graphs. The column ∥A~∥2 A _2 reports the estimated spectral norm prior to clipping. The column ∥A~′∥2 A _2 reports the spectral norm after clipping. The quantity TconvT_conv denotes the number of residual steps required to reach the convergence tolerance. Dataset ∥A~∥2 A _2 Acc (%) ∥A~′∥2 A _2 Acc after clip (%) TconvT_conv wikiCS[50] 1.27 81.22 0.980 80.19 18 Chameleon[45] 1.43 76.73 0.980 76.70 22 Squirrel[45] 1.51 41.05 0.980 41.02 25 B.5 Remarks on directed or non-symmetric adjacency If a non-symmetric adjacency AnsA_ns is used in propagation then replace the spectral norm ∥A~∥2 A _2 by the largest singular value of the corresponding operator. This largest singular value may be estimated by applying power iteration to Ans⊤AnsA_ns A_ns. As an alternative, one may symmetrize the operator by using the normalized Laplacian or by forming (Ans+Ans⊤)/2(A_ns+A_ns )/2. Appendix C Convergence analysis under spectral clipping We analyse the convergence of the confidence-weighted residual propagation when the symmetrically normalized adjacency operator has spectral norm greater than unity, a regime frequently encountered in heterophilous graphs. We present the spectral-clipping construction, bound the encoder perturbation induced by clipping, derive a linear convergence rate, and relate the need for clipping to simple stochastic graph models. Let A~∈ℝn×n A ^n× n denote the symmetrically normalized adjacency matrix and suppose ∥A~∥2>1 A _2>1. We form the clipped operator A~′=1∥A~∥2+ϵA~, A \;=\; 1 A _2+ε\, A, (37) where ϵ>0ε>0 is a small scalar that prevents numerical instability. Here ∥⋅∥2 · _2 denotes the spectral (operator) norm. Under clipped propagation the residual iteration is written as R(t+1)=(I−diag(c′))R(0)+diag(c′)A~′R(t),R^(t+1)\;=\; (I-diag(c ) )\,R^(0)\;+\;diag(c )\, A \,R^(t), (38) where ci′=min⁡(ci,c¯)c _i= (c_i, c) for each node i and c¯∈(0,1) c∈(0,1) is the chosen confidence ceiling; diag(c′)diag(c ) denotes the diagonal matrix with entries ci′c _i. We now quantify the distortion introduced by spectral clipping in the encoder outputs. Let fθ(A~;X)∈ℝn×df_θ( A;X) ^n× d be the node embeddings produced by an encoder with parameters θ on input feature matrix X∈ℝn×dinX ^n× d_in. If the encoder is Lipschitz with respect to adjacency perturbations with constant Lf>0L_f>0, then ‖fθ(A~′;X)−fθ(A~;X)‖2≤Lf‖A~′−A~‖2∥X∥2=Lf(1−1∥A~∥2+ϵ)∥A~∥2∥X∥2, f_θ( A ;X)-f_θ( A;X) _2\;≤\;L_f\, A - A _2\, X _2\;=\;L_f (1- 1 A _2+ε ) A _2\, X _2, (39) where ∥X∥2 X _2 denotes the operator norm (largest singular value) of X; the final equality follows from definition (43). Define the clipped contraction factor κ′=c¯∥A~′∥2.κ \;=\; c\, A _2. (40) Because ∥A~′∥2≤1 A _2≤ 1 by construction and c¯<1 c<1 by choice, it holds that κ′<1κ <1. Let R⋆R denote the unique fixed point of the affine mapping in (38). Then the iterates enjoy the linear convergence guarantee ‖R(t)−R⋆‖2≤(κ′)t‖R(0)−R⋆‖2, R^(t)-R _2\;≤\;(κ )^t\, R^(0)-R _2, (41) where the matrix norm is the spectral norm. This inequality expresses geometric convergence with rate κ′κ . The clipping operation therefore trades representation distortion for convergence speed. Larger deviations of ∥A~∥2 A _2 from one yield smaller ∥A~′∥2 A _2, reducing κ′κ and accelerating convergence while increasing ∥A~′−A~∥2 A - A _2 and consequently the encoder distortion in (39). To connect the spectral behaviour with graph topology, consider a simple stochastic block model in which cross-community connection probability is php_h. Under mild technical assumptions on feature magnitudes, one may upper-bound the expected spectral norm of the normalized adjacency by [∥A~∥2]≤nph(1+cσX2)+(n1/4log⁡n),E [ A _2 ]\;≤\; n\,p_h (1+c\, _X^2 )\;+\;O (n^1/4 n ), (42) where n is the graph size, σX _X denotes the largest singular value of X, and c>0c>0 is a topology-dependent constant. This bound indicates that increasing cross-community connectivity php_h or feature diversity σX _X typically enlarges ∥A~∥2 A _2, motivating the use of spectral clipping in heterophilous regimes. Table 8: Convergence behaviour versus spectral contraction factor κ on heterophilous benchmarks. “Clip” denotes whether spectral clipping was applied. The quantity κ is estimated as ∥A~∥2⋅maxi⁡ci A _2· _ic_i. Divergence is declared when ∥R(t+1)−R(t)∥2>1 R^(t+1)-R^(t) _2>1 for t≥50t≥ 50. Dataset ∥A~∥2 A _2 maxi⁡ci _ic_i κ Clip Steps to conv. Acc. (%) wikiCS 1.27 0.95 1.21 No — 81.22 wikiCS 0.98 0.95 0.93 Yes 18 80.19 Chameleon 1.43 0.92 1.32 No — 76.73 Chameleon 0.98 0.92 0.90 Yes 22 76.70 Squirrel 1.51 0.94 1.42 No — 41.05 Squirrel 0.98 0.94 0.92 Yes 25 41.02 Table 8 supports the theoretical narrative: when κ>1κ>1 the unclipped iteration typically fails to converge, while spectral clipping restores contractivity and achieves convergence with negligible accuracy degradation. C.1 Theorem: Formal Statement of Spectral Stability Theorem C.1 (Convergence under Spectral Clipping). Consider a symmetrically normalized adjacency matrix A~∈ℝn×n A ^n× n where the spectral norm satisfies ∥A~∥2>1 A _2>1. Let ϵ>0ε>0 denote a small positive regularizer, and define the clipped adjacency operator A~′ A according to the following mapping: A~′=A~∥A~∥2+ϵ A = A A _2+ε (43) where ∥⋅∥2 · _2 denotes the spectral norm of the matrix. Furthermore, let c′∈(0,1)nc ∈(0,1)^n represent a vector of clipped node-level confidence scores, where each element is defined as ci′=min⁡(ci,c¯)c _i= (c_i, c) for a global ceiling c¯∈(0,1) c∈(0,1). Assuming the encoding function fθ(⋅;X)f_θ(·;X) maintains LfL_f-Lipschitz continuity with respect to its structural argument, the following properties are established: The modified residual transformation, as defined in the propagation module, constitutes a contraction mapping within the spectral domain. The associated contraction coefficient is given by κ′=c¯∥A~′∥2<1κ = c A _2<1. Under these conditions, the sequence of residual iterates is guaranteed to converge linearly toward a unique fixed point R⋆R at a geometric rate. The distortion introduced to the encoder by the clipping operation remains bounded by the structural perturbation magnitude. Specifically, the discrepancy in the feature space increases monotonically with the original spectral radius ∥A~∥2 A _2, while it vanishes as ∥A~∥2→1+ A _2→ 1^+ or as the regularization parameter ϵ→0+ε→ 0^+. C.2 Proof of Convergence and Stability Proof. To establish the contractive property of the clipped residual mapping, we first analyze the norm of the modified adjacency operator. By the construction of A~′ A in (43), the spectral norm is strictly bounded as follows: ∥A~′∥2=∥A~∥2∥A~∥2+ϵ<1 A _2= A _2 A _2+ε<1 (44) where ϵε ensures that the denominator strictly exceeds the numerator. We then examine the composite operator ′=diag(c′)A~′C =diag(c ) A , which governs the iterative update of the residual matrix. Applying the sub-multiplicative property of the spectral norm yields the following inequality: ∥diag(c′)A~′∥2≤∥diag(c′)∥2∥A~′∥2 (c ) A _2≤ (c ) _2 A _2 (45) where ∥diag(c′)∥2=maxi⁡ci′ (c ) _2= _ic _i corresponds to the maximum entry of the confidence vector. Given that ci′≤c¯c _i≤ c for all i, we obtain the contraction factor κ′≤c¯∥A~′∥2κ ≤ c A _2. Since both c¯<1 c<1 and ∥A~′∥2<1 A _2<1, it follows that κ′<1κ <1. Consequently, according to the Banach Fixed-Point Theorem, the affine transformation defined for the residuals possesses a unique stable equilibrium, and the convergence error at step t decays according to (κ′)t(κ )^t. Regarding the stability of the representations, the encoder distortion is evaluated by considering the perturbation in the adjacency matrix. The spectral difference between the original and clipped operators is expressed as: ∥A~′−A~∥2=‖A~∥A~∥2+ϵ−A~‖2=(1−1∥A~∥2+ϵ)∥A~∥2 A - A _2= \| A A _2+ε- A \|_2= (1- 1 A _2+ε ) A _2 (46) where the identity follows from the scaling of the spectral norm. By invoking the LfL_f-Lipschitz assumption of the encoder fθf_θ, the resulting change in the output feature distribution is bounded by Lf∥A~′−A~∥2L_f A - A _2. Furthermore, the concentration of the spectral norm under stochastic block models allows us to estimate the magnitude of ∥A~∥2 A _2 as a function of the graph density and feature amplification terms. This demonstrates that the clipping mechanism effectively regularizes the propagation dynamics while maintaining a controllable bound on the structural approximation error. ∎ Appendix D Analysis of spectral norm bounds under heterophily This section examines the condition ∥A∥2≤1 A _2≤ 1 used in the contraction arguments, where A∈ℝn×nA ^n× n denotes the adjacency matrix and n denotes the number of nodes. The spectral norm ∥A∥2 A _2 is the largest singular value of A. We analyze how heterophily, defined as the tendency for dissimilar nodes to connect, affects this assumption in realistic graph models. Consider a random-graph model (n,ph)G(n,p_h) in which edges preferentially connect nodes whose feature vectors differ by more than a threshold. Let X∈ℝn×dX ^n× d denote the node feature matrix and let σX _X denote its largest singular value. Under a simple probabilistic approximation the expected spectral norm of the adjacency admits the bound [∥A∥2]≤nph(1+cσX2)+(n1/4log⁡n),E [ A _2 ]≤ np_h (1+c\, _X^2 )\;+\;O (n^1/4 n ), (47) where php_h denotes the heterophily probability and c is a topology-dependent constant. where σX _X denotes the largest singular value of the feature matrix X, php_h denotes the probability of heterophilous edges under the ensemble, and the remainder term accounts for higher-order fluctuations. This bound implies two practical observations. First, in high-heterophily regimes the right-hand side can grow on the order of n n and therefore exceed unity for large graphs. Second, larger feature diversity, as measured by σX _X, amplifies the spectral norm. Empirical experiments on stochastic block model instances configured to be heterophilic confirm that a substantial fraction of sampled graphs with n in the thousands violate the condition ∥A∥2≤1 A _2≤ 1. These findings motivate the preprocessing prescriptions described previously, including spectral clipping, edge dropout, and degree-preserving renormalization. The ensemble bound is informative but it describes average-case behavior. For single-graph concentration bounds one may apply matrix concentration inequalities such as Tropp’s matrix Bernstein inequality to obtain tail bounds on ∥A−[A]∥2 A-E[A] _2 under explicit modeling assumptions. Figure 2: Comparative embedding shifts under hybrid perturbations: (a) Original graph (b) GCN embeddings (c) Graphormer embeddings (d) AdvSynGNN embeddings. Color intensity indicates ‖hi−hpert,i‖2\|h_i-h_pert,i\|_2 magnitude. D.1 Structural Interpretation of GAN-Generated Perturbations To enhance the explainability of our adversarial propagation mechanism, we analyze edge modifications induced by the generator network G. For a given perturbation level δ∈5%,10%,15%δ∈\5\%,10\%,15\%\, we sample modified edges ℰmod=(i,j):(z)ij>0.9E_mod=\(i,j):G(z)_ij>0.9\ and compute the following topological metrics: Degree Centrality Ratio: ξd _d =|(i,j)∈ℰmod:max⁡(deg⁡(i),deg⁡(j))>degmed||ℰmod| = | \(i,j) _mod: ( (i), (j))> _med \ ||E_mod| (48) Feature Divergence: ξf _f =1|ℰmod|∑(i,j)∈ℰmod‖xi−xj‖2 = 1|E_mod| _(i,j) _mod\|x_i-x_j\|_2 (49) Homophily Disruption: ξh _h =|(i,j)∈ℰmod:yi≠yj||ℰmod| = | \(i,j) _mod: y_i≠ y_j \ ||E_mod| (50) where degmed _med denotes the median node degree. Table 9 reveals consistent patterns across OGB-Proteins and DBLP datasets: Table 9: Edge modification characteristics (δ=10%δ=10\%) Dataset ξd _d ξf _f ξh _h OGB-Proteins 0.73 1.82 ± 0.31 0.86 DBLP 0.68 1.45 ± 0.28 0.79 Key observations reveal three consistent patterns. First, high-degree nodes are disproportionately targeted, with ξd>0.65 _d>0.65. Second, the modified edges tend to connect nodes with dissimilar features, as indicated by ξf>1.4 _f>1.4. Third, heterophilous connections are preferentially altered, with ξh>0.75 _h>0.75. These results demonstrate that G systematically focuses on structurally critical and semantically ambiguous links, which explains its effectiveness in improving model robustness. D.2 Qualitative analysis Visual inspection of the learned representations and attention patterns corroborates the quantitative findings. Figure 3 shows t-SNE projections of embeddings before and after GAN-enhanced training, where clusters become more coherent post-regularization. Figure 4 illustrates attention allocation on a heterophilous subgraph: AdvSynGNN allocates weights that better discriminate informative from noisy neighbors. The multi-panel visualization in Figure 5 further demonstrates how incremental structural perturbations impact the embedding geometry and how adversarial training stabilizes the latent layout. Figure 3: t-SNE visualization of GAN-enhanced embeddings Figure 4: Attention patterns on heterophilous subgraph Figure 5: Visualization of GAN-induced structural perturbations: original structure and embedding, three perturbation levels and corresponding perturbed embeddings Appendix E Theoretical justification: adversarial perturbations as sensitivity control and uniformity regularizer We provide a concise theoretical account that connects adversarial graph perturbations, as produced by a learned generator, to two mechanisms that improve robustness on low-homophily graphs. First, adversarial training enforces distributional robustness and thereby controls the encoder’s sensitivity to structural perturbations. Second, when the perturbation distribution has sufficiently high entropy, the training signal implicitly encourages representation uniformity across geometric directions. Together these effects reduce the model’s reliance on immediate-neighbor label agreement and improve generalization in heterophilous settings. E.1 Setup and notation Let G=(V,E,X)G=(V,E,X) denote an undirected attributed graph with n=|V|n=|V| nodes, adjacency A∈ℝn×nA ^n× n, and node attributes X∈ℝn×dinX ^n× d_in. Let A~ A denote the (possibly normalized) adjacency operator used by the encoder. Let fθ(A~;X)∈ℝn×df_θ( A;X) ^n× d be the encoder parameterized by θ that maps the graph to node representations, and let ℓ(h,y) (h,y) be the supervised loss for a node with representation h and label y. A generator produces randomized structural perturbations Δ whose realizations are additive operators on the adjacency, so that the perturbed operator is A~+Δ A+ . where A~ A is the encoder’s adjacency operator, Δ is a random perturbation produced by the generator, fθf_θ denotes the node encoder mapping, and ℓ denotes the supervised loss. E.2 Distributional robust objective We formalize adversarial training as minimizing a distributional worst-case risk over a generator-induced perturbation set U. The adversarial risk is ℛadv(θ)=(x,y)∼[supΔ∈ℓ(fθ(A~+Δ;x),y)].R_adv(θ)\;=\;E_(x,y) [ _ (f_θ( A+ ;x),y ) ]. (51) where D denotes the data distribution over node features and labels and U denotes the support of the generator’s perturbation distribution. E.3 Sensitivity control via Lipschitz continuity Assume the encoder is Lipschitz continuous with respect to the operator norm perturbation of A~ A. Specifically, suppose there exists L>0L>0 such that for any admissible perturbation Δ , ‖fθ(A~+Δ;x)−fθ(A~;x)‖2≤L∥Δ∥2. f_θ( A+ ;x)-f_θ( A;x) _2≤ L\, _2. (52) where ∥⋅∥2 · _2 denotes the spectral (operator) norm for matrices and the Euclidean norm for vectors, and L is the encoder Lipschitz constant with respect to adjacency perturbations. Assume furthermore that the scalar loss ℓ(h,y) (h,y) is Lipschitz in the representation h with constant Cℓ>0C_ >0. Then for any Δ∈ , ℓ(fθ(A~+Δ;x),y) (f_θ( A+ ;x),y ) ≤ℓ(fθ(A~;x),y)+Cℓ‖fθ(A~+Δ;x)−fθ(A~;x)‖2 ≤ (f_θ( A;x),y )+C_ \, f_θ( A+ ;x)-f_θ( A;x) _2 ≤ℓ(fθ(A~;x),y)+CℓL∥Δ∥2. ≤ (f_θ( A;x),y )+C_ \,L\, _2. (53) where CℓC_ denotes the loss Lipschitz constant with respect to the representation and L is defined in Equation (52). Taking the supremum over Δ∈ and the expectation over the data distribution yields the following upper bound: ℛadv(θ)≤ℛ(θ)+CℓLsupΔ∈∥Δ∥2,R_adv(θ) (θ)+C_ \,L\, _ _2, (54) where ℛ(θ):=(x,y)∼[ℓ(fθ(A~;x),y)]R(θ):=E_(x,y) [ (f_θ( A;x),y) ] (55) is the clean expected risk without perturbations. where ℛ(θ)R(θ) denotes the expected clean risk, CℓC_ is the loss Lipschitz constant, L is the encoder Lipschitz constant with respect to adjacency perturbations, and supΔ∈∥Δ∥2 _ _2 is the maximal operator-norm magnitude of admissible perturbations. Inequality (54) shows that minimizing the adversarial risk implicitly controls the encoder sensitivity measured by L and the perturbation budget. In particular, adversarial training imposes an effective regularizer that penalizes representations that change rapidly under small spectral perturbations. E.4 High-entropy perturbations and representation uniformity Beyond worst-case sensitivity control, the distributional shape of perturbations matters. Let PΔP_ denote the generator’s perturbation distribution and assume it satisfies a lower bound on its Shannon entropy: ℋ(PΔ)≥ℋ0>0,H (P_ ) _0>0, (56) where ℋ(⋅)H(·) denotes differential (or discrete) entropy as appropriate. High entropy implies that the generator explores many directions in the perturbation space rather than concentrating on a few modes. Heuristically, when PΔP_ has large entropy and typical perturbation magnitudes are small, the expected perturbation behaves like an approximately isotropic noise component in the effective subspace seen by the encoder. Under this isotropic approximation, adversarial training resembles adding a noise-based data augmentation that forces the encoder to distribute representations more evenly across directions, which we refer to as improving representation uniformity. Formally, consider the pairwise similarity measure S(θ):=v∼Vu∼V[s(hv(θ),hu(θ))],S(θ):=E_v VE_u V [s (h_v(θ),h_u(θ) ) ], (57) where hv(θ)h_v(θ) denotes the representation of node v under fθf_θ and s(⋅,⋅)s(·,·) is a bounded similarity kernel such as cosine similarity. where PΔP_ denotes the generator distribution, ℋ0H_0 is the lower bound on entropy, and s(⋅,⋅)s(·,·) is a similarity kernel used to measure representation concentration. Under mild regularity assumptions, increasing the diversity of perturbations reduces the expected pairwise similarity S(θ)S(θ) of learned representations, thereby increasing uniformity. This effect counteracts the tendency of models to collapse representations locally in response to strong local homophily signals and thus benefits heterophilous graphs where local neighbors are often semantically dissimilar. E.5 Theoretical Framework of Adversarial Regularization The following proposition formalizes the intuition regarding the dual regularizing effects of spectral adversarial perturbations. Proposition E.1 (Adversarial perturbations induce sensitivity control and uniformity regularization). Assume the encoding function fθf_θ and the associated loss ℓ satisfy the Lipschitz conditions defined in Equations (52) and (53) with constants L and CℓC_ , respectively. Let PΔP_ denote the distribution of the generator producing perturbations within a support U, where the spectral norm is bounded by supΔ∈∥Δ∥2≤ρ _ _2≤ρ and the distribution maintains an entropy lower bound ℋ(PΔ)≥ℋ0H(P_ ) _0. Under these conditions, the minimization of the adversarial objective in Equation (51) is equivalent to optimizing the standard risk augmented by two distinct regularizers: a sensitivity penalty bounded by CℓLρC_ Lρ and a uniformity-inducing term whose magnitude is a function of ℋ0H_0. In scenarios characterized by low node-label homophily, this adversarial framework mitigates the model’s dependence on local label consistency, thereby enhancing its generalization capability. E.6 Formal Proof of Proposition E.1 Proof. The proof proceeds by decomposing the adversarial risk into terms representing local stability and global distribution properties. First, we address the sensitivity component. Given the Lipschitz continuity of the encoder and the loss function, the discrepancy between the adversarial and clean risk is bounded by the product of the respective Lipschitz constants and the perturbation magnitude. This relationship is established as: ℛadv(θ)≤ℛclean(θ)+CℓLρ _adv(θ) _clean(θ)+C_ Lρ (58) where ℛadv(θ)R_adv(θ) represents the risk under adversarial conditions, ℛclean(θ)R_clean(θ) denotes the risk on unperturbed data, CℓC_ is the Lipschitz constant of the loss, L is the Lipschitz constant of the encoder, and ρ signifies the maximum spectral norm of the perturbation Δ . Consequently, minimizing the adversarial objective implicitly constrains L, ensuring that the latent representations remain stable under admissible structural fluctuations. Second, we consider the regularization effect stemming from the entropy of the generator. When the distribution PΔP_ maintains a high entropy ℋ0H_0, the expected representation across the perturbation ensemble acts as a diversity-promoting mechanism. By applying a first-order Taylor expansion to the encoder fθf_θ around the original adjacency A~ A, the variance of the perturbed embeddings is governed by the covariance of Δ . This is expressed as: Δ∼PΔ[∥fθ(A~+Δ)−fθ(A~)∥2]≈Tr(∇A~fθ⊤ΣΔ∇A~fθ) _ P_ [ f_θ( A+ )-f_θ( A) ^2 ] ( _ Af_θ _ _ Af_θ ) (59) where ΣΔ _ denotes the covariance matrix of the perturbation distribution and ∇A~fθ _ Af_θ represents the Jacobian of the encoder with respect to the graph structure. High entropy ℋ0H_0 implies that ΣΔ _ is high-rank and tends toward isotropy, forcing the encoder to distribute node embeddings more uniformly across the hypersphere. This dispersion effectively lowers the average pairwise similarity: pair=1n2∑i,jcos⁡(zi,zj) _pair= 1n^2 _i,j (z_i,z_j) (60) where ziz_i and zjz_j are the normalized embeddings of nodes i and j, and cos⁡(⋅,⋅) (·,·) is the cosine similarity. By reducing pairS_pair, the model prevents representation collapse into neighborhood-dictated clusters. The synthesis of these mechanisms demonstrates that adversarial training simultaneously enforces spectral stability and feature uniformity. These effects decouple the representations from an over-reliance on immediate neighbor labels, which is particularly beneficial for heterophilous graphs where neighboring nodes often belong to different classes. ∎ E.7 Practical Diagnostics The theoretical framework presented in Proposition (E.1) suggests three empirical metrics for validating the impact of adversarial training. The first metric involves calculating the empirical Lipschitz response through the following ratio: Γsens=∥fθ(A~+Δ)−fθ(A~)∥2∥Δ∥2 _sens= f_θ( A+ )-f_θ( A) _2 _2 (61) where Γsens _sens quantifies the sensitivity of the encoder to infinitesimal structural changes. A lower value indicates higher robustness. The second diagnostic evaluates global embedding uniformity via the kernel density or average similarity metrics. Finally, one must monitor the entropy of the generator distribution to verify that PΔP_ does not converge to a singular point, which would negate the uniformity benefits. The simultaneous observation of reduced Γsens _sens and enhanced uniformity provides robust evidence for the proposed theoretical account. Appendix F Causal interpretation of GAN-induced perturbations This section quantifies the causal contribution of GAN-synthesized edges to the out-of-distribution generalization performance of the full model. The analysis treats the retention of the GAN-generated edge set ℰGANE_GAN as a binary treatment and measures its necessity and sufficiency for achieving near-peak test AUC. The statistics are estimated from multiple experimental runs reported in the paper. Table 10: Causal effect of GAN perturbations. Probability of necessity (PN) and sufficiency (PS) are computed for retaining ℰGANE_GAN with respect to achieving test AUC within 1%1\% of the full model. Counterfactual ΔAUC is obtained by removing ℰGANE_GAN via an explicit do intervention. Higher PN/PS and more negative ΔAUC indicate stronger causal benefit. Bold marks the strongest effect per column. Dataset AdvSynGNN Ablation (w/o GAN) PN PS ΔAUC↓ PN PS ΔAUC↓ OGBN-ArXiv 0.910.91 0.880.88 −3.7%-3.7\% 0.52 0.49 −1.2%-1.2\% OGBN-Products 0.930.93 0.900.90 −4.1%-4.1\% 0.50 0.48 −1.0%-1.0\% GOOD-Motif[22] 0.950.95 0.920.92 −5.9%-5.9\% 0.55 0.53 −1.8%-1.8\% F.1 Estimands and computation The binary treatment variable T∈0,1T∈\0,1\ indicates whether GAN-generated edges ℰGANE_GAN are retained (T=1T=1) or removed (T=0T=0). The binary outcome Y∈0,1Y∈\0,1\ indicates whether the test AUC is “adequate”, defined as being within 1%1\% of the full-model AUC. Following the standard lower-bound estimators for PN and PS, we compute PN ≥max⁡(0,P(Y=0∣T=0)−P(Y=0∣T=1)), \;≥\; \! (0,\;P(Y=0 T=0)-P(Y=0 T=1) ), (62) PS ≥max⁡(0,P(Y=1∣T=1)−P(Y=1∣T=0)), \;≥\; \! (0,\;P(Y=1 T=1)-P(Y=1 T=0) ), (63) where P(⋅∣⋅)P(· ·) denotes conditional probability estimated empirically from repeated experimental splits. In Eq. (62) and Eq. (63), PN stands for probability of necessity and PS for probability of sufficiency. Probabilities are estimated using the empirical frequencies observed over the held-out runs and random splits described in the experimental protocol. The counterfactual effect ΔAUC is estimated via an explicit do-style intervention: we remove all edges in ℰGANE_GAN (i.e., set T=0T\!=\!0), keep the encoder weights frozen, and re-evaluate the test AUC. The reported ΔAUC in Table 10 is the difference (AUCdo(T=0)−AUCfull)(AUC_do(T=0)-AUC_full) expressed as a percentage point change, negative values indicate performance degradation under the removal intervention. F.2 Interpretation The estimates in Table 10 indicate that retaining GAN-induced, heterophily-oriented perturbations substantially increases the likelihood of achieving near-peak test AUC. High PN values show that in many observed runs the removal of ℰGANE_GAN is closely associated with a failure to reach full-model performance; high PS values indicate that keeping ℰGANE_GAN often suffices to recover near-peak accuracy. The counterfactual ΔAUC corroborates this: removing GAN edges produces larger negative drops than ablating other modules in isolation, which aligns with the unified ablation results reported earlier. The combination of high PN/PS and sizable negative counterfactual effects supports the interpretation that the GAN module is not merely a heuristic augmenter but contributes causally to generalization in the evaluated regimes. This causal statement complements the structural analysis in Appendix E where GAN modifications are shown to preferentially target heterophilous and feature-dissimilar connections. The two lines of evidence together suggest a mechanism in which the generator discovers and proposes topological adjustments that mitigate harmful local homophily bias while preserving or reinforcing signal-bearing long-range relationships. F.3 Practical notes on estimation All PN/PS lower bounds and counterfactual ΔAUC values were estimated from the same set of experimental runs used for the ablation and robustness studies. Probabilities were computed from empirical frequencies across ten random train/validation splits and five random seeds per split. The counterfactual evaluations re-used the frozen encoder to avoid confounding from re-training. For transparency, the experimental logs and the small script used to compute PN/PS and counterfactual effects are included in the supplementary material accompanying this submission. Appendix G GAN training stability analysis To validate the reliability of the adversarial propagation module, we analyze the GAN training dynamics across multiple diagnostic signals. Beyond the structural statistics of synthesized perturbations reported in the main text, we track three complementary quantities during training: the discriminator and generator loss traces under a Wasserstein objective with gradient penalty, the per-epoch ℓ2 _2 norm of gradients flowing into the final convolutional block of each network, and the entropy of the generator’s edge-flip distribution. Together, these diagnostics assess convergence behaviour, gradient stability, and perturbation diversity, and they help detect failure modes such as mode collapse or exploding gradients. Loss curves Figure 6 shows the smoothed Wasserstein adversarial losses (with gradient penalty) for the discriminator and the generator on OGBN-Proteins. The discriminator loss decreases progressively and reaches an approximate plateau after about eighty epochs, while the generator loss follows a complementary but bounded trend. The absence of large oscillations or abrupt spikes indicates a stable adversarial game under the chosen optimization schedule and regularizers. Gradient norms To quantify the smoothness of back-propagation, we compute the epoch-wise ℓ2 _2 norm of gradients with respect to the parameters of the last convolutional block in each network. Let gD(t)g_D^(t) and gG(t)g_G^(t) denote the ℓ2 _2 norms of these gradients at epoch t for the discriminator and generator respectively. Figure 7 reports the observed ranges 0.08≤gD(t),gG(t)≤0.420.08≤ g_D^(t),g_G^(t)≤ 0.42. These magnitudes remain well below a conservative clipping threshold of 1.01.0 that is commonly used in mixed-precision training, and their bounded variance across epochs supports the conclusion that gradient propagation is numerically stable. Perturbation diversity. We measure the diversity of generated topological perturbations by computing the average Bernoulli entropy of the generator’s flip probabilities over the candidate edge set. Let Pij∈[0,1]P_ij∈[0,1] denote the flip probability assigned by the generator to candidate pair (i,j)(i,j) and let EcandE_cand be the set of candidate pairs considered by the generator. We define the edge-flip entropy as ℋ=−1|Ecand|∑(i,j)∈Ecand[Pijlog⁡Pij+(1−Pij)log⁡(1−Pij)].H\;=\;- 1|E_cand| _(i,j)∈ E_cand [P_ij P_ij+(1-P_ij) (1-P_ij) ]. (64) where |Ecand||E_cand| denotes the cardinality of the candidate set and logarithms are taken in base e (natural units). Figure 8 plots ℋH across epochs. The observed entropy remains near 0.690.69–0.730.73 nats (approximately the values reported in the main experiments), which is substantially above a conservative collapse threshold near 0.30.3 nats; this indicates that the generator continues to explore a broad set of perturbations rather than repeatedly proposing the same sparse subset of edges. Figure 6: Wasserstein adversarial losses (smoothed with a 5-epoch moving average) for discriminator and generator during training on OGBN-Proteins. Figure 7: Per-epoch ℓ2 _2 gradient norms of the final convolutional block for discriminator and generator. The dashed horizontal line indicates a conservative clipping threshold of 1.01.0. Figure 8: Edge-flip entropy ℋH (Eq. 64) computed over the candidate edge set across training epochs. Higher values indicate richer perturbation diversity. Summary The stable adversarial loss trajectories, bounded gradient norms, and persistently high edge-flip entropy jointly demonstrate that the adversarial propagation module trains in a numerically stable, non-degenerate regime, supporting the robustness improvements reported in the paper. Appendix H Comparative Robustness Visualization We extend Figure 4 with baseline comparisons using embedding consistency metrics: Absolute Shift: ℛabs=‖H−Hpert‖F _abs=\|H-H_pert\|_F (65) Relative Consistency: ℛrel=1N∑i=1N‖hi−hpert,i‖2‖hi‖2 _rel= 1N _i=1^N \|h_i-h_pert,i\|_2\|h_i\|_2 (66) where H∈ℝN×dH ^N× d and HpertH_pert denote clean/perturbed embeddings. Quantitative results for OGBN-Proteins (δ=10%δ=10\% hybrid perturbation): Table 11: Embedding consistency metrics Method ℛabsR_abs ℛrelR_rel GCN[28] 27.34 0.38 ± 0.12 Graphormer[76] 19.67 0.27 ± 0.09 AdvSynGNN (w/o GAN) 15.02 0.21 ± 0.07 AdvSynGNN 8.91 0.13 ± 0.04 The integrated GAN module reduces embedding distortion by 40.7% compared to the ablated version, confirming that adversarial training preserves representational stability under structural noise. Figure 2 visually demonstrates tighter cluster preservation in AdvSynGNN, particularly for low-degree nodes (light blue regions). Appendix I Analysis of Negative Sampling Strategies This section provides a detailed account of the negative sampling procedure used in the contrastive alignment module and presents an empirical comparison of alternative strategies. The goal is to illustrate how different choices of negative samples influence both node classification accuracy and robustness under topological perturbations. For each anchor node, we construct a set of sixty-four negative examples drawn from three complementary sources designed to capture heterogeneous semantic relations. Half of the negatives are chosen uniformly from nodes that do not share an edge with the anchor, which maintains structural diversity while avoiding near-duplicate samples. A further portion is drawn from feature-level neighbors that exhibit high cosine similarity with the anchor but are known to possess different labels, which yields structure-aware negatives that are informative yet label-inconsistent. The remaining fraction consists of representations of unrelated anchors within the same batch. These serve as quasi-positive distractors that increase contrastive difficulty and encourage the model to learn sharper decision boundaries. All negative samples are strictly required to be non-adjacent to the anchor and semantically incompatible, ensuring that the contrastive objective is not contaminated by accidental positives. This hybrid scheme is particularly beneficial in heterophilous graphs, where local neighborhoods may not reliably reflect semantic proximity. Table 12: Comparison of negative sampling strategies on OGBN-Proteins under a 5% hybrid perturbation budget. Reported are node accuracy (%) and changes in AUC (in percentage points) relative to the mixed strategy. Negative Sampling Strategy Node Acc (%) Δ (p) Random negatives only 85.12±0.2385.12± 0.23 −1.88-1.88 Structure-aware negatives only 85.90±0.2185.90± 0.21 −1.21-1.21 Mixed (proposed) 86.40±0.1886.40± 0.18 0 Table 12 shows that the mixed design achieves the highest accuracy and robustness. Random-only negatives introduce substantial variability but lack semantic challenge, whereas structure-aware negatives alone tend to over-focus on a narrow subset of the feature space. Combining both types with a small fraction of within-batch distractors yields a more balanced distribution of negative samples and results in consistent performance gains across perturbed evaluation settings. I.1 Summary The hybrid negative sampling procedure improves contrastive discrimination by balancing structural diversity, feature-level difficulty, and batch-level variability. The empirical results confirm that this design enhances robustness without requiring additional architectural modifications. Appendix J Sensitivity analysis on key hyper-parameters We study the sensitivity of AdvSynGNN on OGBN-Proteins under a 5% hybrid perturbation budget. The hyper-parameter grid considered is δ∈0.05, 0.10, 0.20,T∈10, 20, 50,γ∈0.3, 0.5, 0.8,δ∈\0.05,\,0.10,\,0.20\, T∈\10,\,20,\,50\, γ∈\0.3,\,0.5,\,0.8\, (67) where δ denotes the generator perturbation strength, T denotes the number of residual propagation steps, and γ denotes the diffusion (residual mixing) strength. Table 13: Sensitivity analysis on key hyper-parameters (OGBN-Proteins, 5% hybrid perturbation). Results report mean ± standard deviation over five random seeds. ΔAUC is reported in percentage points relative to the default configuration δ=0.10,T=20,γ=0.5\δ=0.10,T=20,γ=0.5\. Configuration Node Acc (%) ΔAUC (p) Notes δ=0.05δ=0.05 (perturbation strength) 85.92±0.2085.92± 0.20 −1.42-1.42 Weaker perturbation; small drop in robustness. δ=0.10δ=0.10 (default) 86.40±0.1886.40± 0.18 0 Default setting; balances accuracy and robustness. δ=0.20δ=0.20 85.71±0.2285.71± 0.22 −1.03-1.03 Stronger perturbation; modest accuracy decrease. T=10T=10 (residual steps) 85.63±0.2485.63± 0.24 −1.55-1.55 Under-propagation; residuals not fully propagated. T=20T=20 (default) 86.40±0.1886.40± 0.18 0 Default setting; sufficient convergence. T=50T=50 86.38±0.1986.38± 0.19 −0.08-0.08 Marginal improvement; larger compute cost. γ=0.3γ=0.3 (diffusion strength) 85.90±0.2185.90± 0.21 −1.21-1.21 Diffusion too weak; under-correction. γ=0.5γ=0.5 (default) 86.40±0.1886.40± 0.18 0 Default setting; trade-off between correction and smoothness. γ=0.8γ=0.8 85.77±0.2385.77± 0.23 −0.97-0.97 Excessive diffusion; labels over-smoothed. As shown in Table 13, AdvSynGNN exhibits moderate sensitivity to the perturbation strength δ, the residual step count T, and the diffusion coefficient γ. The triplet δ=0.10,T=20,γ=0.5\δ=0.10,\;T=20,\;γ=0.5\ defines a stable operating region that achieves a favorable balance between clean accuracy and robustness to structural perturbations. Settings that are substantially smaller or larger than these defaults incur mild performance degradation, indicating that extensive hyper-parameter tuning is not necessary in practice; selecting values within the central range yields reliably robust behaviour with modest computational cost. Appendix K Per-Module Time Complexity Analysis We provide asymptotic time complexity for each AdvSynGNN component with respect to the number of nodes N, edges E, node feature dimension d, propagation iterations T, GAN critic steps K, number of contrastive negatives k, attention heads h, and per-head dimension dhd_h. In our experiments we use h=8h=8 and dh=64d_h=64. Table 14: Per-module time complexity in the sparse-graph regime. Module Time Complexity Dominant Operation Multi-scale feature synthesis (T(N+E)d)O\! (T(N+E)d ) sparse propagation with d-dim vectors Contrastive pretraining (Nkd)O\! (Nkd ) negative sampling and similarity computation GAN generator (K(N+E)dg)O\! (K(N+E)d_g ) small edge-MLP and sampling (generator dim dgd_g) GAN discriminator (K(N+E)d)O\! (K(N+E)d ) message passing on perturbed graphs Confidence estimator (Nd)O\! (Nd ) per-node MLP projections Residual propagation (T(N+E)d)O\! (T(N+E)d ) sparse mat-vec per iteration Heterophily transformer (Ehdh+Nhdh2)O\! (Ehd_h+Nhd_h^2 ) sparse attention + per-node projections Combining the dominant terms yields the following per-epoch training cost: epoch _epoch =((T+K)(N+E)d)+(Nkd)+(Ehdh+Nhdh2). =O\! ((T+K)(N+E)d )+O\! (Nkd )+O\! (Ehd_h+Nhd_h^2 ). (68) Here N denotes the number of nodes, E denotes the number of edges, d is the node feature dimension, T is the number of propagation iterations used across multi-scale and residual modules, K is the number of GAN critic steps per training iteration, k is the number of negative samples per node for contrastive pretraining, h is the number of attention heads, and dhd_h is the dimension of each attention head. In typical large-scale sparse-graph regimes where E=Θ(N)E= (N), the following approximation is often faithful in practice: epoch≈((T+K)(N+E)d).C_epoch \! ((T+K)(N+E)d ). (69) This approximation holds when the contrastive term NkdNkd and the transformer projection term Ehdh+Nhdh2Ehd_h+Nhd_h^2 are small relative to the propagation and GAN loop term (T+K)(N+E)d(T+K)(N+E)d. The memory footprint scales linearly with graph size: adjacency storage is (N+E)O(N+E) for sparse representations and feature/activation storage is (Nd)O(Nd). Table 15: Resource efficiency comparison on large-scale graph benchmarks Method Parameters Time/Epoch (s) Dataset Graphormer[76] 119.5M 563 OGB-Proteins GraphGPS[55] 138.1M 480 OGB-Proteins NodeFormer[72] 86.0M 5.37 ogbn-papers100M GraphGPS++[48] 138.5M 465 PCQM4Mv2 SGFormer[73] 113.6M 2.48 ogbn-papers100M RoofGAN[62] 127.3M 318 RoofNet AdvSynGNN (Ours) 110.2M 210 OGB-Proteins Table 16: Key hyperparameters for AdvSynGNN. All values apply to both node-level and graph-level tasks unless noted. Component Parameter Value GAN Training Generator learning rate 1e-4 Discriminator learning rate 1e-4 Critic steps per generator step (ncn_c) 5 Gradient penalty coefficient 10.0 Residual Propagation Max residual iteration steps (T) 20 Confidence ceiling (c¯ c) 0.98 Spectral clipping tolerance (ϵε) 1e-4 Contrastive Pretraining Temperature (τ) 0.3 Negative samples per anchor 64 Augmentation dropout rate 0.2 Projection head hidden dim 256 Attention Module Number of heads (H) 8 Per-head dimension (dhd_h) 64 Dropout rate 0.1 Training Setup Batch size 1024 Optimizer AdamW Weight decay 1e-5 Diffusion Diffusion strength (γ) 0.5 Max diffusion steps 50 K.1 Justification of the approximation We now justify the approximation in Equation (69) by comparing the magnitudes of the constituent terms under practical assumptions. Expanding Equation (68) gives epoch _epoch =(T+K)(N+E)d+Nkd+Ehdh+Nhdh2 =(T+K)(N+E)d+Nkd+Ehd_h+Nhd_h^2 =(T+K)(N+E)d⋅[1+Nkd(T+K)(N+E)d+Ehdh+Nhdh2(T+K)(N+E)d]. =(T+K)(N+E)d· [1+ Nkd(T+K)(N+E)d+ Ehd_h+Nhd_h^2(T+K)(N+E)d ]. (70) Assume a practical regime where the graph is sparse so E=Θ(N)E= (N), the per-node feature dimension d is comparable to or larger than per-head projections hdhhd_h, and T, K, k, and hdhhd_h are design constants chosen small in practice (for instance, T≤4T≤ 4 for multi-scale encoding, T≤20T≤ 20 for residual steps, K≤5K≤ 5 for GAN critic iterations, k small for contrastive learning, and h and dhd_h modest). Under these conditions, the fractional factors in the square brackets of Equation (70) remain bounded by small constants. Consequently the leading term (T+K)(N+E)d(T+K)(N+E)d dominates and the approximation in Equation (69) is justified for runtime bookkeeping in large sparse graphs. If one moves into non-sparse regimes (dense graphs) or chooses large head dimensions or full dense attention (so that EhdhEhd_h grows superlinearly), the transformer-related term EhdhEhd_h can dominate; in such cases Equation (68) should be used without approximation and attention should be re-engineered (for example, via sparse attention, locality restrictions, or low-rank projections) to restore tractability. K.2 Summary In implementations we observe that constant-factor engineering choices such as compact multi-scale encodings, low-dimensional generator projections (dgd_g), mixed precision training, and gradient checkpointing substantially reduce wall-clock time and memory while leaving asymptotic complexity unchanged. The expressions above provide transparent accounting for trade-offs between propagation depth (T), adversarial regularization effort (K), and transformer expressivity (h,dh)(h,d_h), enabling practitioners to tune components according to available compute and target graph regime. Appendix L Computational efficiency We report parameter counts and per-epoch runtimes on representative large-scale benchmarks in Table 15. AdvSynGNN attains a favorable trade-off between model capacity and throughput: by leveraging multi-hop embedding fusion and structure-aware attention we reduce dense parameter overhead while preserving or improving accuracy, yielding substantial runtime improvements relative to several transformer baselines. Appendix M Key Hyperparameters We summarize the key hyperparameters used in AdvSynGNN training and inference in Table 16. These values are fixed across all main experiments unless otherwise stated.

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