Structure-Feature Aligned Graph Learning via Alternating Constrained Optimization Chengcheng Yan ∗†, Qingsong Wang ‡
arXiv:2607.11577v1 [cs.LG] 13 Jul 2026
Abstract We introduce a constrained two-view framework for node prediction that aligns structureconditioned GNN embeddings with a structure-free feature prior learned by an anchor model. Conventional Graph Neural Networks (GNNs) couple feature transformation and neighborhood aggregation, which renders them vulnerable to topology noise and heterophilous connections. To decouple this dependency, our framework utilizes an independent anchor network to capture intrinsic attribute features via a self-supervised reconstruction objective. Furthermore, we propose a Channel-Split Adaptive Gated GNN (CSAG-GNN) that dynamically routes representations between global spectral smoothing and local spatial discrimination through a node-wise gating mechanism. We propose a stable cyclic alternating optimization strategy to solve the resulting coupled bi-level objective, preventing mutual representation drift during training. Empirical results on both homophilous and heterophilous benchmarks show balanced performance gains and structural robustness over competitive baselines.
Keywords: Graph Neural Network; Alternating Optimization; Feature-Structure Alignment; Topological Robustness; Representation Learning
1
Introduction
Graph Neural Networks (GNNs) [4] have emerged as a powerful paradigm for graph representation learning. Given the topological structure A and the node feature matrix X of a graph, the conventional training objective of GNNs is typically formulated as the following unconstrained optimization problem: min L(Ψ(A, X; θ), Y ), (1) θ
where Ψ(·; θ) denotes the neural network model for node-level prediction and θ = {W1 , W2 , . . . , WL } denotes all trainable parameters of the GNN. Symbol L denotes the task-specific loss function, and Y denotes the labels of nodes. Given an L-layer GNN, for problem (1), Ψ denotes the composition of L message-passing layers defined as follows: Ψ(A, X; θ) = HθL ,
Hθ0 = X,
Hθl = σ(AHθl−1 Wl ),
l = 1, . . . , L,
(2)
where σ is an activation function, Hθl denotes the hidden representation at the l-th layer, and Wl are the weight parameters of GNN. By adopting the aggregation operator σ(AHθl−1 Wl ), the model ∗ College of Artificial Intelligence, Shaoxing Institute of Technology, Shaoxing, 312000, China. Email: [email protected] † School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China. ‡ School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China. Email: [email protected]
1
�, �
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lθ
Gate Layer
Inference
Concat CSAG-GNN
Multiply
CSAG-Layer ���
Legend & Symbols Graph Node
1−
Step 2：Structure View
lθ
CSAG-GNN
Node Feature
task align
Node Relation Fixed Weight
Loss Function
lθ
Graph �, �
Feature-Structure Alignment
Cyclic Update Feature Encoder
Aggregation Adaptive Gate Sigmoid lθ
Middle Feature
Node Features �
Feature Decoder
rec
Step 1：Feature View
Figure 1: An overview of the proposed CSAG-GNN framework. The model decouples graph learning into a Feature View (Step 1) and a Structure View (Step 2) optimized via a cyclic alternating scheme. The feature encoder distills a structure-agnostic prior Fϕ , which acts as an alignment anchor (Lalign ) for the intermediate embeddings of the task-driven CSAG-GNN. The embedded CSAG-Layer dynamically balances dual-branch aggregations using an adaptive gating mechanism. performs spatial smoothing over node representations in the non-Euclidean graph domain. This coupling between neighborhood aggregation and feature transformation enables GNNs’ ability to learn stronger feature representations. However, the above procedure implicitly assumes that the observed topological structure A is reliable and semantically consistent with the node labels [19]. In practice, real-world graphs are often noisy, incomplete, and may even contain spurious or misleading edges [35]. Moreover, heterogeneous graphs violate many of the homogeneity assumptions underlying message-passing architectures. Therefore, structural aggregation may propagate erroneous signals across neighborhoods, thereby degrading representation quality and yielding unreliable predictions [29, 27]. Under these circumstances, the standard training paradigm fails to fully exploit the inherent discriminative information embedded in the original feature X. Several studies attempt to improve robustness through graph structure learning [12, 34], data augmentation [33], or regularization techniques [6, 3, 28]. However, these methods still optimize structure-based representations and remain inherently dependent on the given topology A. When the topology is unreliable or corrupted, the learned embedding vectors may still drift [24]. To mitigate this problem, we observed that node features X typically encode more abundant semantic information and are less sensitive to topological noise. This motivates us to construct an anchor model Fϕ from the independent observation dimension (i.e., feature view) that learns structure-agnostic representations via an auxiliary loss function Lrec , which is a self-supervised reconstruction loss for the encoder-decoder procedure. In this manner, Fϕ serves as a feature semantic prior, capturing the intrinsic attribute manifold of the graph nodes, for aligning representations from structure to feature view.
2
From the above discussions, the align style explicitly couples two complementary information sources, ensuring that the semantics distilled from X is injected into the task representation, while the task supervision prevents the feature prior from drifting away from the discriminative structure. In this work, we reformulate the training of GNN as a constrained optimization problem by introducing the consistency constraint: min Ltask (Ψ(A, X; θ), Y ) + Lrec (Fϕ (X)) θ,ϕ
s.t.
Hθl̄ = Fϕ ,
¯l = ⌈L/2⌉,
(3)
where ⌈·⌉ denotes the ceil operator. The consistency constraint encourages the middle-structured representation Hθl̄ produced by the GNN to be aligned with the feature representation Fϕ , which can be interpreted as imposing a regularization on the structural manifold induced by the feature manifold. This design avoids using the final-layer representation, which tends to fit the task-specific discriminative space and discards generalized structural semantics, thereby providing a more stable and structure-preserving alignment signal. An intuitive way to solve problem (3) is to incorporate the consistency constraint condition through a penalty term and perform joint end-to-end optimization with both the task loss and the reconstruction loss. However, under joint backpropagation, gradients from Ltask are propagated to ϕ through Lalign . When topological structure A contains substantial noise, these structural inaccuracies may be transferred into ϕ, which can weaken the effectiveness of the learned feature prior from the feature view. Therefore, we reinterpret the alignment constraint from a bi-level perspective: the anchor network learns an informative feature representation by optimizing Lrec , then the GNN aligns its structural representation to this feature prior while optimizing the main task. To this end, this can be interpreted as solving a relaxed bi-level problem: min Ltask Ψ(A, X; θ), Y + Lalign (Hθl̄ , Fϕ∗ ) θ (4) s.t. ϕ∗ ∈ arg min Lrec Fϕ (X) , ϕ
where the feature align loss function Lalign = D(Hθl̄ , Fϕ∗ ), here D(x, y) = ρ2 ∥x − y∥2 is a quadratic penalty function measuring the discrepancy between two representations. In practice, ϕ∗ is not solved exactly. Instead, we maintain an iterative estimate ϕk , which is updated via a few steps of stochastic gradient descent on Lrec and serves as an online approximation to ϕ∗ . The subsequent structure-view update then aligns Hθl̄ with the current anchor model Fϕk . Formally, under the approximation solving style, (4) naturally suggests an alternating optimization strategy. Specifically, as shown in Figure 1, we alternately update the feature-view parameters ϕ to strengthen a structure-free semantic prior, and the structure-view parameters θ to optimize the task objective while aligning to the current prior, to obtain more balanced task prediction results. In this paper, our main contributions are summarized as follows: • We propose a multi-view perspective for node prediction that explicitly bridges a structure encoder and a structure-free feature prior. Specifically, we introduce an anchor (feature-view) model that learns semantic representations from node attributes alone and aligns these prior informations with intermediate structural embeddings produced by the GNN, resulting in a constrained optimization method for graph learning. • To optimize the relaxed bi-level problem, we develop a cyclic alternating optimization scheme that updates the feature-view parameters to refine the semantic prior and updates the structureview parameters to minimize the downstream task loss while maintaining representation consistency. This block-wise update procedure mitigates mutual drift between the two views.
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• Extensive experiments on benchmarks spanning both homophilous and heterophilous graphs demonstrate that the proposed method improves node classification performance over baselines, validating its effectiveness across diverse structural regimes.
2
Related Work
2.1
Graph Neural Networks and Topological Robustness
GNNs are widely used for modeling non-Euclidean graph-structured data. Their core mechanism is message passing, where node representations are updated by aggregating information from neighboring nodes. In methods such as GCN [13] and GAT [21], the learned representations are closely coupled with the input adjacency matrix A, making model performance sensitive to the quality of graph topology. In practice, graph structure may contain noise arising from missing edges, spurious connections, or adversarial perturbations. Such imperfections can affect information propagation across layers and lead to over-smoothing [23]. To address these challenges, existing methods can be broadly categorized into Graph Structure Learning (GSL) [12] and robustness-oriented regularization approaches [3, 28]. GSL methods aim to refine or reconstruct the adjacency matrix during training, often by learning pairwise similarities or optimizing structural consistency with task objectives. Regularization-based approaches, on the other hand, introduce additional constraints or data augmentation strategies to improve robustness under structural perturbations. Despite these efforts, most approaches rely on joint optimization of model parameters and graph structure within a coupled framework. This coupling makes them sensitive to the initial graph, which may be unreliable. In cases of severely corrupted topology or high heterophily graphs [36], structure-dependent optimization may lead to suboptimal representations.
2.2
Multi-View Learning and Feature–Structure Alignment on Graphs
To overcome the limitations of relying solely on topology, multi-view learning and graph contrastive learning have been widely adopted. These methods typically treat the feature space and the structure space as parallel sources of information. For example, the Deep Graph Infomax (DGI) [22] method enhances model generalization by maximizing the feature information between structureaware node representations and representations derived from a pure-feature Multi-Layer Perceptron (MLP). Furthermore, related work in graph knowledge distillation includes GLNN [32], which transfers structural knowledge from a GNN teacher to an MLP-based student model, enabling inference without explicit graph structure. However, existing multi-view alignment strategies predominantly employ a joint optimization paradigm with multiple objectives, formulated as Ltotal = Ltask + λLalign . In scenarios where the graph topology is unreliable, this single-level joint optimization is prone to feature drift. Due to the lack of independent constraints on the feature view, clean node feature representations can become contaminated by erroneous gradients stemming from noisy topology, causing the two views to align within an incorrect semantic space [9].
2.3
Alternating Optimization in Neural Network Models
From the perspective of optimization theory, alternating optimization and bi-level optimization offer effective approaches for decoupling complex dependencies. In deep learning [15], alternating optimization is frequently employed to solve complex objective functions involving multiple sets of non-convex parameters, such as in meta-learning [11] and generative adversarial networks (GANs) [7]. Unlike traditional stochastic gradient descent (SGD) [20], which directly minimizes an unconstrained joint loss, alternating optimization [31] minimizes one set of variables while holding another 4
fixed. This effectively constrains the parameter search space and provides more stable convergence guarantees. While a limited number of studies in the field of graph learning [25] have explored alternating optimization to update graph structures and network parameters, few have applied this technique to the consistent alignment of feature and structural representations. The approximate bi-level solution framework proposed in (4) leverages alternating optimization to isolate gradient interference between Fϕ and Hθ . This block-coordinate update mechanism [1, 2] ensures that the feature prior model, while independently capturing the attribute manifold, serves as a stable anchor for structural view optimization, thereby theoretically preventing mutual drift during the representation alignment process.
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Methodology
3.1
Notations
Let G = (V, E) denote a graph, where V = {v1 , v2 , · · · , vn } is the set of n nodes and E is the set of edges. The topological structure of G is represented by an adjacency matrix A ∈ Rn×n . Specifically, Aij > 0 indicates the presence of an edge between nodes vi and vj , while Aij = 0 otherwise. We define the input node feature matrix as X ∈ Rn×d , where d is the dimension of the node attributes. Furthermore, let l ∈ {1, · · · , L} denote the layer index. The symbols Hgl ∈ Rn×dl and Hsl ∈ Rn×dl denote the latent representations on the l-th layer, extracted by the Graph Convolutional Network (GCN) branch and the GraphSAGE branch, respectively. Algorithm 1 Channel-Split Adaptive Gated GNN Framework 1: Input :
Adjacency matrix A, Node features X, Node label Y , Max iteration K, Hyperparameter ρ, and Latent representations Hθl̄ . 2: Initialize : Anchor network weights ϕ0 , CSAG-GNN weights θ 0 . 3: for k = 0, · · · , K − 1 do 4: Step 1: Update weights ϕ of anchor network 5: Compute the reconstruction loss to refine the feature prior: Lrec (Fϕk (X)) = ∥F(X; ϕk , ·) − X∥2 6:
Update ϕ by taking stochastic gradient descent steps: ϕk+1 ← ϕk − η∇ϕ Lrec (Fϕk (X))
7: 8: 9:
Step 2: Update weights θ of CSAG-GNN Freeze the updated anchor network Fϕk+1 . Compute the feature alignment constraint: Lalign (Hθl̄ , Fϕk+1 ) = D(Hθl̄ , Fϕk+1 ) =
10:
ρ l̄ ∥H − Fϕk+1 ∥2 2 θ
Update θ by minimizing loss via gradient descent: θk+1 ← arg min Ltask (Ψ(A, X; θ), Y ) + D(Hθl̄ , Fϕk+1 ) θ
11: end for 12: Output :
Optimal model weights θK , ϕK .
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3.2
Overview of CSAG-GNN Framework
In the above optimization framework, the original GNN is responsible for fitting the target labels, while the model F serves to guide the GNN using representations derived solely from the node features X. The overall update procedure is summarized in Algorithm 1.
3.3
Feature View: Anchor Network Fϕ
In our analysis, to generate a reliable anchor feature representation for Fϕ , inspired by the idea of the AutoEncoder method [10], we designed an AutoEncoder network F to obtain the final node feature. Therefore, the corresponding reconstruction loss has min ϕ,ϑ
∥F(X; ϕ, ϑ) − X∥2 ,
(5)
where F(ϕ) := Fϕ denotes the encoder and F (ϑ) := Fϑ denotes the decoder in the AutoEncoder network F. Under this formulation, from the feature perspective, the reconstruction loss encourages the encoder Fϕ to extract the intrinsic representation of the original graph nodes, thereby providing effective guidance for downstream task training. As the primary objective of an AutoEncoder is to reconstruct the input data, the reconstruction procedure consists of two stages: encoding and decoding. From Step 1 in Figure 1, the encoder learns to extract informative latent representations from the input features, whereas the decoder reconstructs the input from these representations. In our framework, we are primarily interested in the representations learned by the encoder. Therefore, when training the anchor network, we explicitly define the training objective of the encoder Fϕ as follows: Lrec (Fϕ (X)) = ∥F(X; ϕ, ·) − X∥2 .
(6)
3.4
Structure View: Adaptive Dual-Branch Aggregation for CSAG-Layer
3.4.1
Dual-Branch Aggregation
To simultaneously capture global spectral dependencies and local spatial context, since GCN excels at modeling smooth aggregation patterns, while SAGE offers more flexible neighborhood representations, a natural approach is to integrate both GCN and SAGE branches into a unified framework. To this end, we first consider a dual-branch architecture. In this framework, given the input middle feature Hθl , the output feature Hθl+1 ∈ Rn×dl+1 is concatenated from the two aforementioned features. As shown in Figure 1 and (7), two branch outputs with complementary operation: Ugl = GCN(A, Hθl ) ∈ Rn×dg ,
Usl = SAGE(A, Hθl ) ∈ Rn×ds ,
where dg + ds = dl+1 is controlled by the channel allocation ratio γ. Therefore, by joining the output feature Ugl from GCN layer and the output feature Usl from SAGE layer at l-th layer, the output feature Hθl+1 can be formulated Hθl+1 = Ugl ∥ Usl , dl+1 = dg = ds , (7) where the symbol ∥ denotes the channel-wise concatenation operation and the two branch outputs Ugl ∈ Rn×dg and Usl ∈ Rn×ds respectively. Symbol n denotes the total number of nodes, and d means the number of channel. Nevertheless, according to (7), if the inputs to the network layers are kept identical, concatenating the outputs of two layers along the channel dimension results in a doubled feature dimensionality, thereby increasing the parameter cost. To alleviate this issue, we let the output channels of the two layers are proportionally allocated, such that the output dimensionality matches that of the 6
original layer, while the number of parameters lies between the two individual layers. Based on this analysis, (7) can reformulated as Hθl+1 = Ugl ∥ Usl , dg = ⌊γ · dl ⌋, ds = dl − dg , (8) where the floor operation denoted as ⌊·⌋ and the channel allocation ratio γ ∈ (0, 1). By this style, compared to enforcing a single aggregation mechanism, we decompose the output representation space into two subspaces, allowing the model to allocate specific channel capacities to different topological views. 3.4.2
Dual Aggregation with Adaptive Gate Layer
Although the dual-branch style (8) successfully generates complementary representations (spectral smoothing via GCN and spatial preservation via SAGE), a simple concatenation provides no explicit mechanism to adaptively regulate their contributions. For example, in heterogeneous graphs, the preference over smoothing versus discrimination is node-dependent: nodes inside homophilous clusters often benefit from stronger smoothing to denoise, whereas boundary nodes or heterophilous neighborhoods require more discriminative, locality-preserving messages to avoid over-smoothing and neighbor contamination [18]. To resolve the above conflict, we need a layer-specific controller that dynamically modulates the two branches based on the current node representation, enabling soft routing between global smoothing and local discrimination. To do this, we introduce a lightweight node-wise gate αl ∈ (0, 1) computed from the layer input Hθl . Therefore, the Gate layer can be formulated as: αl = σ(Wα Hθl + bα ) ∈ Rn×1 , where σ(·) denotes the sigmoid function and Wα is a learnable weight vector of the Gate layer, bα means the bias. The gate introduces negligible overhead compared to the O(dl dl+1 ) parameters in graph convolutions. It can adaptively modulates the contributions of the two branches before fusion: Ũgl = αl ⊙ Ugl , Ũsl = (1 − αl ) ⊙ Usl , where ⊙ denotes the Hadamard product, implemented via the auto-broadcast mechanism. This design enables per-node soft routing between smooth spectral aggregation and flexible neighborhood modeling. We empirically validate the necessity of the gating mechanism in the ablation study. From the aforementioned discussions, the output Hθl+1 of the CSAG layer is merged by channelwise concatenation. Thus, for (8), which can be rewrite as: h i Hθl+1 = Ũgl ∥ Ũsl . (9) The gating coefficient α carries a clear physical interpretation: it quantifies the demand for global smoothing versus local preservation. A higher α indicates the node prioritizes the spectral filter (GCN), whereas a lower α shifts focus to the spatial neighbor aggregation (SAGE). Remarkably, this dynamic adaptability is achieved with negligible parameter overhead (only O(dl )).
3.5
Self-Consistent Cyclic Learning
As an inexact solution to the bi-level problem (4), we adopt the following approximate alternating update scheme, which is widely used in bi-level learning:
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Step 1: Update weights ϕ of anchor network We update the weight parameters ϕ of the anchor network by taking the stochastic gradient descent steps to minimize the auxiliary objective: ϕk+1 ← arg min Lrec (Fϕ (X)), ϕ
(10)
which can be viewed as approximating ϕ∗ in (4) by taking a few gradient steps on the reconstruction loss function Lrec at each iteration and provides a progressively refined feature prior, then updating θ with respect to both task loss and alignment to the current anchor. Step 2: Update weights θ of CSAG-GNN Given the updated frozen anchor network ϕ, we update the GNN parameters θ by minimizing the task loss regularized by the alignment constraint, which utilizes the fixed weight Anchor network Fϕ . Therefore, the arg min procedure can be formulated as θk+1 ← arg min Ltask (Ψ(A, X; θ), Y ) + D(Hθl̄ , Fϕk+1 ), θ
(11)
where Ltask denotes the cross-entropy classification loss, we define the distance metric function as the squared Euclidean distance: D(Hθl̄ , Fϕk+1 ) = ρ2 ∥Hθl̄ − Fϕk+1 ∥2 . Additionally, (11) can be solved by the classic gradient descent algorithm.
4
Experimental
4.1
Dataset
In our work, we evaluate our method on six widely-used node classification benchmarks: Cora, CiteSeer, CS, Wisconsin, Texas, and Cornell. They cover both homophilous and heterophilous settings, these datasets enable a systematic evaluation across diverse structural regimes. Cora and CiteSeer are citation networks where nodes denote publications and edges represent citations. Node features are sparse bag-of-words vectors, and labels correspond to research areas. Their strong homophily makes them standard benchmarks for evaluating neighborhood aggregation. CS is a co-authorship graph from the Microsoft Academic Graph, with authors as nodes and collaborations as edges. Its larger scale and higher density provide a more challenging testbed for representation learning. Wisconsin, Texas, and Cornell are heterophilous WebKB graphs in which nodes represent web pages and edges correspond to hyperlinks. The frequent connections between dissimilar nodes make them particularly challenging for conventional GNNs.
4.2
Experimental Setup
During the training phase, all learnable parameters are optimized by the Adam optimizer with an initial learning rate of 0.1. We fix the hidden representation dimension at 64 across all evaluations. Furthermore, for the specific hyperparameters introduced in our framework, the constraint coefficient ρ is set to 1 × 10−4 , and the update interval τ is defaulted to 1. All experiments are implemented in Python 3.8 using PyTorch 1.9 [16] and conducted on an Ubuntu 20.04.6 workstation equipped with a Hygon C86 7381 64-core CPU and 64 GB of memory.
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Table 1: Comparison experiment results for graph dataset: Node Classification Accuracy (%). The best and second-best results are in bold and underlined, respectively. Homophily
Methods SAGE [8] GCN [13] APPNP [14] GIN [26] ChebNet [5] GAT [21] CSAG-GNN
4.3
Heterophily
Average
Cora [30]
Citeseer [30]
CS [30]
Wisconsin [17]
Texas [17]
Cornell [17]
0.636±0.150 0.724±0.139 0.539±0.170 0.291±0.079 0.614±0.149 0.770±0.037 0.794±0.023
0.344±0.095 0.589±0.158 0.621±0.108 0.162±0.073 0.312±0.116 0.608±0.080 0.678±0.040
0.732±0.154 0.817±0.073 0.610±0.107 0.228±0.016 0.889±0.046 0.830±0.072 0.854±0.022
0.300±0.170 0.294±0.171 0.323±0.172 0.245±0.150 0.462±0.153 0.258±0.184 0.626±0.102
0.464±0.213 0.327±0.246 0.459±0.199 0.245±0.158 0.418±0.273 0.429±0.218 0.632±0.049
0.227±0.087 0.289±0.122 0.221±0.087 0.270±0.109 0.375±0.147 0.248±0.071 0.541±0.060
0.451 0.507 0.462 0.240 0.512 0.524 0.688
Results
Table 1 reports a comprehensive performance comparison between various GNN methods and the proposed CSAG-GNN model on node classification tasks. The evaluation is conducted on six publicly available benchmark datasets, including three homophily graphs (Cora, Citeseer, and CS) and three heterophily graphs (Wisconsin, Texas, and Cornell). Performance is measured in terms of the mean accuracy and standard deviation across multiple runs. The proposed CSAG-GNN achieved the highest average accuracy in the six datasets, outperforming all baseline models. On homophily datasets, CSAG-GNN shows balanced performance. Specifically, it achieves the highest accuracy on Cora and Citeseer, with scores of 0.794 and 0.678, respectively. On the CS dataset, CSAG-GNN obtains the second-best performance, achieving an accuracy of 0.854, slightly below ChebNet, which reaches 0.889. These results suggest that the proposed method can effectively leverage both local structural information and node attributes in assortative graphs, yielding competitive results compared with representative GNN models such as GCN and GAT. On heterophily graphs, traditional GNN models such as GCN, SAGE, and GIN often suffer from performance degradation when dealing with highly heterogeneous structures. This limitation is largely attributed to their message-passing design, which is based on the homophily assumption that connected nodes tend to share similar labels. As a result, their performance on the Wisconsin, Texas, and Cornell datasets is relatively limited, with accuracy generally ranging between 0.2 and 0.4. In contrast, CSAG-GNN achieves better improvements on these heterophily datasets. Specifically, it reaches accuracies of 0.626 on Wisconsin and 0.541 on Cornell, corresponding to absolute gains of 16.4 and 16.6 percentage points over the second-best model, ChebNet. These results indicate that CSAG-GNN can better adapt to heterogeneous connectivity styles compared with the baseline methods.
4.4
Ablation Analysis
To validate the effectiveness of the core components within the CSAG-GNN model and the mechanisms of their interaction, we conducted extensive ablation studies on six graph datasets. Table 2 presents a performance comparison between the full model and various variants on node classification tasks. The experiments primarily evaluated the individual contributions of the alignment loss function (Lalign ) and the gated fusion mechanism (Gate). (1) Contribution of the Alignment Loss Function Lalign The experiments first investigated the role of Lalign within a single graph encoder branch in Figure 1. The results demonstrate that the loss function exhibits highly complementary effects for different encoders:
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Table 2: Ablation experimental results for the graph dataset. Homophily
Method
Heterophily
Average
Cora
Citeseer
CS
Wisconsin
Texas
Cornell
GCN + Lalign (w/o SAGE) Improv. (%)
0.724±0.139 0.792±0.049 +9.39
0.589±0.158 0.661±0.053 +12.22
0.817±0.073 0.871±0.055 +6.61
0.294±0.171 0.231±0.167 -21.43
0.327±0.246 0.289±0.209 -11.62
0.289±0.122 0.308±0.129 +6.57
0.507 0.525 +3.68
SAGE + Lalign (w/o GCN) Improv. (%)
0.636±0.150 0.669±0.145 +5.19
0.344±0.095 0.356±0.118 +3.49
0.732±0.154 0.727±0.135 -0.68
0.300±0.170 0.331±0.177 +10.33
0.464±0.213 0.527±0.187 +13.58
0.227±0.087 0.260±0.091 +14.54
0.451 0.478 +6.18
w/o Lalign w/o Gate CSAG-GNN
0.422±0.154 0.681±0.140 0.794±0.023
0.299±0.135 0.434±0.160 0.678±0.040
0.349±0.103 0.867±0.017 0.854±0.022
0.176±0.159 0.402±0.154 0.626±0.102
0.318±0.217 0.441±0.214 0.632±0.049
0.356±0.124 0.284±0.142 0.541±0.060
0.320 0.518 0.688
For the GCN branch (w/o SAGE), introducing Lalign improves performance on homophilic graphs, including gains of 9.39% on Cora and 12.22% on Citeseer. However, it also increases the sensitivity of GCN to the homophily assumption, leading to performance drops on heterophily datasets such as Wisconsin and Texas, with decreases of 21.43% and 11.62%, respectively. For the SAGE branch (without GCN), adding Lalign improves performance on heterophily graphs, with an average gain of more than 12% on the three corresponding datasets. In contrast, the improvements on homophilic graphs are limited, and a slight decrease is observed on the CS dataset, with a reduction of 0.68%. When Lalign is removed entirely, the dual-branch model shows a substantial drop in average accuracy, decreasing to 0.320. This indicates that Lalign plays an important role in aligning representations learned by the two encoders within a shared feature space. (2) The Necessity of the Gating Mechanism The “w/o Gate” variant removes the dynamic gating mechanism and replaces it with a standard feature fusion strategy. As shown in Table 2, this leads to a reduction in average accuracy from 0.688 to 0.518. This suggests that simple operations such as concatenation or summation are not sufficient to effectively combine representations from the two branches in CSAG-Layer. In particular, the GCN and SAGE branches capture different aspects of the graph structure, and direct fusion may introduce interference from less reliable representations, especially under heterophily settings. By introducing the gating mechanism, CSAG-GNN learns adaptive weights for features from different branches conditioned on local node context, allowing a more flexible integration of information from both homophilic and heterophily patterns. (3) Stability of the Model Performance Beyond the gains in accuracy, the ablation studies also examine the stability of the proposed model. The results of standard deviation across runs show that CSAG-GNN generally has lower variance on most datasets, such as 0.023 on Cora and 0.049 on Texas. In comparison, removing key components leads to less stable performance, particularly for the variant without Lalign , which shows higher variance on Texas. These results suggest that the combination of the proposed modules helps improve not only performance but also consistency across different data splits and random initializations.
4.5
Analysis of Topological Perturbation Experiments
To evaluate model robustness under structural noise, we conduct edge perturbation experiments on both a homophilic dataset (Citeseer) and a heterophily dataset (Cornell). The graph topology 10
is modified by randomly adding or removing 10%, 30%, and 50% of edges. Tables 3 and 4 present the node classification results of CSAG-GNN and baseline models under edge addition and edge removal settings, respectively. Specifically, we utilized standard training procedures during the training phase, while in the testing phase, a proportion of edges is randomly added or removed to assess model robustness under topology perturbations. (1) Edge Addition Analysis Table 3: Edge addition perturb experimental results for the graph dataset on the three perturbation ratios. Citeseer
Methods APPNP ChebNet GAT GCN GIN SAGE CSAG-GNN
Cornell
10%
30%
50%
10%
30%
50%
0.521±0.100 0.288±0.089 0.568±0.057 0.566±0.147 0.170±0.081 0.313±0.097 0.629±0.036
0.436±0.083 0.293±0.059 0.500±0.048 0.524±0.124 0.158±0.066 0.300±0.082 0.601±0.034
0.356±0.069 0.289±0.075 0.474±0.048 0.484±0.115 0.148±0.056 0.307±0.069 0.573±0.036
0.224±0.086 0.427±0.172 0.267±0.097 0.300±0.129 0.291±0.118 0.251±0.051 0.459±0.092
0.240±0.092 0.410±0.161 0.291±0.142 0.291±0.130 0.294±0.121 0.316±0.116 0.464±0.053
0.251±0.109 0.386±0.136 0.237±0.089 0.286±0.131 0.289±0.118 0.273±0.103 0.451±0.065
In this experiment, adding random edges introduces spurious connections in the graph, which may cause GNN models to aggregate less relevant or noisy neighbor information during message passing and further affect performance. As shown in Table 3, models that rely more heavily on the homophily assumption tend to experience noticeable performance drops as the perturbation ratio increases from 10% to 50%. For example, on the Citeseer dataset, the accuracy of APPNP decreases from 0.521 to 0.356, while on the Cornell dataset, GAT and GCN maintain relatively low performance across all perturbation levels. In comparison, CSAG-GNN shows more stable performance under edge addition perturbations. On Citeseer, it achieves an accuracy of 0.573 even at a perturbation ratio of 50%, outperforming the other evaluated models. On the Cornell dataset, CSAG-GNN also maintains the best results across all settings, with accuracies of 0.459, 0.464, and 0.451 under perturbation ratios of 10%, 30%, and 50%, respectively. These results suggest that the proposed feature fusion and gating design can reduce the impact of noisy edges by limiting the influence of less reliable structural information during aggregation. (2) Edge Removal Analysis Table 4: Edge removal perturbation experimental results for the graph dataset on the three perturbation ratios. Citeseer
Methods APPNP ChebNet GAT GCN GIN SAGE CSAG-GNN
Cornell
10%
30%
50%
10%
30%
50%
0.567±0.138 0.310±0.087 0.608±0.078 0.585±0.162 0.160±0.072 0.331±0.097 0.670±0.037
0.610±0.101 0.317±0.097 0.617±0.065 0.580±0.151 0.167±0.085 0.343±0.100 0.659±0.045
0.572±0.140 0.276±0.097 0.595±0.058 0.570±0.149 0.150±0.064 0.327±0.101 0.650±0.035
0.262±0.076 0.340±0.138 0.254±0.093 0.329±0.113 0.289±0.115 0.256±0.098 0.497±0.082
0.262±0.076 0.418±0.176 0.294±0.110 0.302±0.130 0.282±0.115 0.227±0.064 0.462±0.077
0.281±0.114 0.386±0.139 0.229±0.068 0.300±0.129 0.283±0.110 0.251±0.055 0.467±0.106
11
Unlike edge addition, edge removal reduces graph connectivity and leads to the loss of structural information, which can hinder message passing and feature aggregation. As shown in Table 4, this setting has a negative effect on all baseline models. For instance, on the Citeseer dataset, when the edge removal ratio increases to 50%, the accuracy of GCN and GAT drops to 0.570 and 0.595, respectively. On the Cornell dataset, most traditional GNNs achieve accuracy below 0.350 under the same conditions. In comparison, CSAG-GNN shows more stable performance under edge removal. On Citeseer, it achieves an accuracy of 0.650 even when half of the edges are removed, outperforming all baseline models. On the Cornell dataset, CSAG-GNN also maintains relatively stable results, with an accuracy of 0.467 and a standard deviation of 0.106, and performs better than methods such as ChebNet under most settings. These results indicate that the proposed model is less sensitive to reduced connectivity and can better utilize the remaining structural and feature information in sparse graphs. The results in the two tables show that under stronger topological perturbations, such as 30% and 50% edge modification, the variance of models like GCN and APPNP often exceeds 0.1 on the Citeseer dataset. This suggests that their performance is more sensitive to randomness in the perturbation process. In contrast, CSAG-GNN maintains a lower standard deviation. This stability is mainly related to the dual-branch architecture and the gating mechanism. When the graph structure is heavily perturbed, the gating module adjusts the contribution of each branch and reduces the influence of less reliable structural signals, relying more on relatively stable node-level information. This helps maintain more consistent predictions under both edge addition and edge removal settings.
4.6
Hyperparameter Analysis
(1) Effect of the different hyperparameter ρ and τ To evaluate the sensitivity of CSAG-GNN to hyperparameters, we conduct experiments on six benchmark datasets. The analysis focuses on two parameters: the constraint hyperparameter ρ, which controls the relaxation strength of feature–structure alignment, and the update interval τ for alternating optimization between the encoder and decoder in the feature view. The results are shown in Figure 2, where the solid curves represent average classification accuracy under different settings, and the shaded regions indicate the corresponding standard deviations.