Learning Adaptive Distribution Alignment with Neural Characteristic Function for Graph Domain Adaptation
Authors: Wei Chen
Xingyu Guo
Shuang Li
Zhao Zhang
Yan Zhong
Fuzhen Zhuang
deqing wang
Venue: ICLR 2026 Poster (2026)
Keywords: Graph Domain Adaptation
Graph Neural Networks
Characteristic Function
Primary area: transfer learning, meta learning, and lifelong learning

Abstract: Graph Domain Adaptation (GDA) transfers knowledge from labeled source graphs to unlabeled target graphs but is challenged by complex, multi-faceted distributional shifts. Existing methods attempt to reduce distributional shifts by aligning manually selected graph elements (e.g., node attributes or structural statistics), which typically require manually designed graph filters to extract relevant features before alignment. However, such approaches are inflexible: they rely on scenario-specific heuristics, and struggle when dominant discrepancies vary across transfer scenarios.
To address these limitations,
we propose \textbf{ADAlign}, an Adaptive Distribution Alignment framework for GDA. Unlike heuristic methods, ADAlign requires no manual specification of alignment criteria. It automatically identifies the most relevant discrepancies in each transfer and aligns them jointly, capturing the interplay between attributes, structures, and their dependencies. This makes ADAlign flexible, scenario-aware, and robust to diverse and dynamically evolving shifts.
To enable this adaptivity, we introduce the Neural Spectral Discrepancy (NSD), a theoretically principled parametric distance that provides a unified view of cross-graph shifts. NSD leverages neural characteristic function in the spectral domain to encode feature-structure dependencies of all orders, while a learnable frequency sampler adaptively emphasizes the most informative spectral components for each task via minimax paradigm.
Extensive experiments on 10 datasets and 16 transfer tasks show that ADAlign not only outperforms state-of-the-art baselines but also achieves efficiency gains with lower memory usage and faster training.

Published as a conference paper at ICLR 2026
LEARNING ADAPTIVE DISTRIBUTION ALIGNMENT WITH NEURAL CHARACTERISTIC FUNCTION FOR GRAPH DOMAIN ADAPTATION
Wei Chen',
Xingyu Guo',
Shuang Li
Zhao Zhang?,
Yan Zhong',
Fuzhen Zhuang
Deqing Wang?
'School of Artificial Intelligence, Beihang University, Beijing. China
Schooo of Computer Science and Engineering, Beihang University, Beijing, China
Schooo of Mathematical Sciences, Peking University, Beijing, China
"Zhongguancun Laboratory, Beijng, China
(chenwei23, shuangliai, chuangfuzhen, @buaa edu, on
ABSTRACT
Graph Domain Adaptation (GDA) transfers knowledge from labeled source graphs unlabeled target graphs but is challenged by complex, multi-faceted dis- tributional shifts. Existing methods attempt to reduce distributional shifts by align- ing manually selected graph elements (e.g., node attributes or structural statistics), which typically require manually designed graph filters to extract relevant features before alignment. However, such upproaches are inflexible: they rely scenario- specific heuristics, and struggle when dominant discrepancies vary across trans- fer scenarios. To address these limitations, Distribution Alignment framework for GDA. Unlike heuristic methods, ADAlign requires no manual specification of alignment criteria. It automatically identifies the most relevant discrepancies in each transfer and aligns them jointly, capturing the interplay between attributes, structures, and their dependencies. This makes ADAlign flexible, scenario-aware and robust to diverse and dynamically evoly- ing shifts. To enable this adaptivity, we introduce the Neural Spectral Discrepancy (NSD), a theoretically principled parametric distance that provides unified view of cross-graph shifts. NSD leverages neural characteristic function in the spectral domain to encode feature-structure dependencies of all orders, while learnable frequency sampler adaptively emphasizes the most informative spectral compo- nents for each task via minimax paradigm. Extensive experiments on 10 datasets and 16 transfer tasks show that ADAlign not only outperforms state-of-the- baselines but also achieves efficiency gains with lower memory usage and faster training. Code is available at: https:/ http///tttttttttt thub. com/ gx ngyu/ ADA lign.
propose ADAlign, Adaptive
INTRODUCTION
Graph-structured data (Shao et al., 2024; Zhang et al., 2024; Chen et al., 2024; Yuan et al., 2025; Chen al., 2025b), characterized by intricate dependencies among nodes and edges, often under- goes substantial distribution shifts across domains (Li et al., 2022a; Guo et al., 2024). Such shifts break the inductive assumptions underlying conventional graph neural networks (GNNs) (Liu al., 2021), leading to severe performance degradation when key attributes or topological relations vary. To mitigate this problem, graph domain adaptation (GDA) (Wu et al., 2023; Liu et al., 2024b) has emerged as promising paradigm, aiming to learn domain-invariant representations that transfer knowledge from source to target domains and improve generalization across heterogeneous graphs.
In GDA, a popular line of work is cross- cross-graph feature alignment (Chen et al., 2019; You al., 2023; Liu et al., 2024b;a; Shao et al., 2024; Chen et al., 2025a), which directly minimizes distributional distances, such KL divergence, Maximum Mean Discrepancy (MMD), or Wasserstein distance
'Coreesponingg authors: Shuang Li and Fuzhen Zhuang.

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(Panaretos & Zemel, 2019), to reduce gaps between source and target graphs, offering both inter- pretability and empirical effectiveness. Early approaches typically employed a GNN to encode node attributes and structural features into unified embeddings, which were then aligned across domains using these distance metrics. However, alignment at this coarse global level often overlooks fine- grained distributional shifts, resulting in limited adaptation and increased risk of negative transfer.
Recently, several studies (Fang et al., 2025b;a; Yang et al., 2025) have decomposed graph distribution shifts into distinct components (e.g., node attributes, degree distribu- tions, or graph homophily) and designed specialized adaptation strategies for each, offer- ing a more fine-grained understanding of the factors hindering cross-domain generalization. Nevertheless, these approaches often rely on heuris- tic strategies that first manually design graph filters
Original Disparity
Heuristic Aligament
Adaptive Aligament
to extract relevant features (such as node attributes or structural statistics) before performing alignment for separable distributional shifts. This process over- looks a fundamental property of real-world graphs: the dominant sources of discrepancy vary across transfer scenarios, with their interplay being inher- ently complex and unpredictable. For example, shown in Figure I, we visualize the distributional discrepancies across the three Airport (USA, Brazil, Europe) (Ribeiro et al., 2017) transfer scenarios, fo- cusing on the top 5 features. We observe that the dominant sources of discrepancy vary across these scenarios. For instance, the B-E scenario, features 2 and 3 exhibit the largest discrepancies, whereas in the U-E scenario, the features with the most notable shifts simultaneously change to 1, 2, and 4. Ex- isting methods, however, often rely on fixed strate- gies that align only limited set of these features, which makes them inadequate for capturing the full range of distributional shifts and the interplay be- tween multiple factors. This underscores the need for adaptive approaches that can jointly capture and prioritize shifting dimensions, facilitating better alignment across transfer scenarios.
E
U
B
I
0
k 16
24
32
16
24
32
8
16
Divergenee
KT. Divennane
Divergence
Figure I: Distributional disparities across different scenarios. Each dimension (e.g.. 1, 2) corresponds to PCA-reduced feature, and the value represents the KL divergence between pairs the top features, computed for each transfer task, e.g., KL((B||U1).
To address these challenges, we introduce ADAlign, an Adaptive Distribution Alignment frame- work for GDA. Unlike heuristic methods that rely on predefined alignment criteria, ADAlign au- tomatically detects the most relevant sources of discrepancy in each transfer scenario and aligns them jointly, capturing the complex interplay between graph attributes, structures, and their depen- dencies. This makes ADAlign flexible and capable of handling wide range of dynamic shifts across domains. To enable this adaptivity, we propose the Neural Spectral Discrepancy (NSD), a novel parametric distance specifically designed for graphs. NSD uses neural characteristic function (Ushakov, 1999; Ghahramani, 2024) in the spectral domain to capture feature-structure dependen- cies at multiple levels. A learnable frequency sampler adaptively prioritizes the most informative spectral components for each task within a minimax optimization framework. This approach of- fers unified and tractable framework to quantify composite shifts, enabling alignment that is both theoretically grounded and adaptively sensitive to the nature of the transfer scenario.
Our contributions are threefold:
(i) We propose an adaptive framework that automatically identifies and aligns the most relevant sources of discrepancy in each transfer scenario, enabling flexible and task-aware adaptation to dynamic graph shifts.
(ii) We introduce NSD, novel parametric distance for graphs that leverages neural characteristic function in the spectral domain to capture multi-level feature-structure dependencies, providing a unified approach to quantifying distributional shifts.
(iii) Extensive experiments on 10 datasets and 16 transfer tasks show that ADAlign outperforms state-of-the-art baselines, significantly reducing memory consumption and training time.
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2 RELATED WORK
Graph Domain Adaptation. GDA (Zhuang et al., 2020; Li et al., 2025) aims to transfer knowledge from a labeled source graph to an unlabeled or sparsely labeled target graph, mitigating the chal- lenges introduced by distribution shifts in graph-structured data. Early approaches (Ma et al., 2019; Wu et al., 2020; Zhang et al., 2021; Dai et al., 2022; Pang et al., 2023; Liu et al., 2024a) typically employed graph neural networks (GNNs) encode node attributes and structural information into low-dimensional embeddings, and then aligned source and target representations using statistical discrepancy measures such as KL divergence (Menéndez al., 1997; Johnson et al., 2001). MMD (Arbel et al., 2019), or Wasserstein distance (Panaretos & Zemel, 2019). Although progressive, these methods struggle to capture fine- graiied coopett-speccfii differences, often leading to suboptimal alignment and degraded performance when target graphs exhibit complex distributional shifts. Sub- sequent studies (You et al., 2023; Liu et al., 2024b; Fang et al., 2025b;a) have dissected graph distribution shifts into factors such as attribute misalignment, degree divergence, and variations in homophily, inspiring targeted adaptation strategies tailored to specific types of shifts. Despite these advances, most existing methods still rely on heuristic metrics, which remain insufficient for mod- eling the entangled, multi-level nature of graph discrepancies. In this work, we propose an adaptive snrribution-alignmee. framework that holistically accounts for composite shifts.
Characteristic Function. Characteristic Function (CF) (Pólya et al., 1949; Yu, 2004; Ghahramani, 2024) establish one-to-one correspondence between probability distributions and their Fourier- domain representations (Ozaktas & Aytür, 1995), providing compact yet complete characterizations that have motivated applications across machine learning- They have been widely used in areas such as generative modeling (Li et al., 2017: 2020; 2022b; Lou et al., 2023), where they measure gaps between real and synthetic data, and knowledge distillation (Wang et al., 2025), where their compactness enables efficient transfer of knowledge from teacher to student models. Despite these advances, the use of CF in domain adaptation remains limited, especially for graph-structur data. Given the complexity of graph distributions, where attribute, structural, and topological shifts are often entangled, CF offer promising foundation for developing flexible adaptation frameworks.
3 PRELIMINARIES
Notation. We consider an undirected graph 9 EV,E,,, X.,, J), where V and & denote the sets of nodes and edges, respectively. The graph structure is encoded by the adjacency matrix A E IR NxN, where N = | is the number of nodes. Each entry Aij = indicates the presence of an edge between nodes and j. while Aug 0 indicates no edge. Each node E V is associated with feature vector, organized in the feature matrix X' E Nxd, where d is the feature dimension. The node labels are represented by y E NXC. with each node belonging to one of C' classes. Following prior works (Liu et al., 2024b; Huang et al., 2024; Fang et al., 2025a), we consider the unsupervised setting with two graphs: a source graph g$ = (VS,, gS AAS, 25, J////} and a target graph GT = (VT, E AAT J,, and focus the node classification task (Xiao et al., 2022), where the goal is to train a GNN classifier / that accurately predicts node labels y for the nodes in graph G
Characteristic Function. The CF (Waller et al., 1995) is fundamental tool in probability theory that represent a random variable in the frequency domain and provide a convenient way to analyze and compare distributions (Xie et al., 2022). For random vector X. the CF is defined as
[ [t =
/ ett a (Fx(x))
V (t)
where t E R. is the frequency vector, x E IR is a realization of X, Fx (æ) is the cumulative dis- tribution function (Drew et al., 2000), T denotes transpose operator and i i= V-I is the imaginary unit. To capture oscillatory strength and directional shifts, the CF can be expressed in polar form:
V xx (t) V (t) (1))
where [t)] denotes amplitude and (x) the phase. This amplitude-phas representation pro- vides intuitive insights into structural and geometric properties of distributions (Chen et al., 2021).
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Labeled source graph
Complex Plane
GCNs
Frequency Sampler
F(G
Classifier
Sowrce
Lulign
to +00
MLP
Σ
Unlabeled targel graph
E
Eq. (12) Eq. (13)
mint Lsource (8) falign
Minimax
max align (g G 8,
Optimization
(1I(
Figure 2: Overall architecture our proposed ADAlign Framework. The model consists of source and target branch, each processed by GNN. A frequency sampler adaptively selects spectral com- ponents via a learnable parameter Φ, while the classifier is optimized with source labels. The align- ment objective Calign is adversarially maximized with respect to & and minimized with respect the model parameters 6, enabling dynamic and efficient distribution matching in the spectral domain.
METHODOLOGY
In this section, we introduce the detailed workings of ADAlign. As illustrated in Figure 2, we first present the characteristic function transformation in $4.1, followed by NSD in $4.2. In $4.3, describe the adaptive frequency sampler for NSD, and finally, in $4.4 we introduce the minimax framework that integrates these components for effective graph domain adaptation.
4.1 CHARACTERISTIC FUNCTION TRANSFORMATION
ADAlign begins with learning node-level embeddings of the source and target graphs. Formally, we denote the source graph G {V'S E3 AA X , and target graph G {V', "T, AA", X"). We adopt the GNN encoder F (Liu et al., 2024a) to map each graph into node-level embeddings:
2 F(G)), z = F(g)),
where S E Nxd and TT= E Nx are the didimensiinal node embeddings of nodes the source and target graphs, respectively. Ns and Nr denote corresponding node numbers. These embeddings serve as compact representations that integrate both attribute and structural information.
Existing approaches (Zhuang al., 2020; Pang al., 2023; Yang al., 2025; Fang al., 2025b; Chen et al., 2025a) address domain shifts by minimizing statistical distances, e.g., MMD aligns low-order moments while KI. divergence matches full densities. Yet, on graph data these criteria often fail: moment matching ignores higher-order dependencies, and density-based measures are unstable high dimensions. To overcome this, we leverage characteristic function (Waller al., 1995), which provide complete and tractable representations in the Fourier domain. We extend them graphs by defining the characteristic functions of the source and target embeddings 2 and ZT:
S (t) = EZ-P(Z)) e
V T(()) = 1F====--P2: RZ-P(ZT)
(4)
where P(ZS) and P(ZT) denote the empirical distributions of node embeddings in the source and target graphs, respectively. Here, is frequency vector in the Fourier domain, /-T the imag- inary unit, and T denotes transposition. The key advantage of CF is that they uniquely characterize probability distributions, offering a complete representation that goes beyond low-order moments or density estimates. The following theorems (Wang al., 2025) formalize these properties:
Theorem (Convergence). (Lévy, 1959; Brézis & Lieb, 1983) Let {Z 80 be sequence of ran- dom vectors in & with characteristic functions V2 (t). If for every Rd the pointwise limit limg--oe zg (t) (t) exists and (t) is continuous at the origin, then the characteristic function of some random vector Z,, and have 7x Z as k Co.
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Theorem 2 (Uniqueness). (Feuerverger & Mureika, 1977; Marcus, 1981) Let IP and Q be nwo random vectors with characteristic function Vy and (t))). Wf()( = w We(t), vt E R then
the two random vectors P and Q have the same probability distribution.
4.2 NEURAL SPECTRAL DISCREPANCY
From the above analysis, probability distributions are uniquely determined by their CFs. This ob- servation suggests that distributional discrepancies in GDA can be naturally assessed in the spectral domain by comparing the CF of the source embeddings 2 and target embeddings z Motivated by prior works on CF-based discrepancies (Li et al., 2017; 2020; Ansari et al., 2020), we define the Neural Spectral Discrepancy (NSD) as a principled metric for gross-graph alignment:
[ JEET V ( (t) 2r ( V (t)(
ASD(Z$,2T)
(5)
where (t) and T Z (t) the CFs of the source and target embeddings, & denotes complex conjugation, and (t) is sampling distribution over the frequency domain T. The introduction of complex conjugate ensures the integrand is always real and non-negative (Wang al., 2025), since
l(t) v 2s (t) v 2 (t)) (t) (t) = 2 (t) (t) (t)(
(6)
To make this discrepancy more interpretable and amenable to optimization (Ansari et al., 2020), further exploit the polar decomposition of the expressed explicitly in terms of amplitude and phase differences:
valued CF in Eq. (2). This allows L(t) to be
E(t) = V 23 (t) + 1 2T (t)) us (t) v 2T (t)
2 () - 0r(())) +2 23
| ( [2)
2 (tss(tt)))) 0T())) (tr)))
(t) (1
(t) clscoos(ss 0T (t)),
amplitude difference
phuse difference
where (t( and (t(t) denote the phases of the CF, respectively. This decomposition highlights that NSD naturally separates graph-domain discrepancies into complementary components: amplitude term, which captures differences in the concentration of embeddings across frequency components (reflecting global structural variations), and phase term, which characterizes structural misalignments such as shifts or rotations in relational patterns between graphs. Such separation provides both interpretability and more effective basis for optimization. To explicitly balance these two aspects, we consider a convex combination of the two terms via parameter K:
L. (t) = x< 2s (t)l - 1 2 (t)l)² + (1 x) 2) V (t)) 2T (t)] (os(Os(t)) (t) (T))) (8)
where & E [o. 1] controls the trade-off between amplitude discrepancy and phase discrepancy. Build- ing on this discrepancy, we define the alignment loss as:
Lalign = -JET (t)
(9)
where (t) denotes the sampling distribution over frequency components. This objective inte- grates weighted discrepancies across spectral domain, thereby capturing global distributional differ- ences between the source and target graphs while adaptively balancing amplitude and phase shifts.
The siciitdde-hhase decomposition is motivated by the distinct roles these components play in spectral domain adaptation. Specifically, the amplitude component governs the spectral energy distribution, encapsulating global structural invariants and low-frequency homophilic pat- terns. In contrast, the phase component encodes the relative positioning and correlation frequency modes, capturing intricate relational dependencies and heterophilic irregularities. By decoupling these factors, ADAlign facilitates a more granular alignment strategy, allowing for adaptive weight- ing that yields more nuanced distribution matching than traditional alignment approaches.
Remark.
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4.3 ADAPTIVE FREQUENCY SAMPLER FOR NSD
A central challenge in spectral alignment is the choice of frequency points t at which the charac- teristic functions are evaluated. Effective alignment of amplitude and phase discrepancies requires sampling informative frequencies t. However, fixed or heuristic grids are either too sparse (missing subtle but critical shifts) or too dense (incurring redundant computation and noise). We therefore learn the frequency sampling distribution by parameterizing Fy with a density pr( Φ) over the spectral domain T. Concretely, we model pT (t: Φ) as normal scale mixture (Ansari et al., 2020; Li et al., 2023; Wang al., 2025), which affords both flexibility and analytical convenience:
PT (t: (): = { pN(t E) (E) dE,
(10)
where (t 0, Σ) is a Gaussian with covariance Σ, and ps: pr: (E; Φ)) a neural mixing distribution. Normal scale mixtures subsume common families (e.g.. Gaussian, Cauchy, Student-(), enabling adaptive emphasis on low frequencies (global variations) or high frequencies (fine-grained shifts).
We use the reparameterization trick to ensure that sampling remains differentiable. The sampler is trained to prioritize frequencies that expose larger spectral discrepancies. Denoting the pointwise NSD contribution by (Lx ( Eq. (8)), we maximize its expectation under pT:
M
max Calign(() &
[ (tt:)) JET
Σ m=1
V (tm),
Im ~ PT(;
(11)
M
where M is the number of sampled frequencies. As M increases, the Monte Carlo approximation converges to the true expectation, and by Theorem the empirical CF becomes more accurate, ultimately leading to higher-quality distributional alignment. This objective shifts probability mass toward frequeney regions that most contribute to source-target discrepancy, yiclding adaptive spectral sampler for NSD and a compact, high-signal set of frequencies for downstream alignment.
4.4 MINIMAX OPTIMIZATION FRAMEWORK
To achieve effective domain adaptation, ADAlign jointly optimizes two complementary objectives: (i) preserving task-specific discriminability on the source domain, and (ii) reducing cross-domain distributional gaps through spectral alignment. Let denote the learnable parameters the GNN. The source objective is the standard cross-entropy loss on labeled source nodes:
min Lsource min CE(GNN .,A)))))).
(12)
where (x'5, the source node features, adjacency matrix, and labels, and CE(-) denotes cross-entropy. parallel, we align the source and target graph representations in the spectral do- main. Let Φ be the parameters of frequency sampler (Eq. (10)), then the alignment objective is:
min Latign = { -/ JEET
(t)
Pr (t( dt.
(13)
Combining Eq. (11), Eq. (12) and Eq. (13), we obtain a minimax formulation:
min max (Lmurc (G:;) + A Catign (GS ;
(14)
where À balances the two objectives. In this framework, the GNN parameters ō are optimized to jointly minimize the classification and alignment losses, while the sampler parameters & are updated adversarially to emphasize spectral regions with large domain discrepancies.
Due space limitations, the training procedure and complexity analysis provided in Appendix C.
4.5 THEORETICAL ANALYSIS
In this section, provide PAC-Bayesian analysis for proposed ADAlign framework (Ma al., 2021; Fang et al., 2025b). To evaluate the generalization performance deterministic GDA clas- sifier structured graph data, begin by introducing the notion of margin loss graph.

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Margin Loss Graph. Consider graph (V, where V and & denote the nodes and edges, A is the adjacency matrix, X' is the feature matrix, and Y the node la- bels. For a classifier h and margin 7 2 O, the empirical margin loss is defined as (h) == AV EKEV (1, S y + maxe4y; (u(X,)))c)) where 1[:] is the indicator function and e ranges over possible labels. The expected margin loss is then C, (h) == E,eV.Y,- 2, (h), with Y; ~ P()' Zj) denoting the conditional label distribution given node embedding Z,.
Theorem 3 (GDA Bound for Deterministic Classifiers). Let H be hypothesis class. For any E H. < > O, and 7 Z let P be prior distribution on H that independent of the source sample Vs. Then, with probability at least - p over yd, for any posterior Q on H. it holds that
(T - +,, & 2(DKL 11)++)
+ +m
e 4Ng + D 1/1 (P:
(15)
In this bound, DKL (P |I Q) denotes the Kullback-Leibler divergence between distributions IP and Q. which serves as model complexity measure. The terms In 1 and 4Ng are standard in PAC-Bayesian bounds, while the (P:: () captures the difference between source nodes ys and target nodes VT.
Proposition 1 Bound for D.. (P:))) For any y 2 O, if the prior over H is fixed, the domain discrepancy T. (P: can be bounded by weighted gap in the spectral domain:
M Σ
D.. (P: S O 7/2
Wm |TS (tm) - vo (t.m))
(16)
where {tm))) 1M _ are sampled frequencies, and {Wm]m=1 are nonnegative weights approxi- mating a spectral measure. This result shows that domain discrepancy can be upper-bounded by the weighted frequency-domain gap between source and target CFs, with wm reflecting the importance of each spectral component. Crucially, learning these weights adaptively enables the model to em- phasize discriminative frequencies that drive cross-domain differences. Our ADAlign framework is directly aligned with this principle: by dynamically prioritizing informative frequencies, it achieves tighter bounds and more effective graph domain adaptation. Full proofs are deferred to Appendix B.
1M
EXPERIMENTS
In this section, we empirically evaluate the ADAlign framework through experiments addressing five key research questions: RQ1: How does ADAligm compare to state-of-the-ar methods? RQ2: What are the computational efficiency and resource overhead of ADAlign? RQ3: Does the adaptive frequency sampler effectively capture meaningful spectral components? RQ4: How do key param- eters impact model performance? RQ5: How can we intuitively understand ADAlign's strengths?
5.1 EXPERIMENTAL SETUP
(1) Datasets. To demonstrate the superiority of proposed ADAlign framework, we evaluate it across four types of datasets, comprising total of ten datasets (Fang al., 2025a): ArnetMiner (ACMv9, Citationvl, 1, DBLPv7) (Dai et al., 2022), Airport (USA, Brazil, Europe) (Ribeiro et al., 2017), Blog (Blog1, Blog2) (Li et al., 2015), and Twitch (England, Germany) (Liu et al., 2024a). (2) Baselines. To assess the effectiveness of our model, we conduct comparisons with two categories of baseline methods: (i) source-only graph neural networks such as GAT, GIN, and GCN, which are exclusively trained on the source domain before being applied to the target domain; and (ii) graph domain adap- tation techniques, including DANE, UDAGCN, AdaGCN, StruRW. SA-GDA, GRADE, PairAlign, GraphAlign, A2GNN, TDSS, DGSDA, GAA and HGDA. (3) Implementation Details. We utilize Micro-FI and Macro-Fl scores as evaluation metries for our experiments. More dataset descriptions, baselines, and implementation details can be found in Appendix D.
5.2 MAIN EXPERIMENTAL RESULTS (RQ1)
To assess the effectiveness of ADAlign framework, we conduct comprehensive evaluations on ten benchmark datasets covering four diverse graph domains, with results summarized in Table 1.
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Methods
A C
A
C-+ A
C-
D-
B1
R2
B2 RI
GAT (ICLR'IS 18)
62.77
g1.29
18.35
67.07 22.05
51.312a7
23.24
21.15
19. 19.46
GIN (ICLR19)
60.95
23
68.54_0.1
58.1820.00
20.51 1
18. 18.00
GCN (ICLR'17)
70.82
65.05
65.44
69.46_0.s
59.9220.7
60.832054
23.17
23.46
DANE (DCAT11)
69.7712.4
15
63.93 13.93
65.05_2.07
58.21
67. 67.41
28.1511.77
30.321000
UDAGCN (W
80.681031
74.66
73. 13
76,97
69 69.361031
77.811000
33.87
31.86g100
AdaGCN (TKDE'22)
68.0720.00
66.72a1.07
63, 63.255
70.97
60.6520.07
63.83
30.75
27.17
StruRW (ICML'23)
60. 60.48
59 15..
56. 18
62.50 1..0
52.25g1.15
57.55
10.37
12.01
SA-GDA (MM'23)
70.36 26.02
10.84
75.48_0.30
63.41g1.12
12.8111.5
19.36
15.07
GRADE (AAA1123)
75.02
68.17
68. 68.9620.10
73.48_0.m 30
61.7220.66
21
48.44
48.78 31
PairAlign (ICML/24)
59.58 60.25
56.0220.55
63.49_0.70
51.8320.00
68.60+0.42
42.64 40.01
GraphAlign (KDD'24)
75. 18..02
68.8 28.8
65.21 #0.55
72.21 =0.45
61.6621.25
68 18.854..56
45.58
43.21
A2GNN (AAAI'24)
20.93
75.91
40.45
07.16-0.25
73.21
79. 19.6.2..3
47. 47.10
44.01
TDSS (AAAI'25)
80. g0.11
71.04.5.:1
72, 12.8
27.23.2.11
79.0113.: 1..38
1
11 20:20
GAA (ICLR'25)
50.0310377
13.322.0.7
T3. 15.35
76,01_0.38
18.3210.43
18.27
48. 48.8012244
43.74
DGSDA (ICML'25)
73.71.0044 28.5
T3.96 73.96
76,56 a5
72.692031
48.612177 77.41
44.12+1.00
HGDA (ICML)25)
72.18 21.0
67.11 2o.sa
73.63_0.79
66.5220.36
75.23
46.3441.33
44.2941.60
ADAlign (Ours)
82.21 10.56
78.06
76.24 20.29
79.51 10.20
74.38 go.0s
11.84.0.0
55.08 1a:25
52.42 to 16
Methods
U-+B
U-+E
B-U
B-B
E-UU
E-BB
DE-EN
EN -D-
GAT (ICLR188
50.53
40.25
44.25
46.67-1.99
43.90 43.90+1.00
50.84
67.65+0.14
63. 63.13
GIN (ICLR
33. 33.18
32. 24.28
35.87
32.98_2.44
36.07:2.44
33.89
55. 05.10
60. 03.04
GCN (ICLR'17)
56,311377
17.11gm
10.22 1121
18.92 1.0
11.67
58.17 48.17
g7.59
62. 62.82
DAÑE (TCC T95
44. 12+100
28.00
42.
44.20
36.45_...9
36. 13.1.5
16.50
55. 55.44
62. 62.00000.0
UDAGCN (W)W)20)
58. 172.07
44.51004
17.17
17.97
42.79
62.29 29.m
29.37
63.6 63.61
AdaGCN (TKDE'22)
15.05
16.8 16.87 50.63
51.41_o.ss
18.62
73.71 73.71
57. 57.81
62.67
StrukW (CCML/23)
62.4 62.44g1.6
39.70g1.
41.45
38.30_1.32
55.11gam 41.21
56.7221.00
61.0110.0
SA GDA/MM
67. 1.555
51.09 on
48, 18.20ga32
56.99
47.78
63.63.1 sa
56.46.1.30
62. 12.18
GRADE (AAAI223)
40.78
43.33 20.58
55.94_0.19
46.64gn.50
63.05
50.16gn.24
64. 20.00
PairATign (ICML'24)
66.26
39.5020.36
41.92+2.30
38. 18.10-2.000 2000
39.63...54
65.57
57.45+0.12
64.22
GraphAlign (KDD'24)
62.54
52.18.0.17
35,23_n.as 50.33
54.35
71.01220.4
52.5819077
60.121001
A2GNN (AAAI'24)
64.24
46.6221.61
47.06
41.67
52.72+1.16
54.81
65.77=0.47
60.9220.00
TDSS (AAA1'25)
67. 1.02
52.05
47. 47.05
51.80=0.12
46.0820.53
65.73.108
55.91
61.00+0.32
GAA (ICLR'2)
64.89
51.70
52.50+1..7
48.22..1.7
18.70+2.54
56.07
57.54
61.98
DGSDA (ICML'25)
61. 61.22
19.35
29.35
54.91
1..56_1.64
49.76 19.7620.s2
60.15
00.81aa.1
63. 12 12:0
HGDA (ICML'25)
67.82 1,30
48.67 12.11
51.9112e8
48.07
46.6211.40
59.77 1
57.08
61.19 1091
ADAlign (Ours)
72.52
53.23
$0.59
58.49
55.41
75.82
61.92+a32
65.69
Table 1: Node classification performance (% = a, Micro-Fl score) under 16 transfer scenarios. The highest scores are highlighted in bold, while the second-highes scores are underlined.
A
C
A-D D
80
80
MeskelTaataats
Speed (s)
Menorr (GB)
Speed (s)
Memeey (GBB)
Micro-F1 (56)
Micro (86)
AZONN (AAAF'3))
0,1812
10.64
0.1781
10:45
60
60
A D
TDSS 25)
0,1684
10.64
0,1661
10.51
40
40
200lg
(ICUR
0.3662
1580
0.3530
1560
105
HGDA (1CM)25)
ADMlign (Ours)
0.08 (2,0%
(31
0.07 (2.3
0,3817
f) 2.60 (4,0%
20
0
50
100
150
H024
200
20
0
50
100 Epochs
150
1004
200
Epochs
Table 2:
Per-iteration runtime and memory consumption comparison on two transfer tasks. Values in parentheses show relative improve- ment: "4" speedup, "1" memory reduction.
Figure 4: The comparative training curves of Micro-F1 scores are illustrated for four repre- sentative baselines across two distinct transfer scenarios over 200 epochs of iterative learning.
ADAlign consistently achieves the best performance across all 16 transfer scenarios, demonstrat- ing its ability to capture and align complex composite distribution shifts. These improvements are attributed to ADAlign's ability to adaptively capture multi-level distribution shifts through NSD modeling, rather than relying on fixed heuristic factors such as GAA and HGDA. In particular, ADAlign yields an average improvement of over 6% in challenging heterogeneous blog-domain transfers, where the extremely high-dimensional features and multiple label categories create more entangled shifts, highlighting its robustness under severe distribution shifts. These results confirm the effectiveness and generality of holistic spectral alignment for graph domain adaptation. More- over, the consistent gains across academic, social, and airport graphs demonstrate that ADAlign is not tied to a specific domain, but rather provides a general-purpos solution. Additional experimen- tal results and significance tests (p < 0.05) are provided in Appendix E. and Appendix F.2.
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EN - DE
80
E
A-
+ EN DE
Micro-F1 (1)
86
Micro F1 (%)
70
666
63
50
Mixro-F1
MicroFF
84
65
80
implitude
plase
0.75
40
amplinade
plase
0.65
0.35
62
0.0
0.2
0.4
0.6
0.8
1.0
0.0
02
0.4
0.6
0.8
1.0
60
32
128
512
2048
60
30-2
Be-2
9e-2
M
A
73
Mer Ft (9)
Micoo-F1 (5g)
A
D
80
-C-
C-+
Micro-F (%)
60
- U
79
70
Micro-11
50
75
77
67
amplitude
phose
01.25
0.75
amplitude
plusec
0.65
0.35
40
0,00
02 0.4
0,6
0.8
1,0
64
0.0
0.2
0.4
0.6
0.8
1.0
70
32
128
512
2048
3e-2
6e-2
9e-2
M
Figure 6: The performances of our ADAlign w.r.t varying parameters on different transfer tasks.
5.3 ANALYSIS ON EFFICIENCY AND STABILITY (RQ2)
We compare runtime, memory, and training dynamics against competitive baselines. As shown in Table ADAlign effectively reduces runtime and memory usage. On the A-C task, for instance, achieves 0.08 s/iter with 3.45 GB memory, representing a 2.0x speedup and a 3. x reduction in memory usage compared TDSS, the strongest competitor in efficiency. This advantage arises from using more efficient metric that avoids costly kernel computations in traditional methods (c.g., MMD). By leveraging neural characteristic function, we avoid the computational burden of large- scale kernel matrices and the high variance introduced by sampling-based estimations, leading to im- provements in both runtime and memory efficiency while preserving the flexibility needed for graph domain adaptation. Furthermore, Figure 4 illustrates that ADAlign trains efficiently and stably. In contrast to adversarial methods that often suffer from instability, ADAlign remains smooth and ro- bust. This combination of low computational cost and stable training dynamics makes ADAlign practical for real-world deployment. The training loss curves are provided in Appendix E.4.
5.4 ABLATION STUDIES ON FREQUENCY SAMPLER (RQ3)
We ablate the adaptive frequency (AF) sampler by comparing ADAlign with three variants: Random Frequencies (RF), Low Frequencies (LF), and High Frequencies (HF). For LF, we select the frequency range [1, 10], and for HF, the range [10. 20]. These ranges on the similarity between transfer scenar- ios, with the highest frequency dynami- cally scaled according to a similarity mea- sure S between the source and target graphs. Specifically, the highest frequency fmux is given by: fmax Sbase x (1 S/SS S/Smaxx where Fuse is the base frequency, S is the similarity measure between the graphs, and Smax is the maximum similarity. This scaling ensures that the frequency range adapts to the graph's 's structure and the level of discrepancy between the domains. As shown in Figure 5, AF consistently achieves the best performance across four transfer tasks, confirming that dominant shifts lie in scenario-specins specific spectral components. The task-level patterns are intuitive: LF performs competitively on U-E and E E-U, reflecting the role of smooth, large-scale structures, while HFF un- derperforms DE-+EN and DE, indicating that ignoring fine-grained cues is detrimental. RF yields unstable and suboptimal results, suggesting that unguided sampling fails to target critical dis- crepancies. Overall, these findings underscore the benefit of minimax formulation: instead of relying on a fixed heuristic band, adaptively identifies and emphasizes the most discriminative frequencies, enabling precise alignment. Additional ablation results provided in Appendix E.5.
66
72
Mirro11 (4)
52+
-50
Mirr-ft
60-
66
Mirr
57-
determined based
Figure 5:
Classification Micro-F1 comparisons be-
tween ADA variants four cross-domain tasks.

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Published as a conference paper at ICLR 2026
(a) A2GNN
(b) GAA
(c) HGDA
(d) Ours
Figure 7: Visualization of node embeddings inferred by four models for the A - C task.
5.5 PARAMETER SENSITIVITY ANALYSIS (RQ4)
In this section, we analyze the effect of three key hyperparameters in ADAlign: the amplitude-phase trade-off x in Eq. (8), the sampled- freelllncyr number M, and the alignment loss weight À in Eq. (14). The corresponding results are shown in Figure 6. Additional results are provided Appendix E.
Study amplitude weight The parameter x controls the relative emphasis amplitude versus phase in the spectral domain. We find that performance is consistently strong when & lies in the range 0.65 and 0.75, highlighting the importance of balancing global structural consistency (amplitude) with fine-grained relational cues (phase). Worth noting is that Pushing i toward either extreme (x 1.0 or K 0.0) leads to clear degradation, confirming the necessity joint alignment.
The parameter M determines the number of sampled frequency components used compute the discrepancy. Increasing M improves performance at first, since richer frequency coverage captures more detailed aspects of distributional shift. However, gains plateau beyond M 2048, suggesting that ADAlign can effectively capture essential spectral characteristics without requiring excessive samples, thereby retaining computational efficiency.
Study on the number of frequencies M.
Study on loss weight x. The hyperparameter X balances the domain alignment loss against the source classification loss. As À increases, performance improves steadily before saturating, in- dicating that moderate alignment weighting suffices to achieve strong transfer. Very small under-emphasizes alignment, while overly large À slightly impairs source discrimination. Overall, ADAlign remains robust, with mid-range values consistently providing an effective trade-off.
5.6 VISUALIZATION OF NODE EMBEDDINGS (RQ5)
To intuitively demonstrate the superiority of ADAlign, we visualize the node embeddings learned by different methods for the A + C task using t-SNE (Maaten & Hinton, 2008), as illustrated in Figure 7. A contrast can be observed across the visualizations. The embeddings generated by baseline models, such as A2GNN, GAA, and HGDA, show significant overlap and intermingling between classes from the source and target domains. This indicates that these methods struggle to learn repre- and minssddisriimiative,, leading to suboptimal adaptation
sentations that are both performance. In contrast, the embeddings produced by our framework demonstrate two desirable characteristics. First, samples from the same class across both domains form compact, well-aligned clusters, highlighting the effectiveness of our spectral alignment mechanism in learning domain- invariant features. Second, the boundaries between different classes are sharp and well-separated, showing that our minimax alignment over characteristic function not only mitigates domain shift but also preserves task-ret relevant discriminative information.
6 CONCLUSION
In this paper, we introduced the ADAlign framework, novel approach to handling composite distribution shifts in GDA. By leveraging characteristic function in the complex Fourier domain, ADAlign dynamically identifies and aligns discriminative spectral components through minimax optimization, eliminating the need for manual reweighting or heuristic metrics. Extensive experi- ments across 16 benchmarks show that ADAlign consistently outperforms tf-heart methods, while also improving training stability and reducing computational overhead. These advantages make ADAlign highly effective and practical solution for real-world graph transfer learning tasks.
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ACKNOWLEDGEMENTS
This work was supported by the National Key Research and Development Program of China under Grant Nos. 2024YFF0722003, the National Natural Science Foundation of China under Grant Nos. 62176014, 62276015, 62206266, the Fundamental Research Funds for the Central Universities.
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Appendix
Table of Contents
A The Use of Large Language Models (ILLss)
14
B Proof of Proposition 1
14
C Training Details
16
C.I. Complexity Analysis
16
C.2 Algorithm of ADAlign
16
D Experimental Setup Details
17
Dataset Description
Baselines.
Implementation Details
E Experimental Results
19
E.I
Main Experimental Results
19
E,2
Significance test (t-test) of ADAlign
20
E.3 Effect of GNN Backbones
20
E,4
Loss Comparison
21
E.5
Ablation Study
21
E.6
Parameter Sensitivity Analysis
21
E.7
Visualization of the learned frequency distributions
22
A THE USE OF LARGE LANGUAGE MODELS (LLMS)
In this work, we utilized LLMs solely for the purpose of English language refinement. These mod- els were employed assist with the proofreading and enhancement of written text, ensuring clarity, coherence, and grammatical accuracy. The LLMs were not used for generating content, and all re- search, analysis, and conclusions presented the result of own work and independent thought.
B PROOF OF PROPOSITION 1
The nodes are partitioned into two disjoint classes, denoted co and Each node feature vector x is sampled from a Gaussian distribution ((1,,,,) for (0, 1}. The source and target graphs each contain nodes from both classes, denoted as c; and c. respectively. The class distribution is balanced, i.e., P(Y = co) = P(Y = =c1)-
Definition 1.
(S) (T),
Theorem 4. For any two nodes s E LS and L VT with aggregated feature representations
CNN(x), the following inequality (Mao al., 2023; Fang et al., 2025a) holds:
2o.il < O (IZZ - Zoll 11 - #1 (5) 11 + 11 14
(1111 11) (17)
P = co Zw.) P(y. c
Proposition 1
(Bound for (P:E)) For any 0, under the assumption that the prior distri- bution IP over the function family H is specified, the domain discrepancy TT as follows:
(P:; {) can be bounded
M
D (P:E) O 7/2
Σ
Wm T (t.)
T (tm))
(18)
where {tm M that approximate target spectral measure.
4d are frequency samples, and {wm (M are nonnegative quadrature weights
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Proof. For notational simplicity, let h h, (2 9) for any i V UVT. Define me(i) P(/// Z,) for {0, 1), and let (h,,, y/) = 1 [h, I y + max [a[[]]]
The difference in expected margin losses between the source and target domains is given by:
(1) (2)h) = EgyT
Bys
Σ
Cy (19)
NT JEVT
Ns
LEVS
Using Definition 1, Eq.. (19) can be rewritten and bounded as follows:
1
Σ
1
Σ JEVT
Bng (2(
-By,f (hs.
(20)
max (Ns, NT)
NT
iEVS
Σ
(na (1)))))))2())))))) - Plu: = fin
(21)
(Nax(Ns, NT)
NT
LEVS
JEVT
k
(n (2) - m (z) (hi: R:))
(22)
max(NNs. NT)
NT
iEVS
JEVT
Σ JEVT
Σ
(ma(i (Cy/2(1g,, - (he, + (mx(i)
A))
max(Ns, NT)
NT
iEVS
(23)
Σ
Σ
Σ
(fm)2(1)3;h) - (,,,,, k.) + (1 - 1/x (1)1B)
(24)
NNa
NT
LEVS
JEVT
Now, applying Theorem 4 to bound the difference in conditional probabilities, obtain:
Σ [|) - m) (
(25)
0
E Zu Z + Σ Vars (Zu Ua =c) Vart(Zo | yv + (1 14 111
(26)
where μe and pr are the class-conditional means for the source and target domains. respec- tively. This bound is second-order nature, as controls the discrepancy through pairwise dis- tances and class-conditional variances means. In principle, one could generalize this to a K-th order metric by incorporating higher-order moments (c.g., skewness (Seglen, 1992), kurtosis (Ba- landa & MacGillivray, 1988)). However, such higher-order terms quickly become cumbersome and may lead to instability high-dimensional settings. To handle this in a more unified and tractable manner, utilize characteristic function, which encode moments of all orders and can uniquely determine a distribution. The characteristic function for random vector X is defined as:
T (t) E it x
(27)
which can be expanded into a power series:
vx = Σ k! (t y
(28)
This expansion makes clear that the characteristic function encapsulates moments of all orders, which naturally subsumes the information contained in second-order bounds while remaining com- putationally efficient for distributional alignment. Motivated by this representation, we lift the second-order bound to a distributional discrepancy on the complex plane via CF:
M
S.T 7/2 (P:E) <
Σ m=
Wm /VS (t.) - v (tmm)
(29)
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Algorithm 1: minimax Optimization for Graph Domain Adaptation
Input:
Source graph g = (2' AA4,, ,,5),, target graph GJ (A", ."). trade-off parameter X,
learning rates 776; Ne.
Trained GNN parameters 6, frequency selector parameters ø. Initialize ò, & randomly;
Output:
for each epoch = 1, 1..... main do
Sample mini- batch from G and g";
/* Step A: Update GNN parameters ô (fix Φ)
Compute source loss: Lioure (ō) = (E(GNN
Compute alignment loss: Lalign(ó, Φ) - (tx PT dt;
Total loss: L = Lsuurce + Caligni
Update d x 8 - 1/8 VoC;
/ Step B: Update frequency sampler & (fix 8)
* /
Compute alignment loss with fixed 6;
Update Φ + Φ + nh V Lalign
return d, Φ
This bound suggests that the domain discrepancy can be controlled by a weighted sum of squared dif- ferences between the characteristic functions of the source and target domains, evaluated at multiple frequency points. The weights Wm allow for adaptive weighting of different frequency components, reflecting the intuition that some frequencies may be more important for generalization than others.
Our proposed adaptive frequency selection mechanism directly leverages this insight. The weights Wm are learned to emphasize frequencies where significant cross-domain discrepancies exist. This allows the model to focus discriminative features while suppressing irrelevant noisy compo- nents, leading to more robust and effective domain adaptation.
Remark. To clarify the relationship between the learned frequency distribution pr(t; Φ) and the discrepancy weights Wm. observe that the adaptive frequency sampler effectively prioritizes the most relevant frequencies based on the transfer scenario. Specifically, the optimization of the align- ment loss with respect to the pr (t; <)) is equivalent or upper-bounded by maximizing weighted sum of individual frequency losses, where the weights win correspond to the importance of each fre- quency in driving the overall distributional alignment. This relationship importance sampling, where py (t; Φ) adjusts the sampling probability of each frequency, naturally aligning the most relevant features without requiring explicit weight assignments.
be formalized through
C TRAINING DETAILS
C.I COMPLEXITY ANALYSIS
In this section, we analyze the time complexity of the proposed ADAlign framework. For one training iteration, the computational cost mainly comes from GNN message passing and spectral alignment. Suppose the source and target graphs together have [V] nodes and |E| edges, the embed- ding dimension is d, the number of layers is L. and M frequency samples are used in alignment. The GNN forward and backward passes take ((L(E) d' + [V] df). reflecting sparse matrix multi- plications and linear transformations each layer. The spectral alignment requires computing M frequency responses for all node embeddings, each involving dd-iimensional inner product, result- ing in O (Ml)) Thus, the overall per-iteration complexity is O (((( d |V] d²)) + O(M Vld), which is linear in the graph size [V], |E] and at most quadratic in the embedding dimension d.
C.2 ALGORITHM OF ADALIGN
The complete training procedure of our framework is summarized in Algorithm 1.
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Types
Datasets
Nodes
# Edges
Feat Dims
# Labells
ACMv9 (A)
9,360
31,112
ArnetMiner
Citationv (C)
8,935
30,196
6,775
5
DBLPv7 (D)
5,484
16,234
USA (U)
I,190
27.198
Airport
Brazil (B)
131
2,148
241
4
Europe (E)
399
11,990
England (EN)
7,126
35,324
Twitch
3,170
2
Germany (DE)
9,498
153,138
Blogl (B1)
2,300
66,942
Blog
8,189
6
Blog2 (B2)
2,896
107,672
Table 3: Dataset statistics.
D EXPERIMENTAL SETUP DETAILS
D.1 DATASET DESCRIPTION
We conduct node classification experiments across four distinct migration settings, outlined in Table 3. In each setting, our model is trained on one graph and its performance is evaluated on the remaining graphs. The datasets used these experiments are follows:
ACMv9 (A), Citationv I (C), DBLPv7 (D): These datasets are citation networks collected from distinct academic sources. In these networks, each node represents research paper, and the edges denote citation relationships between papers. The goal of the task is to classify each research paper into its corresponding category. The datasets are sourced from different time periods and repositories: ACM (before 2008), DBLP (from 2004 to 2008), and Microsoft Academic Graph (after 2010). We perform experiments under six different transfer scenarios, namely: - C, A + D, C A, C D, D - A, and D -
USA (U), Brazil (B), Europe (E): These datasets are derived from transportation statistics, specif- ically focusing on aviation data. In these datasets, each node corresponds to an airport, and the edges represent direct flight connections between two airports. The objective of the task is to pre- dict the category each airport based on its characteristics and network relationships. We explore the following six transfer scenarios: U - B, U E, B U, B E, E - U, and E + B.
Blog1 (B1), Blog2 (B2): These datasets come from the BlogCatalog dataset, where each node represents a blogger, and the edges represent the friendship relationships between bloggers. The node attributes consist of keywords extracted from the bloggers' deff-descriptions, and the task is to predict the group or category to which each blogger belongs. We set up two specific transfer scenarios: B1 + B2 and B2 B1.
Germany (DE), England (EN): These datasets sourced from the Twitch Social Networks, where nodes represent streamers, and edges denote the follower relationships between stream- Each node is associated with attributes such categories or tags that describe the streamer's profile. The task is to predict the category of each streamer based on their attributes and relation- ships within the network. We investigate two migration scenarios: DE EN and EN DE.
D.2 BASELINES
We compare our method with the following baseline methods, which can be divided into two cat- egories: (1) Graph Neural Networks (GNNs) on the source graph only: GAT (Veličković et al., 2018), GIN (Xu et al., 2019), and GCN (Kipf & Welling, 2017) are trained end-to-end on the source graph, allowing direct application to the target graph; (2) Graph Domain Adaptation Meth- ods: DANE (Zhang et al., 2019). UDAGCN (Wu et al., 2020), AdaGCN (Dai et al., 2022), StruRW (Liu et al., 2023), SA-GDA (Pang al., 2023), GRADE (Wu al., 2023), PairAlign (Liu et al., 2024b), GraphAlign (Huang et al., 2024), A2GNN (Liu et al., 2024a), TDSS (Chen et al., 2025a), DGSDA (Yang et al., 2025), GAA (Fang al., 2025a) and HGDA (Fang et al., 2025b) are specifi- cally designed to address the GDA problem.
Below, we provide descriptions of the baselines used in the experiments.
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ERM (GAT). ERM (GIN), ERM (GCN) These methods train using GAT (Graph Attention Networks), GIN (Graph Isomrrphism Networks), and GCN (Graph Convolution Networks) under the standard Empirical Risk Minimization (ERM) framework.
DANE employs shared weight GCNs to generate node representations. To address distribution shifts, it leverages a least-squares generative adversarial network, which promotes alignment be- tween the source and target distributions of node representations.
UDAGCN is a model-centric method that combines adversarial learning with GNNs for domain adaptation. It aims to reduce the discrepancy between source and target graphs by using an adver- sarial objective, promoting the learning of domain-invare
features across different domains.
AdaGCN is model-centric method designed for GDA. It incorporates adversarial domain adap- tation techniques along with graph convolution to align node representations across domains, us- ing the empirical Wasserstein distance as a regularization term to minimize distribution shifts between the source and target domains.
STRURW modifies graph data by adjusting edge weights based on domain adaptation strategies. This adjustment is informed by the data-centric GDA techniques, ensuring that the model effec- tively adapts to distribution shifts while preserving the structural properties of the graph.
SA-GDA observes that nodes from the same class across different domains share similar spectral characteristics, while different classes remain spectrally separable. Based on this, it aligns source- target feature spaces in the spectral domain at the category level, avoiding confusion caused by global feature alignment.
GRADE introduces a graph subtree discrepancy metric to measure distribution shifts. By con- necting graph neural networks with the Weisfeiler-Lehmaan subtree kernel, it evaluates how node embeddings differ across domains, enabling more effective transfer under distributional variations.
PairAlign is method that recalibrates the influence between neighboring nodes using edge weights to address conditional structure shifts. Additionally, it adjusts the classification loss with label weights to account for label shifts, ensuring that the model can handle both structural and label distribution shifts effectively.
GraphAlign generates a small, transferable graph from the source graph, which is then used to train a GNN under Empirical Risk Minimization (ERM). This approach focuses on leveraging smaller graphs to improve domain adaptation by aligning the distribution of node representations across domains while preserving graph structure.
A2GNN proposes a simple yet effective GNN architecture that increases the number of propa- gation layers in the target graph. This increase in depth allows the model to capture complex dependencies within target domain, improving its adaptation performance in domain- tasks.
TDSS applies structural smoothing directly to the target graph to mitigate local variations. This method helps enhance model robustness by preserving consistency node representations, which improves the transfer of knowledge across domains while reducing the impact of structural dis- crepancies between the source and target graphs.
DGSDA addresses the challenge domain adaptation for graph-structured data by disentangling the alignment of node attributes and graph topology. This framework uses Bernstein polynomial approximation to align graph spectral filters efficiently. avoiding expensive eigenvalue decompo- sition. By separating attribute and topology shifts, DGSDA enables the use of flexible GNNs, improving model performance and adaptability across domains.
GAA focuses addressing both graph topology and node attribute shifts in Graph Domain Adap- tation (GDA). Unlike traditional methods that primarily address topology discrepancies, GAA emphasizes the critical role of node attributes in aligning domain- specific graph features. The method introduces a cross-cies channel module that uses attention based learning for attribute align- ment, allowing the model to focus significant attributes.
HGDA highlights the impact homophily distribution discrepancies between source and target graphs in GDA tasks. By investigating the homophilic and heterophilic patterns in graph nodes, HGDA proposes a novel method that uses mixed filters capture and mitigate homophily shifts. The method separates homophilic, heterophilic, and attribute signals in graphs, enhancing domain alignment across different graph types.
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D.3 IMPLEMENTATION DETAILS
We utilize Micro-Fl and Macro-FI as evaluation metries for our experiments. For each experimental result, we conducted five runs and calculated the average performance across these runs. Fach run was executed for a total of 150 epochs. 2015) optimizer with learning rate ranging from 0,001 to 0.01. The hyperparameter search ranges are follows: the dropout rate is in the range [0.1, 0.5]. the propagation steps take values from the set {0, 1, 10, 15), the frequency number is chosen from {32, 64, 128, 256, 512, 1024, 2048, 4096), the amplitude weight coefficient is in the range [0, 1he experiments were conducted on a Linux server, running Ubuntu 18.04.6 LTS. For GPU resources, we utilized single NVIDIA GeForce RTX 3090 graphics card with 24GB of memory. The Python libraries employed for implement- ing our experiments include Python 3.8, PyTorch 2.4.0, PyTorch Geometric 2.6.1, PyTorch Sparse 0.6. 18, and Py Torch Scatter 2.1.2. The complete parameter space is provided in Table 7.
train our model by utilizing the Adam (Kingma & Ba.,
E EXPERIMENTAL RESULTS
Methods
A AC
>
C
C
D
D
>
D
C
B1
B2
B2 B1
GAT (ICLR'88)
59.60s28u
06.09.1.16
58.06_158
03.82
52.99_2.6
59.964280
11.18
9.43_a.04
GIN (ICLR'19)
62.34.108
53 23.08.206
6..
21.36_2.3
63.00
11.44.2.8
9.57
GCN (ICLR"17
66.494221
59.5340.1
65.06-1.8
65.80
58.95-nst
64.664101
13.80
13.76.148
DANETUCATTI
66.89+2.0
70.09-2m
64.0442.1
60.34410
57..8-1.47
02.37+247
23.20-2.13
28.37=009
UDAGCN (WWW'20)
78.74g035
72.50 22.00
24.37
75.50
70.10_n.0
76.091078
28 28.83
28.3 28.95
AdaGCN (TKDE'22)
64.15++18 188
61.97+1.25
63.22-1.25
66.
45
58,99-142
58. 58.27+1.4
20.69=1.00
18.73+221
StruRW (ICMI?23)
54.49
51.77=201 22.54
56.74 66.741201
14.68
50.47
39. 39.6220.77
41 41.3620047
SA-GDA (MM'23)
72.10+200
66. 13.198
71.40-159
70.97+278
56.83-1.0
69.08+288
40.26
44.59
GRADE (AAAT'23)
71.66
63. 23.05
68.43_n.14
69.762202
56.45_n..0
66.544am
42.00 44.46
PairAlign (ICML'24)
55.11
53. 20.8.
51.57
69.02
16.47
39.57 53.66
41.98
GraphAlign (KDD'24)
a1.09
65.51 51...1
61.37
71.18 98.32
62.33
65.56
43 14
42.29
A2GNN (AAAT'24)
78.06
g0.31
76.41
23.28
74.48
45.4224.7 26.01
43.14
TDSS (AAAI'225
19.10
71.59.3.0s
76.09_1.06
74.96
73.23_a.00
77.70
45.31 22:00
14.31
DGSDA (ICML'25)
77.03+032
72.634402
71.61=0.50
75.52+a31
71.82-0.45
76.13+223
48.01=1.7
42,06+330
GAA (ICIR'25))
75.851065
68.07
73.57_n.3
71.40
62.66_n.50
72.744078
45.66
40.59
HGDA (ICML'25)
74.28+071
69.03+4.12
66.98-057
70.59+0.6
66.47-0.57
82
45.32+2.0
43.35+1.1
ADAlign (Ours)
80.65 0.30
77.62 76.00.117
-0.3
77.48 +0.35
75.83-0.0
79.95+0.4
54.89....0 -0.50
51.85-000
Methods
U B
U
B U
B + R
B U
F. B
GF. + EN
F.N GF
GAT (ICLR'88) 18)
40.0
35.57.xts
41.25_1.37
46. 18
43.13
49.02
57.222..2
57.692111
GIN (ICLR'D 19)
23.29+4.6
22. 22.50+1.12
28.06-1.88
26.03++.30
28.03-2.11
26.21+128
51.96=2m
50.32++.6
GCN (ICLR'I7)
12.124
42.63
31.21
46.52
37.39-4.1
57.274313
56.71
58.431047,
DANE (TCAAP10
41.23
34.241476
35.733n..1
"35.37 23:0
28.26
28.26_7.4
39.942080
48.6024.72
49.012259
UDAGCN (WWW'20
56.82+250
41. 10.85
34.65-1.00
47.60+1.7
41.83-1.0
61.97+415
58.31-0.0
59.45+135
AdaGCN (TKDE'22)
66. 15.15
51.45goos
43.89_1.07
54.922am
73.34 24.37
56. 28..68n..1
59.862a.4
StruRW (ICML'23)
62.25
37 97.26
40.02_1.66
36.841154
38.10
32. 32.15gaa7
52, 72..6
50.51
SA-GDA (MM'23)
66.51 11
50. 50.30.m.v
47.94
53.92,1.4
45.34-0.0
64.63
54. 54.20.241
57.24 57.244082
GRADE (AAAI'23)
56. 16.16
46 864222
38,68
38.68-n.41
53.932oas
42.47=043
60.13
58, 58.36
59. 59.622a.18
PairAlign (ICML'24)
65.351265
35.61.9.45
37.57 40.89_2.3
1300
37.71
33.26
55.04.0.14
60.681028
GraphAlign (KDD'24)
61.33+121
50 60.09
47.11=1.77
53. 19
49.31=n.s7
69.97+1.2
48. 14=0.7
59,33+a.15
A2GNN (AAAT'24)
61.081257
13.4
38.60_7.7
42.80
48.39
46.761247
54.25 54.25
51.782316
TDSS (AAA1'25)
63.36+133
48. 48.17+1.50
36.46-243
40.38+1.15
36.55-2.1
45.27+112
55.01=p.57
53.19++.4
DGSDA (ICML'25)
60.28+1.5
48. 18.20.+1.6
53.76
42.58+1.78
47.59-a.0
59.30+aos
59. 59.76.p.12
61.58+a2s
GAA (ICLR 25)
59.28
50.1621.
50.18_1.0
42.76
47.1721.0
55.34
56.882m.20
59.45
HGDA (ICML'25)
63.27
+1.98.1.9
50.25-2.1
43.17
43.44-3.0
56.74,206
56.61
54.84,284
ADAlign (Ours)
71.96 1.10
52.74 1952
54.47
-0.33
59.10, 11
51.37
10.75
75.74 on
60.77 10:54
62.11 2:20
Table Node classification performance (% + a, Macro-F1 score) under 16 transfer scenarios. The highest scores are highlighted in bold, while the second-highest scores are underlined.
E.1 MAIN EXPERIMENTAL RESULTS
The macro-F1 scores reported in Table 4.
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E.2 SIGNIFICANCE TEST (T-TEST) ADALIGN
We conduct paired two-sided t-tests comparing ADAlign with all baselines across A A-C and A-+D. In Figure 8 and Figure 9, all bars lie well above the (0.05) threshold, indicating statistically significant improvements (p < 0.05) every setting.
Significance Test: ADAllgn Baselines (A C)
23.3
Porskign
1000
Baseline Methods
Figure 8: t-test on the A-C task with significance level 0.05.
Significance Test: ADAlign Baselines (A D)
31.3
62
Stute
Figure 9: t-fsst on the A A-D task with significance level 0.05.
E.3 EFFECT GNN BACKBONES
To examine the influence of different GNN backbones, we instantiated ADAlign on two widely used architectures (GCN and GAT) and evaluated it against strong baselines As shown in Table 5 and Table 6, ADAlign consistently outperforms the corresponding source-only models and all competing methods under both backbones, often by clear margin. These findings demonstrate that ADAlign is not tied to particular encoder and can be reliably integrated with diverse GNN architectures.
transfer scenarios.
Method
A-CCMiiFI)
A-C (Ma-FI)
A-D (Mi-FI)
A-D (Ma-FI)
GCN (ICLR'17)
70.82 + 1.26
66.49 + 2.21
65.05 + 2.15
59.53 + 0.44
GAA (ICLR'25)
73.50
0.85
70.20 1.30
67.30
1.50
62.80 0.90
HGDA (ICML'25)
75.40 + 0.60
72,80
+ 0.90
68.65 + 1.10
64.50 + 1.15
ADAlign (Ours)
76.99 + 0.37
75.11 0.45
70.11 + 0.80
66.33 1.33
Table 5: Results on the GCN backbone.
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Method
(Mi-Fi
(Ma-F)
A-D (Mi-FI)
A-+ (Ma-F1)
GAT (ICLR' 18)
62.77 = 2.24
59.60 2.82
60.29 1.33
55.09 1.16
GAA (ICLR'25)
67.63 + 1.42
64.53 = 1.74
63.33 + 1.03
59.26 + 2.06
HGDA (ICML'25)
65.20 1.80
62.10 2.20
61.80
1.20
57.30 1.60
ADAlign (Ours)
70.45 0.95
69.80 1.10
67.45
0.85
65.90 1.25
Table 6: Results the GAT backbone.
E.4 LOSS COMPARISON
To evaluate the stability of ADAlign, we track the training loss over epochs for A-x and A-D. As shown in Figure 10, the loss decreases smoothly and rapidly during the early epochs, eventually plateauing at a low level with minimal oscillations. This indicates a well-behaved and stable opti- mization process. Notably, both transfers exhibit nearly identical convergence trajectories, further demonstrating that ADAlign maintains stability across different domain shifts.
C
1.5
A D
Loss
1.0
0.5
0.0
50
100
150
200
Epochs
Figure 10: Training loss curves of ADAlign on two tasks.
E.5 ABLATION STUDY
89
85
58
54
NONSS RF
leazz HF
NONS RF Reaz HF
83
80
Micro-Fi (%)
Micro-F (%)
54
-50
Micro-FI (%)
Micro-1 F1
71
70
46
42
4.0
65
01
65
42
38
A
B2
Figure 11: Classification Mi-Fl comparisons between ADAlign variants on four cross-domain tasks.
E.6
PARAMETER SENSITIVITY ANALYSIS
60
60
60
82
-
Micro-F1 (8))
Micro-F1 (94)
Micro (96)
555
Micro-F1 (8))
60
80
-0
50
40
45
30
128
2048
128
2048
40
3e-2
6e-2
96-2
30
3e-2
6e-2
96-2
Figure 12: The performances of our ADAlign w.r.t varying parameters different transfer tasks
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E.7 VISUALIZATION OF THE LEARNED FREQUENCY DISTRIBUTIONS
A-C
C-D
-0.05
0.00
Figure 13: Learned frequency sampling distributions for three transfer scenarios.
In this section, we visualize the learned frequency distributions across several transfer scenarios. As shown in Figure 13, ADAlign learns distinct dominant frequency regions for different datasets, con- firming that the adaptive sampler does not rely on a fixed or heuristic grid. Instead, it automatically emphasizes spectral components that reveal the most salient discrepancies for each source-target pair. For example, the A-C scenario concentrates more probability mass on higher-frequency com- ponents, whereas A and C-D favor lower or mid-range frequencies. These patterns support the claim that ADAlign tailors its spectral emphasis the underlying structural and attribute-level shifts of each task, aligning with the motivation of NSD and its minimax training strategy.
Dataset
Transfer Scenarios
Hyperparameter
Search Space
learning rate
[0.001, 0.005, 0.01]
weight decay
[0.001, 0.005, 0.01]
frequency number
128, 1024, 2048, 4096]
pnums
[0, 1, 10, 15]
C
lambda
[0, 0.2, 0.4, 0.6, 0.8, I]
dropout ratio
[0.1, 0.25, 0.5]
feature dimension
128
amplitude weight
0~1
epochs
150
learning rate
[0.001, 0.005, 0.01]
weight decay
[0.001, 0.005, 0.01]
frequency number
128, 1024, 2048, 4096]
pnums
[0, 10, 15]
A D
lambda
[0. 0.2, 0.4, 0.6, 0.8. IJ
dropout ratio
[0.1, 0.25, 0.5]
feature dimension
128
amplitude weight
0 0~]
epochs
150
learning rate
[0.001, 0.005, 0.01]
weight decay
[0.001, 0.005. 0.01]
frequency number
128, 1024, 2048, 4096]
pnums
[0, 1, 10, 15]
C-A A
lambda
[0, 0.2, 0.4, 0.6, 0.8, IJ
dropout ratio
[0.1, 0.25, 0.5]
feature dimension
128
amplitude weight
0~1
epochs
150
Citation
learning rate
[0.001, 0.005, 0.01]
weight decay
[0.001, 0.005, 0.01]
frequency number
128, 1024, 2048, 4096]
pnums
[0. 1, 10, 15]
C+D
lambda
[0. 0.2, 0.4, 0.6, 0.8. I]
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Dataset
Transfer Scenarios
Hyperparameter dropout ratio
Search Space
[0.1, 0.25, 0.5]
feature dimension
128
amplitude weight
0~1
epochs
150
learning rate
[0.001, 0.005, 0.01
weight decay
[0.001, 0.005, 0.01]
frequency number
[128, 1024, 2048, 4096]
pnums
[0. I, 10, 15]
D-A
lambda
[0, 0.2, 0.4, 0.6, 0.8, II
dropout ratio
[0.1, 0.25, 0.5]
feature dimension
128
amplitude weight
0~1 1
epochs
150
learning rate
[0.001, 0.005, 0.01]
Weight decay
00.001, 0.005, 0.01]
Irequency number
128. 1024, 2048, 4096]
pnums
[0. 1, 10, 15]
D D-C
lambda
[0, 0.2, 0.4, 0.6, 0.8, IJ
dropout ratio
[0.1, 0.25, 0.5]
feature dimension
128
amplitude weight
0~1 1
epochs
150
learning rate
[0.001, 0.005, 0.01]
weight decay
[0.001, 0.005, 0.01]
Irequency number
128, 1024, 2048, 4096]
pnums
[0. 1, 10, 15]
U B
lambda
[0, 0.2, 0.4, 0.6, 0.8, 1]
dropout ratio
[0.1, 0.25, 0.5]
Teature dimension
128
amplitude weight
0 0~1 1
epochs
150
learning rate
[0.001, 0.005, 0.01]
weight decay
[0.001, 0.005, 0.01]
frequency number
[32, 64, 128, 1024, 2048, 4096]
pnums
[0, 1, 10, 15]
U- E
lambda
[0, 0.2, 0.4, 0.6, 0.8. I]
dropout ratio
[0.1, 0.2, 0.25, 0.5]
feature dimension
128
amplitude weight
0~1 1
epochs
150
learning rate
[0.001, 0.005, 0.01]
weight decay
[0.001, 0.005, 0.01]
Irequency number
[32, 64, 128, 1024, 2048, 4096]
pnums
[0, 1, 10, 15]
B 3+0 U
lambda
[0, 0.2, 0.4, 0.6, 0.8. 1]
dropout ratio
[0.1, 0.25, 0.5)
feature dimension
128
amplitude weight
0 N~
epochs
150
Airport
learning rate
[0.001, 0.003, 0.005, 0.01]
weight decay
[0.001, 0.005, 0.01]
frequency number
[32, 64, 128, 1024, 2048, 4096]
pnums
[0, 1, 10, 15]
B ++0 E
lambda
[0, 0.2, 0.4, 0.6, 0.8, 1]
dropout ratio
[0.1, 0.25, 0.5]
feature dimension
128
amplitude weight
epochs
150
23

---

Published as a conference paper at ICLR 2026
Dataset
Transfer Scenarios
Hyperparameter
Search Space
learning rate
[0.001, 0.005, 0.01]
weight decay
[0.001, 0.005, 0.01]
frequency number
[32, 64, 128, 1024, 2048, 4096]
pnums
[0. I, 10, 15]
E-U
lambda
[0, 0.2, 0.4, 0.6, 0.8, 1]
dropout ratio
[0.1, 0.25, 0.5]
feature dimension
128
amplitude weight
0 0~
epochs
IS0
learning rate
[0.001, 0.003, 0.005, 0.01]
weight decay
[0.001, 0.005, 0.01]
frequency number
[32, 64, 128, 1024, 2048, 4096]
pnums
[0, 1, 10, 15]
E B
lambda
[0, 0.2, 0.4, 0.6, 0.8, 1]
dropout ratio
[0.1, 0.25, 0.5]
Teature dimension
128
amplitude weight
U
epochs
150
learning rate
[0.001, 0.005, 0.01]
weight decay
(0.001, 0.005, 0.01]
frequency number
[32, 64, 128, 1024, 2048, 4096]
pnums
[0, 1, 10, 15]
B1 + B2
lambda
[0. 0.2, 0.4, 0.6, 0.8, 1]
dropout ratio
[0.1, 0.25, 0.5)
Teature dimension
128
amplitude weight
0~1
epochs
150
Blog
learning rate
[0.001, 0.005, 0.01]
weight decay
[0.001, 0.005, 0.01]
frequency number
[32, 64, 128, 1024, 2048, 4096]
pnums
[0, 10, 15]
B2 + B1
lambda
[0. 0.2, 0.4, 0.6, 0.8, 1]
dropout ratio
[0.1, 0.25, 0.5]
feature dimension
128
amplitude weight
epochs
150
learning rate
[0.001, 0.005, 0.01]
weight decay
[0.001, 0.005, 0.0T]
frequency number
[32, 64, 128, 1024, 2048, 4096]
pnums
[0, 10, 15]
DE - EN
lambda
[0, 0.2, 0.4, 0.6, 0.8, 1]
dropout ratio
[0.1, 0.25, 0.5]
feature dimension
128
amplitude weight
epochs
150
Twitch
learning rate
[0.001, 0.005, 0.01]
weight decay
[0.001, 0.005, 0.01]
frequency number
[32, 64, 128, 1024, 2048, 4096]
pnums
[0, 10, 15]
EN + DE
lambda
[0, 0.2, 0.4, 0.6, 0.8, I]
dropout ratio
[0.1, 0.25, 0.5]
Teature dimension
128
amplitude weight 0~1
epochs
150
Table 7: Parameter search space list.
24


Reviews:
[{"review_id": "JkTEYZOSbN", "reviewer": "Reviewer_A6vr", "summary": "This paper introduced the ADAlign framework, a novel approach to handling composite\ndistribution shifts in GDA. By leveraging characteristic function in the complex Fourier domain, ADAlign dynamically identifies and aligns discriminative spectral components through minimax optimization, eliminating the need for manual reweighting or heuristic metrics. Extensive experiments across 16 benchmarks show that ADAlign consistently outperforms state-of-the-art methods, while also improving training stability and reducing computational overhead. These advantages make ADAlign a highly effective and practical solution for real-world graph transfer learning tasks.", "questions": "1. Regarding Weakness 1: Did the authors consider making the amplitude weight k adaptive?\n2. If you replace the GNN backbone with GCN or GAT, will the advantages of your model be offset?\n3. Could the authors visualize the learned frequency sampling distributions across different datasets?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Novel methodology: This paper introduces NSD, a novel parametric distance\nfor graphs that leverages neural characteristic function in the spectral domain to capture multi-level feature-structure dependencies, providing a unified approach to quantifying distributional shifts.\n2. Effective adaptive mechanism: This paper proposes an adaptive framework that automatically identifies and aligns the most relevant sources of discrepancy in each transfer scenario, enabling flexible and task-aware adaptation to dynamic graph shifts.\n3. Exceptional Empirical Performance and Significant Computational Efficiency: Extensive experiments in this paper show that ADAlign outperforms state-of-the-art baselines, significantly reducing memory consumption and training time.", "weaknesses": "1. The paper's core philosophy is adaptivity, yet the amplitude weight k, which balances amplitude and phase, is a hand-tuned hyperparameter. The analysis in Figure 6 shows an optimal range, but it's possible that it is also scenario-dependent. A fixed k seems slightly at odds with the \"adaptive\" goal.\n2. It would be helpful if the paper could further justify modeling the sampler as a normal scale mixture, and clarify whether simpler parameterizations such as a single Gaussian, a fixed finite mixture would already work in practice.\n3. The paper adopts the GNN encoder  F (Liu et al., 2024a) to map each graph into node-level embeddings,  but it remains unclear how sensitive ADAlign is to the choice of backbone such as GCN and GAT.\n4. There is a minor inconsistency in the wording. For example, the main text states ‘We adopt the GNN encoder F (Liu et al., 2024a),’ whereas Figure 2 describes the components as ‘each processed by a GCN.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "yn6j4aR2FE", "reviewer": "Reviewer_2AFp", "summary": "This paper proposes ADAlign, an adaptive distribution alignment framework for Graph Domain Adaptation to address complex distributional shifts between source and target graphs. It introduces Neural Spectral Discrepancy (NSD), a parametric distance leveraging neural characteristic functions in the spectral domain to capture multi-level feature-structure dependencies. Equipped with a learnable frequency sampler and minimax optimization, ADAlign automatically identifies and aligns discrepancies without manual criteria.", "questions": "See Weaknesses", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Eliminates manual specification of alignment criteria by dynamically prioritizing relevant spectral components for each transfer scenario.\n2. NSD integrates amplitude and phase differences for a comprehensive view of cross-graph shifts.\n3. Performs consistently well across domains.", "weaknesses": "1. Regarding the acceleration aspect, I understand the paper adopts the spectral method. However, this approach does not have low computational complexity. What methods are used for acceleration or approximation in the paper? It would be better if the authors could provide a proof of the complexity.\n2. The perspective proposed in the paper is good, but I am not very familiar with the statement in the introduction: \"these approaches often rely on heuristic strategies that first manually design graph filters to extract relevant features\" and feel that it is not the mainstream method in Graph Domain Adaptation. Proposing a model based on this point does not seem to be an appealing argument. Besides, it would also be helpful if the paper provides some descriptions of this pipeline to assist readers' understanding.\n3. The KL divergence in Figure 1 is used to measure the difference between features. Can this be regarded as assuming that the dataset satisfies Ps(Y|X) = Pt(Y|X)? Otherwise, merely exploring the difference between Ps(X) and Pt(X) seems irrelevant to transfer learning.\n4. There is a lack of experiments on large datasets. All datasets used here are too small, such as Brazil.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RFd6M2NPQV", "reviewer": "Reviewer_HQfr", "summary": "This paper introduces ADAlign, a framework for graph domain adaptation (GDA) that addresses complex distributional shifts between source and target graphs. It proposes the neural spectral discrepancy (NSD), which measures cross-graph gaps in the spectral domain using neural characteristic functions, and adaptively samples informative frequencies via a minimax optimization. The method is theoretically supported by PAC-Bayesian analysis and achieves state-of-the-art performance across 16 transfer scenarios with lower computational and memory costs.", "questions": "Look at the Weaknesses.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1. Figure 1 clearly illustrates the limitations of existing approaches, providing strong motivation for introducing adaptive alignment.\n2. The introduction of NSD based on neural characteristic functions is theoretically grounded. By performing adaptive alignment in the spectral domain, the method provides a more flexible framework.\n3. The experiments are comprehensive, covering 13 strong baselines across 16 transfer scenarios, with ablation studies and parameter sensitivity analyses supporting the method’s validity.\n4. ADAlign achieves significant improvements in computational efficiency, being faster and using less memory than baselines.", "weaknesses": "1. Although the paper explains that low-frequency captures large-scale structures and high-frequency captures fine-grained structures, it does not clearly clarify what specific graph structural characteristics the learned frequency distributions reflect.\n2. There is no analysis or experiment showing how replacing the GNN backbone affects the NSD-based alignment performance.\n3. While the paper provides a complexity analysis, the paper lacks experiments on the scalability of ADAlign to very large graphs or how its performance changes with graph size.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8eRhcKcqDW", "reviewer": "Reviewer_TwjK", "summary": "This paper introduces ADAlign, an adaptive distribution alignment framework for graph domain adaptation (GDA). The authors propose to address the challenge of complex, multi-faceted distributional shifts between source and target graphs by leveraging neural characteristic functions (CF) in the spectral domain. The key contribution is the Neural Spectral Discrepancy (NSD), which uses characteristic functions to capture feature-structure dependencies and employs a learnable frequency sampler to adaptively identify discriminative spectral components through minimax optimization. The framework is evaluated on 10 datasets across 16 transfer tasks, demonstrating superior performance compared to state-of-the-art baselines.", "questions": "See Weaknesses.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. **Clear Problem Formulation**: The paper effectively motivates the need for adaptive alignment by showing how dominant discrepancy sources vary across transfer scenarios (Figure 1), making a compelling case against fixed heuristic alignment strategies.\n\n2. **Theoretical Grounding**: The use of characteristic functions is theoretically well-motivated, with proper mathematical formulation including convergence and uniqueness theorems. The PAC-Bayesian analysis in Section 4.5 provides theoretical justification for the approach.\n\n3. **Comprehensive Experiments**: The experimental evaluation is extensive, covering 16 transfer scenarios across diverse graph domains (citation networks, airports, social networks) with consistent improvements over 15 baseline methods.", "weaknesses": "1. The paper fails to clearly distinguish its characteristic function-based approach from existing spectral alignment methods in GDA. Most critically, the relationship to SA-GDA (Pang et al., 2023), which also performs spectral augmentation for graph domain adaptation, is not discussed. SA-GDA already demonstrated that spectral domain alignment can effectively handle distribution shifts in graphs by augmenting the graph Laplacian spectrum. \n\n2. The paper emphasizes the amplitude-phase decomposition (Equation 7) but doesn't convincingly demonstrate why this complex-valued representation is necessary.\n\n3. The adaptive frequency sampler (Section 4.3) appears to be a standard importance sampling approach: The normal scale mixture formulation (Equation 10) is not well justified - why this specific parametrization? Minimax optimization can lead to instability, though the authors claim stability without showing the training dynamics of the frequency sampler itself—missing comparisons with simple fixed-frequency grids or other sampling strategies beyond the limited ablation in Section 5.4.\n\n4.  The baselines don't include recent spectral methods like SA-GDA (Pang et al., 2023) or other Fourier-based domain adaptation approaches. The significance tests (Appendix E.2) compare against only three baselines, not the full set. The visualization in Figure 10 is not particularly convincing - the claimed \"sharp boundaries\" in ADAlign's embeddings are not clearly superior to baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)