Disentangled Graph Spectral Domain Adaptation
Authors: Liang Yang
Xin Chen
Jiaming Zhuo
Di Jin
Chuan Wang
Xiaochun Cao
Zhen Wang
Yuanfang Guo
Venue: ICML 2025 poster (2025)
Keywords: graph neural networks
unsupervised domain adaptation
disentangled learning
spectral domain
Primary area: deep_learning->graph_neural_networks

Abstract: The distribution shifts and the scarcity of labels prevent graph learning methods, especially graph neural networks (GNNs), from generalizing across domains. Compared to Unsupervised Domain Adaptation (UDA) with embedding alignment, Unsupervised Graph Domain Adaptation (UGDA) becomes more challenging in light of the attribute and topology entanglement in the representation. Beyond embedding alignment, UGDA turns to topology alignment but is limited by the ability of the employed topology model and the estimation of pseudo labels. 
To alleviate this issue, this paper proposed a Disentangled Graph Spectral Domain adaptation (DGSDA) by disentangling attribute and topology alignments and directly aligning flexible graph spectral filters beyond topology. Specifically, Bernstein polynomial approximation, which mimics the behavior of the function to be approximated to a remarkable degree, is employed to capture complicated topology characteristics and avoid the expensive eigenvalue decomposition. Theoretical analysis reveals the tight GDA bound of DGSDA and the rationality of polynomial coefficient regularization. Quantitative and qualitative experiments justify the superiority of the proposed DGSDA.

Disentangled Graph Spectral Domain Adaptation
Liang Yang'
Xin Chen
Jiaming Zhuo' Di Jin? 2 Chuan Wang? Xiaochun Cao
Zhen Wang
Yuanfang Guo
Abstract
Model
The distribution shifts and the scarcity of la- bels prevent graph learning methods, especially graph neural networks (GNNs), from generaliz- ing across domains. Compared to Unsupervised Domain Adaptation (UDA) with embedding align- ment, Unsupervised Graph Domain Adaptation (UGDA) becomes more challenging in light of the attribute and topology entanglement in the representation. Beyond embedding alignment, UGDA turns to topology alignment but is limited by the ability of the employed topology model and the estimation of pseudo labels. To allevi- ate this issue, this paper proposed Disentangled Graph Spectral Domain adaptation (DGSDAA) by disentangling attribute and topology alignments and directly aligning flexible graph spectral filters beyond topology. Specifically, Bernstein polyno- mial approximation, which mimics the behavior of the function to be approximated to a remark- able degree, employed to capture complicated topology characteristics and avoid the expensive eigenvalue decomposition. Theoretical analysis reveals the tight GDA bound of DGSDA and the rationality of polynomial coefficient regulariza- tion. Quantitative and qualitative experiments justify the superiority of the proposed DGSDA.
Source
Attritune and apoolse Desompoosnn
(Topology)
Alignment
Source
Domain
(Supervised)
Attribute Alignment
aligned model
aligned attributes
Spectral Alignment
Target
Domain (Unsupervssed
Target
disentanglement
Figure 1. The proposed Disentangled Graph Spectral Domain Adaptation (DGSDA). The disentanglement between attrihute and model/topology alignments.
1. Introduction
Graphs are ubiquitous as language for modeling complex relational data in diverse fields, ranging from social net- works to traffic to the sciences (Hu et al., 2020; Li et al., 2022). Research on graphs has been conducted in many dis- ciplines, including graph theory in mathematics (Bondy Murty, 2008), network science in physics (Barabási, 2013), and graph representation Icarning in artificial intelligence (Cui et al., 2019; Guan et al., 2024; Zhuo et al., 2023; Fang et al., 2022). Unfortunately. their essential complex- ity makes labeling graphs more difficult and requires more domain knowledge than labeling images, text, and speech. Although the quest for the foundation model has come a long way in many fields (Bommasani et al., 2021), e.g., LM in NLP and CV (Zhao al., 2023), the distribution shifts and scarcity of labels prevent graph learning methods, es- pecially graph neural networks (GNNs), from generalizing across domains, and impede the design of graph foundation models (Li et al., 2024; Fan et al., 2024; Liu et al., 2024b).
'Hebei Province Key Laboratory Big Data Calculation, School of Artificial Intelligence, Hebei University of Technology, Tianjin. China "College of Intelligence and Computing. Tianjin University, Tianjin, China SShoool of Computer Science and Tech- nology, Beijing JiaoTTong University, Beijing, China "School of Cyber Seience and Technology, Shenzhen Campus of Sun Yat- sen University, Shenzhen, China 'School of Artificial Intelligence, OPtics and ElectroNics (iOPEN), School of Cybersecurity, North- western Polytechnical University, Xi' an, China "School of Com- puter Science and Engineering. Beihang University, Beijing, China. Correspondence to: Di Jin <jindi @twu.cdu.cn..........
Unsupervised graph learning methods have demonstrated the potential learn transferable representations without relying on labeled data (Yang et al., 2025; Zhuo al., 2024). However, these methods often assume a single- domain setting, overlooking cross-domain distribution shifts. Unsupervised Domain Adaptation (UDA) transfers knowl- edge from label-rich domains to unlabeled target domains with distribution discrepancies. Embedding alignment
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Disentangled Graph Spectral Domain Adaptation
widely adopted methodology in UDA (Liu et al., 2022), and many alignment strategies have been proposed (Ganin et al., 20;6b; Shen et al., 2018). Unsupervised Graph Do- main Adaptation (UGDA) introduces the UDA problem to the graph domain (Liu et al., 2024b; Shi et al., 2024), and embedding alignment strategies are employed to han- dle distribution shifts (Zhang et al., 2019; Wu et al., 2020; Pang et al., 2023). However, graph data contains two differ- ent types of information, i.e., topology and node attributes, both of which may suffer from distribution shifts (Liu et al., 2024b; Shi et al., 2024; Fang et al., 2025). Thus, UGDA becomes more challenging in light of the attribute and topol- ogy entanglement in the representation compared to UDA (Ma et al., 2019).
learned instead of the employed polynomial. Besides, the analysis also justifies the polynomial coefficient regulariza- tion in the model alignment loss.
The main contributions are summarized as follows:
We introduce a novel UGDA pipeline by disentangling attribute and topology alignments and replacing topol- ogy alignment with model alignment.
We propose a novel Disentangled Graph Spectral DA (DGSDA) by directly aligning spectral filters. DGSDA is end-to-end, parameter efficient, and possesses high expressive ability.
We present the tight DA bound for our DGSDA with Bernstein polynomials and justify the alignment loss.
Beyond drawing on UDA, UGDA turns to topology shift and alignment (Liu et al., 2023; 2024c), which is the problem specific to UGDA. The topology alignment needs additional topology structure models, such as Contextual Stochastic Block Models (CSBM) (Deshpande et al., 2018), to effi- ciently align topologies in source and target domains. Thus, the expressive ability of topology structure models is criti- cal for knowledge transfer among graph domains. Besides, robust topology alignment often relies on accurate pseudo- label estimation in the target domain. However, this is often difficult as label prediction is the core task in UDA. In con- clusion, the ability of the employed topology model and the accurate estimation of pseudo-labels hinder the quality of the topology alignment.
We conduct experiments to show the proposed UGSDA achieves new SOTA.
2. Related Work
Traditional domain adaptation approaches use intermediate representations to minimize domain discrepancy, which can be categorized into two main streams: methods that mini- mize pre-defined probability discrepancy metries (Gretton et al., 2012: Zellinger et al., 2017) and those that employ adversarial learning techniques (Ganin et al., 201 6a; Long et al., 2018; Tzeng et al., 2017). However, these methods are not appropriate for graph-structured data.
Recently, several approaches have been proposed to address the unique challenges of UGDA (Zhang et al., 2021; Wang et al., 2023; Shen et al., 2023; Cai et al., 2024; Huang et al., 2024). Notable methods include DANE (Zhang et al., 2019), which uses shared GCNs and a least square generative adver- sarial network. ACDNE (Shen et al., 2020) employs feature extractors and a domain classifier. UDAGCN (Wu et al., 2020) and AdaGCN (Dai et al., 2022) integrates graph con- volution with adversarial training for graph transfer learning. CoCo (Yin et al., 2023) and DREAM (Yin al., 2024) are designed for the classification of graph-level domain adapta- tion. These approaches inherit limitations from conventional domain adaptation: they predominantly address feature- level shifts while neglecting structural misalignment.
To alleviate this issue, this paper proposed a Disentangled Graph Spectral Domain adaptation (DGSDA) as shown in Fig. 1 by disentangling attribute and topology alignments and directly aligning complicated graph spectral filters be- yond topology. Firstly, DGSDA disentangles the UGDA into attribute alignment, which has been widely investi- gated in UDA, and topology alignment based on the aligned mode attribute. Secondly, based on the close relationship between topology and GNNs, especially spectral ones, topol- ogy alignment is converted to model alignment. The model alignment possesses the advantages of (1) end-to-end mod- cling, (2) parameter efficiency, and (3) benefit from a large amount of flexible GNNs. Thirdly, BernNet (He et al., 2021), which uses Bernstein polynomial approximation, adopted, and coefficients of the polynomial are aligned between source and target domains.
Beyond addressing feature distribution shifts, recent stud- ies explore structural adaptation strategies for graph do- main shifts. StruRW (Liu et al., 2023) develops edge reweighting mechanism that mitigates conditional neigh- borhood distribution shifts across domains. Extending this paradigm, PairAlign (Liu et al., 2024c) introduces a dual adaptation framework that simultaneously recalibrates node influence through adaptive edge weighting and counteracts label distribution mismatch via classification loss reweight- ing. JHGDA (Shi et al., 2023) designs multi-level pooling model to extracts hierarchical representations and compute
The final objective function is the combination of the losses for attribute alignment, model alignment, supervised error in the source domain, and clustering regularization in the target domain. Theoretical analysis demonstrates the tight DA bound for the proposed DGSDA. The Lipschitz contin- uous of Bernstein polynomial on mimicking the behavior of the function be approximated makes the spectral Lips- chitz constant determined by the ground truth function to be
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Disentangled Graph Spectral Domain Adaptation
domain discrepancies different levels. KBL (Bi et al., 2023) proposes an alternative paradigm via bridged-graph knowledge transfer. These approaches fundamentally cou- ple node representation learning with structural adaptation in a joint optimization framework. This tight integration risks entangling domain-invariant patterns with copology-specifiie artifacts, potentially amplifying spurious correlations.
are employed train the model g: G + )²
3.3. Spectral Graph Neural Networks
Let = UAU denote the eigen-decomposition of the symmetric normalized Laplacian matrix L, where U is the matrix of eigenveetors and A = diag[>1,... .., is the matrix of eigenvalues. The spectral filter the graph is
Beyond simply using GNNs as node embedding modules, emerging research systematically investigates their intrinsic properties for domain adaptation scenarios. SpecReg (You et al., 2023) establishes theoretical connections between optimal transport-based generalization bounds and GNN spectral characteristics, revealing that the Lipschitz continu- ity of graph filters fundamentally constrains cross-domain risk. A2GNN (Liu al., 2024a) finds that the propagation operation plays a pivotal role and proposes a simple GNN that stacks more propagation layers on the target branch.
([I))=U((AAU") x= Udiag Udiag[M(A))) (MA))...... ((N)) x;
(1)
where X, E [0. for i 1,,.,, N and x E R- stands for graph signal. Spectral graph neural networks aim to design and learn the mapping function h(L), or equivalently (A)). Different polynomial approximations are employed to fit h(A), such as Chebyshev polynomials (Defferrard et al., 2016; Kipf & Welling, 2017: He et al., 2022). Bernstein polynomials (He et al., 2021), and Jacobi polynomial (Wang & Zhang, 2022). Specformer (Bo et al., 2023) considers the set relationships between eigenvalues with Transformer.
3. Preliminaries
3.1. Notations
4. Method
A graph can be represented as G = (V, where and & are the sets of nodes and edges. N = | and M = [|| stand for the numbers of nodes and edges. The node attribute matrix, denoted by X = {x EV} E NNF, contains attribute vector x; for each node n4 where F represents the dimensionality of the attributes. Adjacency matrix of G is represented as A= = [ayy] E RNXN aij- = 1 holds if there is an edge etj E E connecting nodes v, and v/. and asj = 0 otherwise. (u;;) stands for the neighborhood of node u; D = [dx]] N denotes the degree matrix with diagonal element d; EnyeN((x) (V,, Oij as the degree of node U, P D AD is the normalized adjacency matrix. L- A and L I- I-D AD à represent the Laplacian matrix and its symmetric normalized version, where I stands for the identity matrix. Y E R.Nx xCC represents the node label matrix, where C' denotes the number of classes.
This section presents the Disentangled Graph Spectral Do- main Adaptation (DGSDA). Sections 4. and 4.2 elaborate two components Distribution Shift Disentanglement and Graph Spectral Domain Adaptation, respectively. Section 4.3 provides the overall objective function and algorithm followed by the theoretical justification as in Section 4.4.
4.1. Distribution Shift Disentanglement
Unsupervised Graph Domain Adaptation (UGDA) aims to align the embedding condition distributions ps PH I' (H to deal with the data shift (A, Y) * (A X with the feature extractor h h:: g - H. Since P(A, X = P(X|Y) (A|X, Y). the graph DA process can be disentangled into two steps: node attribute alignment w.r.t. IP P(X|Y) and topology alignment w.r.t. P(A(X, Y).
Node attribute alignment. Fortunately, there is much progress in the non-graph domain adaptation (Liu et al., 2022). They achieve 1S (Hx|Y) = p x" |Y) under scenarios of I YX * P with hx : X + H. where X stands for the collection of i.i.d. data and Hx for its representation. Thus, the node attribute can be aligned.
3.2. Problem Definition
Given a labeled source graph G = -VS, .3:9) and an unlabeled target graph G" = " with the data shift that P)gg)) * P( or equivalently P(A ((A5XX51 YS) + P(AT. X ) Superscripts $ and T stand for the source and target domains, respectively. Domain indica- tor can be placed on IP for simplicity, e.g. pd (" (|) = P Y)) for U E (S,T). Unsupervised Graph Do- main Adaptation (UGDA) is to seek a model g g + yr which can be generalized to tasks on unlabeled target graph g Model g g + )' consists of a feature extractor h g + H. and classifier f : H. + y. Here, the node- level classification task is mainly considered. Both the labeled source graph G and the unlabeled target graph S
Topology alignment. Therefore, the graph data shift can be simplified from (A, + P'(A, to PŠ Y) A PT (A[X, Y). Since the node attribute has been aligned in embedding space, node attribute divergence can be ignored and topology alignment can be further con- verted to the data shift scenarios of
S (A|Hx. Y) A ( Y).
(2)
Section 4.2 proposes a flexible topology alignment scheme
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Disentangled Graph Spectral Domain Adaptation
with aligned attribute embedding.
4.2.2. SPECTRAL FILTER ALIGNMENT
Benefiting from the above graph distribution shift disentan- glement, the overall framework of the proposed Disentan- gled Graph Spectral Domain Adaptation (DGSDA) is shown in Fig. 1. Note that DGSDA does not require additional final embedding alignment after topology alignment. Thus, overcomes the drawbacks of methods using entangled node embedding alignment.
Benefiting from the above advantages, highly expressive GNN is required to act as the backbone for GNN alignment. Here, spectral GNN is employed for its ability to capture and model diverse characteristics of graph signals, such as low- passing, high-passing, and bandit-passing. Unfortunately, as shown in Eq. (I), vanilla spectral GNNs are computa- tionally expensive due to the eigenvalue decomposition of the Laplacian matrix. Here, BemNeet (He et al., 2021) is adopted for its simplicity, efficiency, and theoretical support for learning arbitrary graph spectral filters.
4.2. Graph Spectral Domain Adaptation
This section proposes novel Graph Spectral Domain Adaptation (GSDA) algorithm achieve ps (H|Y) PT Y) under data shift scenarios of TPS (A|Hx, Y) + I" (A|HH, Y). The following two subsections show why and how to use GNNs alignment instead of topology one.
BernNet implements h(A) in Eq. (1) with K-order Bern- stein polynomial approximation on E [0, 1] as
K
(1 - -1) (K)
K
hx((1= Σov k=0
On (1)
K
K=U
4.2.1. GNNS ALIGNMENT
(4)
where (t) (t)==()))) (1- is the k-tth Bernstein base, and O2 = =((z) is the function value at K/K, which acts as the coefficient of by (l). Thus, by deflating the input to [0. 1J. the spectral GNN for signal x in Eq. (1) becomes
Note that the relationship between i.dd date feature x and feature extractor hx= X' > H is very different from that between graph topology A and graph learning function h: 9 + H. On the one hand, the feature i..d. data x only acts as the input to model lxx On the other hand, graph topology, represented as adjacency matrix A, is closely related to the graph learning scheme, such as the propagation scheme in GNNs. Taking spectral GNN as an example, graph topology determines the spectral space to filter data with the eigenvectors of its Laplacian matrix as U in Eq. (1). Therefore, different from existing methods (Liu al., 2023; 2024c), which perform topology structures alignment and apply GNNs on the aligned topology, this paper directly aligns the parameterized GNNs across divergent domains shown in Eq. (2):
Bx(A)x
Udiag[h ( (A1/2), ..... hx (An An
K 8k2k
Σ
(2I
K- KLL--X (5)
2k
k
where x represents general graph signal. Since the model alignment is based on the aligned node attributes embedding Hx as shown in Eq. (2), the graph signal in the proposed DGSDA is the aligned node attribute, i.c., the row of Hx.
It is proved that BernNet possesses many good properties (He et al., 2021). Firstly, for an arbitrary continuous filter function h [0, 2] + [0, 1]. the z in Eq. (5) can approx- imate h(L)x as K - co. Secondly, BernNet does NOT need expensive eigenvalue decomposition and is thus ef- ficient. Thirdly, BernNet exactly realizes existing filters, which are commonly used in GNNs by specifying 0₂ 's, such as Linear/Impulse low pass filters, Linear/Impulse high-pass filters, and Impulse band-pass filters. Intuitively, the bases (2I - L) I + and L correspond to smoothing and sharpening operations, respectively.
GNN (A. Hx)
align
GNN (A. Hx).
(3)
where 0 is the parameters related to the graph topology in GNN. The alignment of GNN is equivalent to the parameter alignment. It possesses the following three advantages:
End-to-end model alignment is optimal compared to the topology alignment. Model alignment is equivalent to jointly aligning the topology and selecting proper GNNs. On the contrary, topology alignment also re- quires choosing additional GNNs.
According to Eq. (3), the ht(')'s in Eq. (4) from source and target should be aligned. To this end, its parameters 0g 's which denote the responses to different frequencies, need to be aligned as the following loss term
Model alignment, i.e., parameter alignment, may be ef- ficient. Compared to the number of edges, which need to be aligned and adjusted, that of model parameters is often much smaller and independent of the graph size.
K of
Compared to topology structure models, a larger num- ber of GNNs exist. There are many flexible GNNs designed for different kinds of graphs ranging from homophilous to heterophilic ones. On the contrary, the topology structure model is rare.
Lutign Σ K=0
+
0o + =002)
(6)
k=0
where o and 8T. are the coefficients for source and target domains, respectively. The second term is to regularize the model parameters for adaptation as justified in Section 4.4.
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Disentangled Graph Spectral Domain Adaptation
4.3. Objective Function and Algorithm
where the (empirical) source and target risks are
(9, N= >n lg(GG) g(Gn)) and e" (g 9) = ETT(G) - ( respectively, where 9 : 9 + y is the labeling function for graphs and
As shown in Fig. I, the overall Disentangled Graph Spectral Domain Adaptation (DGSDA) consists of four components: source domain encoder, target domain encoder, node at- tribute alignment, and model alignment. Thus, the overall objective function is as follows
min w Is SCrlBILA, {="(9.0) (9.9) + E (g.9)))) (10)
Leource + Cologggn + mmmmd + LLarge:: (7)
Unfortunately, it isn't easy to instantiate the GNN Lipschitz constant, since the distance metric (G G2) often requires computationally expensive graph matching. As in (You et al., 2023), the numerator, i.e. (lg(G,) - 10ggg9G (G2)ll2 which is related to the GNN stability, is estimated. Recall that K- K-order Bernstein polynomial in Eq. (5) is employed as the graph encoder. One important property of Bernstein polynomials is that they mimic the behavior of the function to be approximated to a remarkable degree. The following theorem formally demonstrates this attractive property.
where Lsource (Eq. (11)), Latign (Eq, (6)), Lmmdd (Eq. (12)) and Ltarget (Eq. (13)) correspond to source domain classification loss, spectral alignment loss, maximum mean discrepancy loss, and target domain unsupervised loss. at 3, and 7 are hyper-parameters to balance these terms. Refer to the Appendix A for a detailed description of each item in the formula.
4.4. Theoretical Analysis
This section provides the domain adaptation (DA) bound of the proposed DGSDA by following the DA bound frame- work for graph-structured data in (You et al., 2023), which extends the DA bound for ii.d. data in (Redko et al., 2017). For clarity, the definition of Lipschitz continuous is as fol- lows.
Theorem 4.3. if f E LIPc. then its K-order Bernstein polynomial approximation hgg (t) for all K 2 1 defined in Eq. (4) with 02 f(x)) belong to LI LIPc,, also.
The proof is given in Appendix B.1. With the above the- orem, (l g(G2)lla can be estimated follows. Given VG1, G2 with size N and L1 A, U,,,U,,, L2 U2 the eigenvalue decomposition for Laplacian matrices L and L2 that A1 = diag (AA..I....AA.NII) A2 = diag ((A2,1.... .2..1.)), with eigenvalues sorted in the descending order. The proposed DGSDA is constructed by composing graph filter Bx (A) in Eq. (5) and non- linear mapping that g (G1) = (Bx X,
Definition 4.1. Given two metric spaces (X,dx) and dy), where dxX denotes the metric on the set X and dY is the metric on set y function ʃ : X )/ is called Lipschitz continuous of μ order, denoted as f E Lc, if there exists real constant C' 2 0 and 0 < I such that, for all i and 22 in X,
(8) Any such C is referred to as a Lipschitz constant for the function f, and may also be referred to as C-Liscchitz for the case of the order ja =1
dy f(22) I Cdx x2),
K
(U1hg (A1) U W) where hg b (t)
R=O
is the polynomial function in Eq. (4) that Bx (A1)
K
Σ O (21- L,)
W e DDD' is the learn-
With the definition of Lipschitz continuous, the DA bound for graph data can be generally expressed as follows.
able weight matrix, and the pointwise nonlinearity holds as (b) (a)) < lb - a). Ya, b E R.
Theorem 4.2. (You et al., 2023) Let's assume that the learned discriminator f is Cy-Lipcchitz continuous where [((X.)-fiZ2l
Theorem 4.4. Suppose the Bernstein polynomial hgg ap- proximate the ground truth one f with 02 = = and
the Lipschitz norm [|f[lzip maxz,,2, p(2,.72) C holds for some distance function p. and the graph fea- ture extractor h (also referred to as GNN) is Lipshhitt that [|A||esp = ma [b(G.)))(G22l== = Cr. for some G2))
X < 1 and ||ll < 1 where || llop stands for oper- ator norm, the following inequality holds:
(Gi( - 9 (G2)
graph distance measure 1j. Let := {g G + y} be the set of bounded real-valued fiancton with the
s <cx (1+
||A - -P'A,P'
Fdim(FF= = d that g = f fo E F, with probability at least 1 - 8 the following inequality holds:
+ +0 (IAA P'A2P- TIE)
max 11/Ks (A2)IH - PP'X2llp
c" (g.9) S (g.
where T = (IIU - U2l1 +12 - 1 stands for the eigenvector misalignment which can be bounded, P = argminpen {|[x} - PXxllF + ||A - PPA.P* 1 is the set of permutation matrices,
4d
eevs d
+ V Ns
In
+
NS
7
+ 2C,CaWi (PS PT (G)) + w, (9)
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co. The Lipschitz. constant of the proposed DGSDA can be estimated as Disentangled Graph Spectral Domain Adaptation
O - [1:)) bounded multipliers, and Ox is the Lipschitz constant of f that Vdd, ^j. (A.) - (A₂)) S Cx 1Ai - ->yl-
is the remainder term with
Citationv| (C) and DBLPv7 (D)), social interactions (Blog- Catalog and Twitch-DE/EN), and transportation systems (Airport: Brazil (B), Europe (E) and USA (U)). Refer to Appendix C for dataset details.
The proof is given in Appendix B.2. Note that Theo- rem 4.4 is different from the Lemma in (You et al,, 2023). In Lemma I in (You et al., 2023), the Lipschitz constant Cx determined by the basic polynomial function, i.e., VY,, AG. Bx (2:) Bx (>₂)1 V Cx lA; - A3] where Bx (A) E== SLA* is the common polynomials. In Theorem 4.4, the Lipschitz constant Cs is determined by the ground truth function f, since the attractive property of Bernstein polynomial in Theorem 4.3.
Baselines. The baselines for comparison can be divided into three categories. (1) Source-only methods, including vanilla GCN (Kipf & Welling. 2017) and GAT (Velickovic et al., 2018). These methods are trained only on the source graph and directly applied to target graph for evaluation. (2) GDA methods using node embedding, containing DANE (Zhang et al., 2019), UDAGCN (Wu et al., 2020) and AdaCGN (Dai et al., 2022). These methods use node embedding to solve the graph domain adaptation problem. (3) GDA methods tailored for graph structure shift, including StruRw (Liu et al., 2023), JHGDA (Shi et al., 2023), KBL (Bi al., 2023) and PairAlign (Liu et al., 2024c). (4) Graph domain method tailored for propagation: A2GNN (Liu et al., 2024a). These models are analyzed in the Related works.
Theorem 4.5. Let's define the matching dis- tance between G2 as (G1 G2) minpen {|| - -X + || || A -PA,P~TIP) [p].
Suppose that the edge perturbation G2, ||| A Pp " S E with the optimal permutation P and there exists an eigenvalue X E R to achieve the maximum l (X*) <
bounded that
Configurations. For reproducibility, the detailed settings of the experiments are described below. All experiments are performed on Nvidia GeForce RTX 3090 (24GB). Our proposed model DGSDA! is implemented with PyTorch (Paszke al., 2017) and PyTorch Geometric library (Fey & Lenssen, 2019). To be fair, we use the source code provided by the authors for each baseline and fine-tune the hyperpa- rameters to achieve optimal values. In all the experiments, we use the Adam optimizer. We run all models five times on each dataset, and the mean accuracy is used as the metric.
Cj = max =mmxx
(1 + eK2, 1hg (X'),,
where K1, K2 is the supremes of (1 (1+ (1+TVN) TVN) and the re-
mainder multiplier in Theorem 4.4.
The proof is given in Appendix B.3. According to Eq. (9) the gap between the target and source errors is bounded with two terms:
H Hyperparameters. For hyperparameter settings, The node representation dimension is selected from { {128, 256}. The learning rate is tuned from {0.01, 0.005, 0.001, 0.0005), weight decay is tuned from (0.0005, 0.005, 0.01] and K is selected from (5, 8, 10, 15).
Term Wi (G), (G)) which captures the distri- bution divergence between the source and target multi- plied by Lipschitz constants Cy;
Term w in Eq. (10) which models the discriminative capability of model to capture invariant knowledge
5.2. Result Analysis
restricted by Lipschitz constants.
Citation Network. The experimental results on citation networks are presented in Tab. 1. From these results, two key observations can be made. First, models tailored for graph structure shifts consistently outperform models that align only node representations on all datasets. This indi- cates the significance of graph structure in graph domain adaptation, which fits the observations that the cross-domain differences often manifest in both citation patterns (struc- ture) and textual features (attribute). More importantly, the proposed DGSDA achieves superior performances over baseline models, particularly, models tailored for graph structure shifts, across all citation networks. To be specific, DGSDA achieves performance improvements of 6.33% over the second-best baseline JHGDA on the (D-C) datasets. This reveals that the decoupled modeling allows mutually
Thus, varying the Lipschitz constant may balance between domain-divergenee and discriminability to vary the DA bound. To tighten the DA bound in Eq. (9) can be imple- mented by regularizing the Lipschitz constant Cy. Recall that C2 is determined by the ground truth function to ap- proximate instead of the polynomial, and thus CA is fixed. Therefore, only [y (A*) need to be regularized. To this end, the absolute values of Bernstein polynomial coeffi- cients, i.e., [6,|'s 'ss are regularized as shown in the second terms in Eq. (6).
5. Experiments
5.1. Experimental Setup
Datasets. The experiment utilizes three types of bench- mark datasets: Citation networks (AmetMiner: ACMv9 (A),
Our code is available at https//gtthb..cm///ccrrrrr////ss
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Disentangled Graph Spectral Domain Adaptation
Table 1. Node classification performance on citation network. The metric is mean accuracy (%) and standard deviation. The best and the second best results are highlighted in bold and underfined, respectively.
METHOD
A-C C
C- A
D A
C+D D
GCN
76.13:2..51
68. 68.52:0..3
68.80=0.22
62.42+0.40
72.97=0.17
72.53-00.25
GAT
74.45=+0.4
68.1540.41
69.43=0.76
62.11=0.70
72.52+0.38
71.65+0.62
DANE
63.80-10.13
65.39100.4
63.98-0.28
60.2810.78
71.291:..44 65.9810.36
UDAGCN
72.06++0.15
70.28+0.17
70.93-0.53
63.42+0.67
73.03+0.19
71.15+0.11
ADAGCN
74.28+00.28
67.8910.34
67.83=0.85
60.52:0..2
0.883+..89
71.98+t0.44
STRURW
77.35+00.16
01.23+0.16
73.51=0.28
65.1940.16
74.33+0.21
74,28+0.16
JHGDA
77.28:1..3
73.72:0.28
73.13=0.28
69.8010.21
74.2510.43
76.59:10334 33
KBL
78.2110.36
71.4910.16
69.28-0.37
63.45:10.28
73.9240.14 74.9620.25
PAIRALIGN
73.76:10.13
70.25
09.18=0.17
62.8810.67
19.
72.35-L0.36
A2GNN
80.68+0.98
05.1240.54
76.13=0.39
73.48+00.88
76.15=0.73
79.89++0.36
DGSDA
83.57+0.22
0..54+0.28
76.90 0.51
74.07+0.56
0.28
02.92+0.15
Table Node classification performance social network. The metric is mean accuracy (%) and standard deviation. The best and the second best results are bold and underlined, respectively.
mains, resulting in the model overly adapting to the dom- inant hub and failing to generalize the whole structure. Anyway, DGSDA still performs better than vanilla GNN- based methods and methods only using node embedding (i.e., UDAGCN, and DANE).
METHOD
DE-EN
EN-+DE
B1 B1-+B2
B2-+B)
GCN
04.42+0.89
01.00=0.30
40.66=4.98 40.15-33.46 28.5515.67
GAT
54.56:10.01
59.3211.28
24.48:15.57
5.3. Effectiveness Study
DANE
$1.84::0.16
$1.69
32.17:2.08
01.86=t0.33
UDAGCN
58.23:t0.61
0.26
33.51
34.16=2.118
To validate the Bersttii polynomial-based topological alignment mechanism, the frequency response curves of the source and target domain filters are visualized as shown in Fig. 2. The x-axis denotes normalized graph signal fre- queney À (A=0 for low-frequency, X=2 for high-frequency components), and the y-axis represents filter gain h(A), Red/blue colors indicate target/source domains, with solid lines denoting filters jointly optimized under method and dashed lines showing single-domain baselines.
ADAGCN
57.18.10.89
58.01 10.67
4..03
37.6911.18
STRURW
58,27=0.188
19:19:0.95 0.39
41.53==0.96
JHGDA
56.50:10.41
63.1710.111
20.89=2.26
23.86=4.18
KBL
58.89:10.38
60.5410.16
35.6312.89
14.89-1.22
PAIRÁLIGN
56.56=t0.28
58.75+2.27
39.83=8.61
45.18-3.67
A2GNN
56.52+0.338 24.5 0.85
16
DGSDA
59.81 0.286+0.22 65.64=1.98
43.99-44.34
non non-disruptive alignment processes for citation networks, where attributes and structures represent complementary se- mantic hierarchies. The disentanglement strategy inherently respects their distinct roles in cross-domain transfer.
From Fig. 2, the following three key observations can made: (1) Independently trained single-domain filters (dashed lines) exhibit consistent trend patterns, revealing structural similarities in underlying spectral distributions across domains. This provides the prerequisite for feasible spectral alignment. (2) The jointly optimized filters (solid lines) demonstrate nearly identical response curves, verify- ing our method's capability to align cross-domain spectral distributions. (3) Optimized filter responses lic between single-domaie baselines while being closer to the target distribution. This suggests that our model preserves infor- mation from the source domain while adaptively adjusting to target spectral characteristics, achieving bidirectional alignment rather than unilateral transfer.
Social Network. Similar conclusions regarding perfor- mance advantages can be drawn from the experiment re- sults on social networks. It can be observed from Tab. 2 that the proposed DGSDA has performance advantages on three of the four datasets, which demonstrates its effective- ness and universality. It is worth noting that on the Blog dataset, DGSDA achieves performance gain of over 17% compared to the second-best baselines. This is mainly at- tributed to the adaptivity of the learnable spectral filters used in DGSDA to graphs with different homophily.
Traffic Network. The experimental results on the airport traffic network datasets are shown in Tab. It can be ob- served that the proposed DGSDA outperforms the baselines on domain adaptation tasks between Brazil and USA, well as between Europe and USA, which highlights the su- periority of DGSDA. While DGSDA demonstrates strong performance in many scenarios, the domain adaptation task between Brazil and Europe presents unique challenges. It can be partially attributed to the limited scale of both do-
5.4. Ablation Study
This experiment aims to evaluate the contribution of each constraint. In Fig. 3, Base denotes the variant model using only source cross-entropy loss Lisorree DGSDA - A and DGSDA - AM represent the variant model that sequen- tially increases Latign and Lmmas respectively. It can be
7

---

Disentangled Graph Spectral Domain Adaptation
Table 3. Node classification performance on traffic network. The metric is mean accuracy (%) and standard deviation. The best and the second best results are highlighted in bold and underlined, respectively.
METHOD
B-+ U
U-
R-
U-+ E
B
GCN
43.28 18
49.49-0.48
45.01=0.98
47.3240.89
44.21+2.05
55.88-3.39
GAT
42.69 98
30.81+3.87
40.92=4.48
38.10#2.21
42.14 34.79+2.29
DANE
41.78.1..99
40.4411.01
32.38-2.36
33.87:10.29
33.0311.299
41.98:10.22
UDAGCN
34.49+2.118
36.78+2.866
41.26=0.78
41.06++0.18
43.23+1.72
42.33+1.88
ADAGCN
44.0411.118
35.3243.18
47.89=3.18
46.89:12.36
1..06+1.21
60.28
STRURW
52.01+33.21
60.01+0.69
48.89=3.67
52.31=1008
62.10++..2 53.23+0.13
JHGDA
51.46:13.85
60.7413.24
49.03=2.96
50.8313.36
54.3915.76
70.12:16.78
KBL
36.6911.63
35.2740.75
44.28=0.63
45.36:10.29
53.03-0..3 46.8940.96
PAIRALIGN
41.76:11.93
37.8611..13
44.38=0069
45.68:1.009
44.8910.18 18
53.13
A2GNN
51,18+1.38
54.47422.67
47.63-2.18
50.47+1.90 196
50.3343.20
59.98+1.99 1.93
DGSDA
53.18+0.81
61.07+ 1.60
49.73
0.20
51.23+0.19
0.99
61.37++4.3
1.6
source
sounce
1.5
source
1,5
source
target
urget
target
target
1.2
Training
1.2
Triining C
Teaining
Training
h(A)
Traiting
h(2)
Training
h(2)
1.0
Traiting
1.0
Training A
0.8
0.8
0.5
0.5
0.4
0.4
0.0
0.0
2
(a)
A-C
(b)
C-A
(c)
A-D
(d) D-*A
Figure 2. Spectral alignment verification via Bernstein polynomial filters. Red/bluc colors indicate target/source domains, with solid lincs
denoting filters jointly optimized under DGSDA and dashed lines showing singlc-domain basclines.
Bau
LKHIDA-M DGSDA
Utec DISSDA-A
DGSDDA-M ENSSDA
84
S0
DCSDA-A
Accuracy(5o
Accumcyy (%)
AccuracyC
80
80-
80
Accuracy
54
76
70
7
72
A-+L
C-+
D-+A
68
A-C
C-+A
C
D
Figure 3. Ablation studies on citation networks.
Figure 4. The sensitivity of polynomial order K.
observed that additional constraints are incorporated, the performance of the models steadily improves. This trend underscores that the enhanced performance is result of the synergistic effect of all the constraints. Moreover, it is particularly noteworthy that the introduction of the pro- posed model alignment loss significantly boosts the model's performance, thereby highlighting its effectiveness.
80
A-CC
C-A
A-D
D-
65 0.1
B
5.5. Hyperparameter Analysis
Figure 5. The sensitivity of attribute alignment weight B.
These experiments are performed to offer an intuitive under- standing for selecting hyper-parameter parameters (including polyno- mial order K and attribute alignment weight :3).
this hyperparameter K.
Polynomial Order K. It can be observed from Fig. that DGSDA exhibits stable performances with respect to the hyperparameter K in the range K 2 5, with variations re- maining within 2%. This indicates that the learnable spectral filter has expressive spectral patterns at this stage, enabling to effectively adapt to these domain transformations. Moreover, this illustrates that DGSDA is not sensitive to
Attribute alignment weight B. From the observations in Fig. 5, it is clear that DGSDA achieve approximately consistent performances across range of parameter choices. This suggests that DGSDA is not sensitive to the parameter B. Therefore, we need not pay excessive attention to the specific values of the above two parameters. See Appendix C for a full analysis of the remaining hyperparameters.
8

---

Disentangled Graph Spectral Domain Adaptation
6. Conclusions
of foundation models. arXiv preprini arXiv: 10Xi::2008.07258 2021.
Unsupervised Graph Domain Adaptation (UGDA) faces inherent challenges due to entangled attribute-topology dis- tribution shifts and reliance on fragile pseudo-labels. This work presents Disentangled Graph Spectral Domain Adap- tation (DGSDA), which addresses these limitations through spectral filter alignment and distribution shift decoupling. By disentangling attribute embeddings from topology via Bernstein polynomial- -ased spectral filters, DGSDA cir- cumvents error-prone topology alignment and pseudo-label estimation. The theoretical analysis further establishes that regularizing polynomial coefficients tightens the domain adaptation bound by constraining the Lipschitz continuity of spectral filters. Extensive experiments on various graphs demonstrate the superior performance of DGSDA.
Bondy, J. A. and Murty, U. S. R. Graph theory. 2008.
Cai, R., Wu, F., Li, Z., Wei, P., Yi, L., and Zhang, K. Graph domain adaptation: A generative view. ACM Trans. Knowl. Discov. Data, 18(3):60: 1-:0244, 2024.
Cui, P., Wang, X., Pei, and Zhu, W. A survey on network embedding. IEEE Trans. Knowl. Data Eng. 31()):833- 852, 2019.
Dai, Q.. Wu, X.-M., Xiao, J., Shen, X., and Wang. D. Graph transfer learning via adversarial domain adaptation with graph convolution. IEEE Transactions Knowledge and Data Engineering, (0::))998-49222 2022.
Impact Statement
Defferarrd, M., Bresson, X., and Vandergheynst, P. Con- volutional neural networks on graphs with fast localized spectral filtering. In NIPS, pp. 3837-3845, 2016.
This paper presents work whose goal is advance the field of Machine learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
Deshpande, Y., Sen, S., Montanari, A., and Mossel, E. Con- textual stochastic block models. In NeurIPS, pp. 8590- 8602, 2018
Acknowledgments
Fan, S., Wang, X., Shi, C.,, Cui, P., and Wang, B. Generaliz- ing graph neural networks ofitt--diitributtin graphs. IEEE Trans. Pattern Anal. Mach. Intell., 46(14 46(1):322-3377 2024.
This work was supported in part by the National Natural Science Foundation of China (No. U22B2036, 62376088, 62272020, 62025604, 92370111. 11, 62272340, 62261 136549), in part by the Hebei Natural Science Foundation (No. F2024202047), in part by the National Science Fund for Distinguished Young Scholarship (No. 62025602), in part by the Hebei Yanzhao Golden Platform Talent Gather- ing Programme Core Talent Project (Education Platform) (HJZDD20200)), in part by the Post-graduate's 's Innovation Fund Project of Hebei Province (CXZZBS2025036), in part by the Tencent Foundation, and in part by the XPLORER PRIZE.
Fang, R., Wen, L., Kang, Z., and Liu, J. Structure-preserving graph representation learning. In 2022 IEEE Interna- tional Conference on Data Mining (ICDM), pp. 927-932. IEEE, 2022.
Fang, R., Li, B., Kang, Z., Zeng, Q., Dashtbayaz, N. H., Pu, R., Wang, B., and Ling, C. On the benefits of attribute- driven graph domain adaptation. In ICLR, 2025.
Fey, M. and Lenssen, J. E. Fast graph representation learning with pytorch geometric. arXiv preprint arXiv:1903.0242: 2019.
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Disentangled Graph Spectral Domain Adaptation
A. Objective Function Details.
this section, detailed description about the objective function Eq. (7) are provided. Each of the loss terms in the overall objective function focuses on different objectives.
Source domain classification losses. The Lsource term constitutes the fundamental supervised learning component of our framework. Formulated cross-entropy loss over labeled source domain data, it directly optimizes the model's prediction accuracy through:
N
Louurce
Di.e log Piçe
(11)
N
Spectral alignment losses. Lalign focuses on aligning spectral coefficients between the source and target domains, which provided in Eq. (6).
Maximum mean discrepancy loss. Maximum mean discrepancy (MMD) is widely-used for non-graph DA (Gretton et al., 2012), aiming to align feature representations to reduce distribution differences. Here is the specific formula.
N S
N N le (
N N
mmmd
Σ
k (H H + (NTY²
ΣΣ j=1
ΣΣ l (H.
HJ)
(12)
(NS
NSNT
Target domain unsupervised loss. It promotes model adaptation to the target domain through unsupervised learning:
N
1 NT
Ltarget
log d:
(13)
where ii denotes the predicted labels in target domain.
B. Proofs for Theorems
B.1. Proof for Theorem 4.3
Theorem 4.3 f E Lip for all 2 1, Br(f) E Li also.
ProofLes æ S 222 be any two points of (0, 1]. We need show that
[Bx(fxx2) - Bn(fizi)) < A(x2 - x1)",
given that f satisfies Eq. (8). From Eq. (4),
B, (f:
(1 x22
(x) + (x2 x1))²'
9
3=0
x2 nー m 1
1
#f (x2 x2 2
1
nlat (xz - ) (1 x22) nー
kl(j k))(n2 j))
0=
On inverting the order of summation and writing & + = j.then
n!
(k+l
B, (f:
klll(n - k 1)! o (xz 21)² x (1 - xz)"~=k1
(14)
0 1
12

---

Disentangled Graph Spectral Domain Adaptation
We now construct a similar double sum for B, (f; Again, from Eq. (4), we have
Bn (J;01) =
n2
k
f
k
((02 - x1) + (1 x2)) x2))== n-k
k
n2
k= 0
a
Fi
k
f
k
n - k) )
(x2 x1) (1 x2) m²)
n
k=
(15)
11 n-k
n!
k - 1)! " (x2 - xz)
k!ll(nn
0=
k
x (1 - 222= f
72
On subtracting Eq. (15) from Eq. (14), we have
Bn x2) Ba (f;z1)
n n-k
klll(n n - 1)! J (I2 -2)= (1-))(11100)) x2)" 122
n!
k= 1=0
k ++
- f k }
x
n-k
n!!
k (x2
m2) (1 x2) (1-22) n-
(1) "
klll(n
k- 1)!
n
k=
1=0
on using Eq. (8),
11
(22 21) n!
(1) 11
Σ
n-1
n
o (1 x2)) )2)-m
[n(n- 1)!
1=0
==
(1) a
A
(122 22)²
(x1 1- -22))
n
1=0
AB, (a": 822 21). by Eq.,
& A (x22 21 )"
Thus we see that Bf) E Li, where A is the Lipschitz constant of f so that the theorem is proved.
B.2. Proof for Theorem 4.4
Theorem 4,4. Suppose the Bernstein polynomial hx approximate the ground truth one f with 02 = =( and ||||po S 1 and ||llop < 1 where || - llop stands for operator norm, the following inequality holds:
llg (G (G2)ll Sc (1+TVN) |A, - -P'AAp-
+0 (0 A1 - -P'APP" TI(T) + max {lAk (A2)l) [X, - -P'X2llF
lp +12 1 stands for the eigenvector misalignment which can be bounded, P argminpen (||X, PX ||A, TI II is the set of permutation matrices, O (IA P' TI(? is the remainder term with bounded multipliers, and CA is the Lipschitz constant of f 'hht VA ,, - S Cs |A; - AAyl-
where 5 (||U, U2l1p
13

---

Disentangled Graph Spectral Domain Adaptation
Proof. Denote the optimal permutation matrix for G1, G2 P we compute the difference of the GNN outputs:
(G1) - y (G2))l2 [Bx (A1) X W) a (Bx (A2) X.
||o (A1) XIW) W) (BK (P' A2 P'X,
[B (A1) X1 W BR (P*A,P'T) P'
||ilop Bl| (Ai) X1 - BR (P" (2'APP"T) X1 + Bx (P" 22 ) X, - -B (P" 12 P'X2 ()
Wlop X X1llop BB (A1) BK (P*A2P"T) 1| + Wo BB (P'A2P" |X1 P"X211F
BK (A1) - Bx (P*A2P*T) T) max (hx (A2)l) [ - -P
Cx (1+TVN) | I' +0 ( P"A,pP'TIP)
- (1:)) + max (lh (A2)l) |X - P'X21FF
where (a) is due to the permutation invariance property of graph filters; (b) is achieved with the triangle inequality and the assumption (b(b) {a) I bb al, Va, b E I; (c) and (d) the fact that for any two matrices A,B, |AB|| F < min[ op B || [ llp)). and (c) further applies the triangle inequality; (e) adopts the assumption |X op < 1 and [ lpp S which in practice can be guaranteed with normalization, and easily extended to the case with||X] lap I [ llop S KK 0, and because By (P* A2P*T) = (P'U2)hx (A2) can be diagonalized, its operator norms equal the spectral radius; (f) is the direct outcome borrowed from (Gama al., 2020) Theorem 1. In common case, Cx is the Lipschitz constant of Bx that Vλ,, >j. [B (A,) Bx (A₃) S Cx [>; - Ayl- According Theorem 4.3, Bernstein polynomial possesses the same Lipschitz constant as the function be approximated, i.e.. f. Therefore, the Cy is the Lipschitz constant of f that that Vd,, If (A:) (A,)] $ Cx [As Ay/- The proof is completed.
B.3. Proof for Theorem 4.5
Theorem 4.5. Let's define the matching distance between G1, G2
(G1, G2)
mimpen { [X| PX2/| + A PA.P- [pp.
Suppose that the edge perturbation is bounded that F S with the optimal permutation P and there exists an eigenvalue X E
YG, A₁ PP*ApP
achieve the maximum [/g (X+) co. The Lipschitz constant of the proposed DGSDA can be estimated
Cj max {CxKı + =K2, hh (A")),
where K1, K2 is the supremes of (1 (1+TVN) and the remainder multiplier in Theorem 4.4.
To calculate the Lipschitz constant Cf w.r.t the matching distance, based upon Lemma 1, we assure the following inequality:
Proof.
lig (G²) - (G2)ll SCs (1+T/NG)) - -h-PPAAp +0 +O(IA: - P'' + |h (A*)1 P'X2llF, <Cnn(G,,
the latter inequality of which can be rewritten
(cx (1+ ||A P' || 1.000
- -P''pP-T 11 -Cyl|A, - -P'AAP'T lF)
+ +(hx (2-) C,)[|x |X, 1 P'Xalle $0
which is necessary for:
Cx 21++TVN) ||A 1I
(1+PV))IA-PPPPP*)++++ +00 - -C, (IIK (A*)I -,011-))) | - P'X2
-P'A.P"T 50
- $5
14

---

Disentangled Graph Spectral Domain Adaptation
which is equivalent to:
2 CxKi +ekg
C 2 1/g (X*)
The bounding of K1, K2 follows (Gama et al., 2020) Theorem 1 and the first minimum solution can be calculated from the quadratic function w.r.t. the edge matching distance || P P'ApP11> Let Cj takes the larger value between them, complete the proof.
C. Experiments Details.
C.1. Dataset Details
In this section, detailed description about the datasets used in our experiments are provided.
Table Dataset Statisties.
DATASET
#NODES
#EDGES
#LABELS
#HOMO
ACMv9
9360
31112
0.7998
CITATIONVI
8935
30196
0.8598
DBLPV7
5484
16234
0.8189
BLOGI
2300
66942
0.3991
BLOG2
2896
107672
0.4002
ENGLAND
7126
35324
0.5560
GERMANY
9498
153138
0.6322
BRAZIL
131
2148
0.4683
EUROPE
399
11990
0.4048
USA
1190
27198
0.6978
ArnetMiner These datasets are three citation networks obtain from AreeMinner (Dai et al., 2022); ACMv9 (A), Citationv I (C), DBLPv7 (D). Nodes are papers, while edges represent citations between papers. The objective is to classify all papers into five distinct research domains: Artificial Intelligence, Computer Vision, Databases, Information Security, and Networking.
Blog These datasets are derived from the BlogCatalog dataset (Shen et al., 2020). In these datasets, each node symbolizes blogger, while the edges denote the friendships among bloggers. The objective is to forecast the group affiliations of these bloggers.
Twitch These datasets are Twitch gamer networks from six regions (Liu et al., 2024a): Germany (DE), England (EN), Spain (ES), France (FR), Portugal (PT), and Russia (RU). Nodes are users, while connections signify friendships among them. In this situation, users are divided into two groups depending on whether they use explicit language. Among these datasets, we pay more attention the two largest datasts, Germany dataset (DE) and England datasets (EN).
Airport These datasets are airport traffic networks from three countries (Ribeiro al., 2017); Brazil (B), Europe (E) and USA (U). Within these datasets, nodes stand for airports, and edges indicate flight links between the airports. The labels classify airports based their activity levels, measured in the number of flights passengers.
C.2. Detailed Hyperparameters.
C.2.1. HYPERPARAMETER
This section provides additional insights into hyperparameters and 7, As shown in Fig. 6, demonstrates strong robustness with less than 1 1% fluctuation across weight initialization schemes. For y, empirical studies reveal 0.05 achieves optimal performance, so experiments consequently adopt this fixed value.
C.2.2. HYPERPARAMETER K
This section extends the analysis K (Bernstein polynomial order) to low-homophily social networks. Experimental results reveal two patterns: (1) Heterophilic graphs require substantially larger (e.g., K>15 vs. K>5 for homophilic
15

---

Disentangled Graph Spectral Domain Adaptation
K3
74 KeaKommay
Acoracyy
39
80
A-C
21
76
50
C-D D-C
41
Blogi-Blogg Blog? Blog2-Dlogg
D--A
0.1 0.4
0.7
65
0,7
n.
0.4
0.7
32
14
201
a
Figure The sensitivity a.
Figure 7. The sensitivity of K.
graphs) to reach saturation accuracy, potentially due to complex neighborhood patterns demanding higher-order filtering; (2)
Performance stabilizes when K2 15, indicating sufficient signal modeling capacity at this threshold.
D. More Experiments.
D.1. Pseudo-labels verification.
This section verifies the feasibility of using predicted pseudo-labels the target domain. A variant model named DGSDA+PL is introduced, which combines pseudo-labels of the target domain. The compared results reveal that pseudo- labels consistently lead to performance degradation in all domain adaptation scenarios, demonstrating the infcasibility of the mentioned scheme.
Table 5. Pseudo-labels experiment on citation network. The metric is mean accuracy (%) and standard deviation.
C
D
+
D -
DGSDA 83.5710.22 PG 81.23+22.5
78.54 74.40+2.22
76.90:0.51 75.36+22.77
74.07 71.36 1.33
78.3810.28 77.03+1.04
82.92:1..15 79.45+1.49
This is primarily due the low reliability of the pseudo-labels generated the early stages of training, which can cause error accumulation in learning processes, and the noise amplification effect in graph neural networks, where erroneous pseudo- labels propagate through pessage-passingg mechanisms. This is also the reason why the proposed method outperforms topology alignment with pseudo-labels.
D.2. Disentanglement effectiveness validation.
To clarify the sources of performance improvement, this experiment introduced variant of DGSDA that uses Bernstein polynomials and directly aligns node representations instead of separately aligning attributes and topology. As demonstrated in Tab. 6, this variant consistently underperforms compared to the full DGSDA model, which suggests that the explicit disentanglement of attributes and topology plays crucial role in enhancing model effectiveness.
Table 6. Disentanglement experimeetteeeeeeeeeeememmm citation network. The metric
accuracy (%) and standard deviation.
A -
C++A
A +DD D
D + A
C D
D - C
DGSDA
83.57+0.22
75.54:0.28
76.90+0.51 73.2540.22
74.07=0.56 72.03=0.24
78.3840.28 76.32+0.17
82.92+0.15 80.25+0.21
VARIANT MODEL
81.01=0.32
73. 19
D.3. Topology pattern capture capability study.
This section verifies that DGSDA able to capture the correct topological patterns even in unsupervised scenarios. In the experiment, the supervised variant of DGSDA employs the supervised loss from 10% labeled data in the target domain, replacing the unsupervised loss: the spectral alignment loss and the entropy loss. The results are shown in Tab. 6.
The results indicate that the unsupervised DGSDA achieves comparable performance to the supervised version, highlighting the effectiveness of the unsupervised losses. This can be attributed to two key factors. First, by regularizing the coefficients of Bernstein polynomials (in Eq. (4)), the method explicitly aligns the spectral filters across different domains. This alignment enables the target filters inherit topology-awan aware patterns from the source domain, even without labels. Second, the entropy
16

---

Disentangled Graph Spectral Domain Adaptation
Table 7. Topology pattern capture capability experiment on citation network. The metric is mean accuracy (%) and standard deviation.
A + C
C A
A - D
D - A
C- D
D -> C
DGSDA
83.57++0.2 03.20
75.54+0.28 76.37+2.75
76.90+0.51 79.49+0.56
74.07+0.56
78.38 +.228
82.92-0.15 03.03
77.10+0.83
80.16+0.66
unsupervised
1,5
unsupervised
1,5
unsupervised
supervised
supervised
supervised
1.2
b(2)
1.0
bC.)
1.0
0.5
0.2
0.0
0.0
2.
2
(a) A-+C
(b) A-+D
(c) C->D
unsupervised
unsupervised
unsupervised supervised
1.5
1.5
supervised
supervised
1.2
h(2)
1.0
b(2)
1.0
0.5
0.5
0.2
0.0 0
0.0
(d) C-+A
(c) D-+A
(f) D-+C
Figure 8. Comparison of filter curves in unsupervised and supervised scenarios.
loss sharpens cluster assignments, which implicitly encourages the model to learn more discriminative topological features.
Moreover, from the observations in Fig. 8, the learned filter curves are similar in both scenarios, which also suggests that the target domain parameters capture consistent topological patterns in the unsupervised scenario as well.
17


Reviews:
[{"review_id": "jOyWtTrJRX", "reviewer": "Reviewer_er9T", "summary": "This paper proposes Disentangled Graph Spectral Domain Adaptation (DGSDA) to alleviate the inaccuracies of pseudo-labels and the limited expressive ability of graph encoders to capture rich topology information. It decomposes the attribute and topology alignments and replaces the topology alignment with the powerful model alignment. To harness the parameter efficiency of spectral GNNs, the Bernstein polynomial is employed, and the polynomial coefficients are aligned. Theoretical analysis shows its rationality and superiority compared to existing ones. Experimental experiments also justify the claims.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1. The symbols, especially the theory part, are too complex to read.\n\nR1. Thanks for your feedback. We will thoroughly review and modify all the symbols to make them easier to read. \n\n--- \n\n> Q2. Some explanations should clarify the theoretical differences from [You et al., 2023].\n\nR2. The theoretical differences between this paper and the mentioned work [You et al., 2023] is  **Polynomial Choice & Lipschitz Properties**: The proposed DGSDA adopts Bernstein polynomials, whose Lipschitz constant $C_{\\lambda}$ is determined by the ground-truth function (in Theorem 4.3), rather than being restricted by the basic polynomial coefficients as in the work [You et al., 2023]. This allows more flexible and accurate spectral domain adaptation. \n\n--- \n\n> Q3. It is better to provide the source code to enhance reproducibility.\n\nR3. According to your suggestion, the source code has been made available at (https://anonymous.4open.science/r/DGSDA) for verification purposes.\n\n---\n\n> Q4. Why is the performance of the proposed method lower than that of JHGDA on the traffic network dataset? \n\nR4. The proposed method generally outperforms JHGDA on most tasks in the traffic network dataset and is less effective than JHGDA on the $B → E$ and $E → B$ tasks. The performance weakness may be attributed to overfitting due to limited training data. The Brazil and Europe datasets contain a relatively small number of nodes, which makes the proposed models with multiple constraints more prone to over-capturing the patterns of individual hub nodes. This, in turn, makes it difficult to effectively generalize to the overall structure of the target domain. Nonetheless, the extensive results illustrate the effectiveness of the proposed method. \n\n---\n\n> Q5. The legend in Figure 2 is not clear. What are the differences between source/target and A/C.\n\nR5.  The solid lines represent the filter curves trained on the domain adaptation tasks, while the dashed lines represent the filter curves obtained by training BernNet only on the corresponding datasets. Thus, “source” and “target” in the legend mean the filter curves of source domain encoder and target domain encoder trained on the domain adaptation tasks; “A” and “C” in the legend denote the filter curves of BernNet on the A and C datasets, respectively. We will change them to “Training on A” or “Training on C” in the revised manuscript to enhance readability. \n\n---\n\n> Q6. Why are the results in Figure 2 continuous curves?\n\nR6. Figure 2 shows the learned polynomials instead of their coefficients, and thus contain continuous curves. The x-axis in Figure 2 denotes the normalized graph signal frequency $\\lambda$, and the y-axis represents filter gain $h(\\lambda)$. They are independent of the polynomial order $K$ and coefficients $\\theta_k$. The Bernstein polynomial is defined as $h(\\lambda)=\\sum_{k=0}^K\\theta_kb^K_k(\\frac{\\lambda}{2})$, where $b^K_k(t)$ is the Bernstein basis function. For any given $\\lambda$, it is first mapped to the [0, 1] interval (i.e., $\\frac{\\lambda}{2}$), and $h(\\lambda)$ is calculated using the learned coefficients $\\theta_k$ and the corresponding Bernstein basis functions $b^K_k$. Thus, by sampling sufficiently many $\\lambda$ values, we can plot the continuous curves shown in Figure 2.", "claims_and_evidence": "The rationality and superiority of the proposed DGSDA are supported by both theoretical and experimental evidence. It is clear and convincing.", "methods_and_evaluation_criteria": "The proposed DGSDA makes sense and is novel by decomposing attribute and topology alignments. The experimental evaluations are reasonable with widely-sed criteria.", "theoretical_claims": "The correctness of theorems is checked, as well as their proofs in the appendix. However, the symbols are very complex.", "experimental_designs_or_analyses": "The soundness of the experiments is checked. The design is based on widely-employed datasets and criteria. The performances are verified on varying datasets. The ablation study and hyper-parameter analysis are conducted.", "relation_to_broader_scientific_literature": "UGDA is an important topic in the graph machine learning field. Previous work focuses on the employment of DA methods in i.i.d. data. This paper is along the line of topology alignment. It alleviates the issue of pseudo-label inaccuracy by adopting spectral model alignment beyond the topology one. Therefore, it is novel. Besides, the decomposition of topology and attribute alignment are also novel and interesting.", "essential_references_not_discussed": "The references are sufficient.", "other_strengths_and_weaknesses": "**Strengths**\n\nThe motivations are interesting and make sense. \n\nThe proposed method is novel and solid.  \n\nThe theoretical justification is rigorous. \n\nThe experimental evaluations are convincing. \n\n**Weakness**\n\nThe symbols, especially the theory part, are too complex to read.", "other_comments_or_suggestions": "1. Some explanations should clarify the theoretical differences from [You et al., 2023].\n2. It is better to provide the source code to enhance reproducibility.", "questions_for_authors": "1. Why is the performance of the proposed method lower than that of JHGDA on the traffic network dataset?\n\n2. The legend in Figure 2 is not clear. What are the differences between source/target and A/C.\n\n3. Why are the results in Figure 2 continuous curves? I think they should be discrete values.", "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": "2CLMgOi6kA", "reviewer": "Reviewer_SNRK", "summary": "This paper introduces a novel pipeline for unsupervised graph domain adaptation by disentangling attribute and topology alignments by considering that attribute alignment has been widely investigated. Based on the aligned node attribute, the topology alignment is converted to the model alignment by taking into consideration the widely developed GNN models. Then, the Bernstein polynomial is employed as the backbone for its approximation property and spectral perspective. Theoretical analysis and experimental evaluations justify the pipeline and proposed models.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1. The main weakness is the lack of source code.\n\nR1. The source code has been made available at (https://anonymous.4open.science/r/DGSDA) for verification purposes. We promise to make the code public once this paper is accepted.\n\n---\n\n> Q2. Whether the predicted pseudo-labels on the target domain can be employed?\n\nR2. To answer your valuable question, we have conducted experiments to verify the feasibility of using predicted pseudo-labels on the target domain. This experiment introduces a variant model named DGSDA+PL, which combines pseudo-labels of the target domain. The compared results reveal that pseudo-labels consistently lead to performance degradation in all domain adaptation scenarios, demonstrating the infeasibility of the mentioned scheme.\n\n|     | A→C | C→A | A→D | D→A | C→D | D→C |\n| --- | --- | --- | --- | --- | --- | --- |\n| DGSDA | 83.57$\\pm$0.22 | 75.54$\\pm$0.28 | 76.90$\\pm$0.51 | 74.07$\\pm$0.56 | 78.38$\\pm$0.28 | 82.92$\\pm$0.15 |\n| DGSDA+PL | 81.23$\\pm$2.52 | 74.40$\\pm$2.22 | 75.36$\\pm$2.37 | 71.16$\\pm$1.33 | 77.03$\\pm$1.04 | 79.45$\\pm$1.49 |\n\nThis is primarily due to the low reliability of the pseudo-labels generated in the early stages of training, which can cause error accumulation in learning processes, and the noise amplification effect in graph neural networks, where erroneous pseudo-labels propagate through message-passing mechanisms. This is also the reason why the proposed method outperforms topology alignment with pseudo-labels.\n\n---\n\n> Q3. Does the decomposition strategy possess solid theoretical findings?\n\nR3. We acknowledge that the current analysis focuses on demonstrating the **feasibility** of disentanglement without providing a precise error bound between the entangled and disentangled representations. This is a common limitation in the graph disentanglement field, where theoretical guarantees are still lacking. We will strive to address this in future work.\n\n---\n\n> Q4. Unsupervised graph domain adaptation is often composed of multiple terms as the objective function, and thus, training it is difficult to balance the impacts of terms. Can these terms be unified to felicitate the training?\n\nR4. We understand your concern for the stability of training. Unfortunately, these terms cannot be unified, as each of the loss terms focuses on different objectives as other GDA methods. To be specific, $L_{source}$ targets minimizing prediction error in the source domain, ensuring effective training on labeled source data. $L_{align}$ focuses on aligning spectral coefficients between the source and target domains. $L_{mmd} $aims to align feature representations to reduce distribution differences. $L_{target}$ promotes model adaptation to the target domain through unsupervised learning. Moreover, the experiments in the hyper-parameter analysis demonstrated that our model is relatively robust to the hyper-parameters used for weighting these terms. We will explore integrating these loss terms in future work to facilitate easier balancing.", "claims_and_evidence": "The correctness of the introduced pipeline is verified by the derivation from the Bayesian theorem. The replacement of topology alignment with model alignment makes sense due to the connection between topology and GNN models. Theoretical analysis and experiments demonstrate the statements.", "methods_and_evaluation_criteria": "As shown in the previous section, I think both the pipeline and the proposed GNN alignment make sense. Besides, the employment of these two strategies reduces the requirement of the pseudo label in the topology alignment, which often relies on the accurate estimation of the node membership.", "theoretical_claims": "I cursorily examined the proof of the theorems in the appendix and believe they are correct.", "experimental_designs_or_analyses": "The experiments are extensive, including quantitative and qualitative analyses. The setting is widely used in this field, and the baselines are recently proposed competitive ones. Thus, the experimental evaluations are convincing.", "relation_to_broader_scientific_literature": "Unsupervised graph domain adaptation is a critical topic in graph learning for the graph foundation model design. Although there exist GNN-based methods, as reviewed in the related work section,  this paper gives a novel methodology by both decomposition and model parameter alignment. This is more powerful and efficient compared to the existing ones by considering the spectral perspective.", "essential_references_not_discussed": "Sufficient. It covers most existing competitive SOTA and important milestones.", "other_strengths_and_weaknesses": "This paper possesses high originality, which may inspire the following variants on model alignment. The claims are justified with a rigorous theory investigation and extensive experiments. The main weakness is the lack of source code. Since the proposed method is flexible and complicated with four terms as the objective function, it is necessary to provide source code to make the read easy to get the implementation details.", "other_comments_or_suggestions": "None", "questions_for_authors": "Although the authors claim the use of model alignment avoids the requirements of pseudo-labels, I wonder whether pseudo-labels also benefit the model alignment since the polynomial coefficients can also be learned from labels as supervision. Whether the predicted pseudo-labels on the target domain can be employed?\n\nThe theoretical analysis focuses on the Bernstein spectral GNN alignment. Does the decomposition strategy possess solid theoretical findings?\n\nUnsupervised graph domain adaptation is often composed of multiple terms as the objective function, and thus, training it is difficult to balance the impacts of terms. Can these terms be unified to felicitate the training?", "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": "iHAh55n3Zh", "reviewer": "Reviewer_Fybq", "summary": "This study addresses the challenge of unsupervised graph domain adaptation in scenarios involving distribution shifts and missing labels by proposing a novel solution that disentangles the distribution shift. Specifically, the method DGSDA refines the topology alignment into GNN alignment and incorporates spectral filter alignment loss.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1. The title appears to be somewhat ambiguous. The title does not reflect the focus on the **unsupervised** problem in graph domain adaptation, which is a key aspect of the study.\n\nR1. Thank you for pointing this out. The primary focus of this paper is indeed on the unsupervised problem in graph domain adaptation. While our architecture can also accommodate target labels when available, the unsupervised scenario remains our main emphasis. We will consider revising the title to better reflect this focus.\n\n---\n\n> Q2. The notation T used in the paper is not clear. For example, X^T represents the node attribute matrix of the target domain, but it can also be interpreted as the transpose of the node attribute matrix.\n\nR2. Thanks for your careful check. We will correct $X^T$ to $X^{\\top}$ to denote the transpose of the matrix.\n\n---\n\n> Q3. The description of the proposed method lacks clarity. While $L_{source}$ and $L_{mmd}$ are described in words, they are not accompanied by clear, formal formulations.\n\nR3. The formal formulations of $L_{source}$ and $L_{mmd}$ are presented as follows:\n\n$\\mathcal{L}\\_{source} = -\\frac{1}{N^{{S}}} \\sum\\_{i=1}^{N^{{S}}} \\sum\\_{c=1}^{C} y\\_{i,c} \\log p\\_{i,c}$\n\n$\\mathcal{L}\\_{mmd} = \\frac{1}{\\left(N^{{S}}\\right)^2} \\sum\\_{i=1}^{N^{{S}}} \\sum\\_{j=1}^{N^{{S}}} k\\left(H\\_i^{{S}}, H\\_j^{{S}}\\right) + \\frac{1}{\\left(N^{{T}}\\right)^2} \\sum\\_{i=1}^{N^{T}} \\sum\\_{j=1}^{N^{T}} k\\left(H\\_i^{T}, H\\_j^{T}\\right) - \\frac{2}{N^{{S}} N^{T}} \\sum\\_{i=1}^{N^{{S}}} \\sum\\_{j=1}^{N^{T}} k\\left(H\\_i^{{S}}, H\\_j^{T}\\right) $\n\nwhere $k(·,·)$ represents the kernel function.\n\nWe will add them to the appendix to enhance the clarity of the manuscript.\n\n---\n\n> Q4. In Section 4.1 Distribution Shift Disentanglement, \"Topology alignment in the graph data shift can be simplified from $P^{S}(A,X| Y) \\neq P^{T}(A,X| Y)$ to $P^{S}(A| X,Y) = P^{T}(A| X,Y)$\" seems to be misstatement. The correct statement should be $P^{S}(A| X,Y) \\neq P^{T}(A| X,Y)$.\n\nR4. Thanks for pointing out this important detail. We will correct this formula in the revised manuscript and ensure that all related discussions are consistent with this accurate representation.", "claims_and_evidence": "This paper provides a comprehensive evaluation of the DGSDA model, supported by both theoretical analysis and extensive experiments, thereby effectively demonstrating its efficacy. Thus, the claims made in the paper are well-substantiated by clear and convincing evidence.", "methods_and_evaluation_criteria": "The proposed method involves (1) disentangling embedding alignment into topology and attribute alignments and (2) exploiting alignments of filter parameters to flexibly implement topology alignment, both of which make sense for the graph-domain adaptation problem.", "theoretical_claims": "After a detailed examination of the proof, I have essentially confirmed its correctness.", "experimental_designs_or_analyses": "I examined all the experimental designs and analyses in Section 5 and Section B, and believe it effectively demonstrates the properties of the proposed model.", "relation_to_broader_scientific_literature": "Current graph domain adaption works focus on proposing topology alignment strategies, including aligning the edge distributions of two domains using the CSBM. DGSDA utilizes a new filter alignment to improve flexibility.", "essential_references_not_discussed": "No other related works that are essential to understanding the (context for) key contributions need to be discussed or cited.", "other_strengths_and_weaknesses": "**Strengths**\n\n1）The idea of GNN alignment is interesting. \n2）The method is simple yet has solid theoretical support.\n\n**Weaknesses** \n\n1) The title appears to be somewhat ambiguous. The title does not reflect the focus on the **unsupervised** problem in graph domain adaptation, which is a key aspect of the study.\n\n2) The notation $T$ used in the paper is not clear. For example, $\\mathbf{X}^{T}$ represents the node attribute matrix of the target domain, but it can also be interpreted as the transpose of the node attribute matrix. \n\n3) The description of the proposed method lacks clarity. While $L_{source}$ and $L_{mmd}$ are described in words, they are not accompanied by clear, formal formulations.", "other_comments_or_suggestions": "1) In Section 4.1 Distribution Shift Disentanglement, “Topology alignment中“the graph data shift can be simplified from $P^S(A, X|Y) \\neq P^T (A, X|Y)$ to $P^S(A|X, Y) = P^T (A|X, Y)$” seems to be misstatement. The correct statement should be $P^S(A|X, Y) \\neq P^T (A|X, Y)$.", "questions_for_authors": "See Weaknesses.", "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": "TdZX05U9dd", "reviewer": "Reviewer_fWcK", "summary": "To break away from the attribute and topology entanglement on Unsupervised Domain Adaptation (UDA), this paper introduces a novel method, DGSDA, directly aligning complicated graph spectral filters. This paper conducts experiments on various types of graph datasets to demonstrate the effectiveness of DGSDA.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Q1. The authors fail to clearly attribute the source of performance improvement in their method. Further empirical results are needed to isolate the specific role of disentanglement.\n\nR1. To address your concerns, we have conducted an additional experiment to clearly identify the source of performance improvement. This experiment introduced a variant of DGSDA that uses Bernstein polynomials and directly aligns node representations instead of separately aligning attributes and topology. The variant model's performance is consistently worse than that of our full DGSDA, as shown in the following table. This indicates that the disentanglement play a crucial role in enhancing the performance of our model.\n\n|     | A→C | C→A | A→D | D→A | C→D | D→C |\n| --- | --- | --- | --- | --- | --- | --- |\n| DGSDA | 83.57$\\pm$0.22 | 75.54$\\pm$0.28 | 76.90$\\pm$0.51 | 74.07$\\pm$0.56 | 78.38$\\pm$0.28 | 82.92$\\pm$0.15 |\n| variant model | 81.01$\\pm$0.32 | 73.25$\\pm$0.22 | 73.15$\\pm$0.19 | 72.03$\\pm$0.24 | 76.32$\\pm$0.17 | 80.25$\\pm$0.21 |\n\n---\n\n> Q2. Given the lack of labels in the target domain, it is unclear how the model ensures that the target domain parameters accurately capture the topological patterns.\n\nR2. We have conducted two additional experiments to demonstrate that the unsupervised loss can provide effects similar to supervised loss in terms of model parameter optimization. In the experiment, the supervised variant of DGSDA employs the supervised loss from 10% labeled data in the target domain, replacing the unsupervised loss: the spectral alignment loss and the entropy loss. The results are shown below.\n\n|     | A→C | C→A | A→D | D→A | C→D | D→C |\n| --- | --- | --- | --- | --- | --- | --- |\n| DGSDA | 83.57$\\pm$0.22 | 75.54$\\pm$0.28 | 76.90$\\pm$0.51 | 74.07$\\pm$0.56 | 78.38$\\pm$0.28 | 82.92$\\pm$0.15 |\n| DGSDA (supervised) | 83.20$\\pm$0.52 | 76.37$\\pm$2.75 | 79.49$\\pm$0.56 | 77.10$\\pm$0.83 | 80.16$\\pm$0.65 | 83.03$\\pm$0.48 |\n\nThe results indicate that the unsupervised DGSDA achieves comparable performance to the supervised version, highlighting the effectiveness of the unsupervised losses. This can be attributed to two key factors. First, by regularizing the coefficients of Bernstein polynomials (in Eq. 6), the method explicitly aligns the spectral filters across different domains. This alignment enables the target filters to inherit topology-aware patterns from the source domain, even without labels. Second, the entropy loss sharpens cluster assignments, which implicitly encourages the model to learn more discriminative topological features.\n\nIn addition, we have compared the learned filter curves in both labeled and unlabeled target domains. The results can be found at https://anonymous.4open.science/r/DGSDA/figure/DC.png. In the supervised learning case, the model parameters capture the homophily topological pattern, characterized by increasing low-frequency information and suppressing high-frequency information. Similarly, in the unsupervised learning case, the target domain parameters can capture the same topological pattern.\n\n---\n\n> Q3. Some equations, such as Eq. (4) and Eq. (9), as well as Theorem 4.4, appear to be slightly misaligned. It is recommended to adjust the line breaks for better formatting.\n\nR3. We will adjust the line breaks in Eq. (4), Eq. (9), and Theorem 4.4 to ensure proper alignment and improve the overall formatting.\n\n---\n\n> Q4. Expression error on line 182 and spelling mistake of model \"PairAlign\".\n\nR4. We will perform a thorough review of the manuscript to correct any expression errors and spelling mistakes.", "claims_and_evidence": "Yes, the claims made in the submission are supported by clear and convincing evidence from both experimental and theoretical aspects.", "methods_and_evaluation_criteria": "The proposed DGSDA is well aligned with the problem of UGDA. The disentanglement of attribute and topology alignments, the use of spectral filter alignment, and the comprehensive experiments on diverse datasets collectively demonstrate its effectiveness.", "theoretical_claims": "Yes, I have checked the correctness of the proofs for the theoretical claims presented in the paper. Specifically, I have verified the proofs for Theorems 4.3, 4.4, and 4.5, which are central to the theoretical analysis of DGSDA.", "experimental_designs_or_analyses": "Yes, I have checked the soundness of the experimental designs (including the compared methods and experimental setups) and analyses.", "relation_to_broader_scientific_literature": "The key contributions are directly related to the broader literature by proposing disentanglement techniques to solve the problems in traditional UGDA. It leverages recent advancements in spectral GNNs and builds on theoretical foundations of Lipschitz continuity and model alignment.", "essential_references_not_discussed": "No, there are no essential related works missing in the paper that need further discussion.", "other_strengths_and_weaknesses": "**Strengths**\n\n1) The paper is well-written and easy to follow.\n2) Experimental results on benchmark datasets are provided.\n\n**Weaknesses**\n\nThe authors fail to clearly attribute the source of performance improvement in their method. First, DGSDA employs Bernstein polynomials, which are not commonly used in comparison methods. This choice alone may contribute to the performance gains, making it unclear how much of the improvement is due to the disentanglement strategy itself. Further empirical results are needed to isolate the specific role of disentanglement. Additionally, given the lack of labels in the target domain, it is unclear how the model ensures that the target domain parameters accurately capture the topological patterns.", "other_comments_or_suggestions": "1) Some equations, such as Eq. (4) and Eq. (9), as well as Theorem 4.4, appear to be slightly misaligned. It is recommended to adjust the line breaks for better formatting.\n\n2) On line 182: It possesses the following **three** advantages instead of **two**.\n\n3) The name \"PairAlign\" is misspelled as \"PariAlign\" in Tables 1 and 2.", "questions_for_authors": "Refer to Weaknesses.", "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:

Meta-review: **Summary:** This paper proposes DGSDA, a new method for unsupervised graph domain adaptation (UGDA) that disentangles attribute and topology alignment and introduces spectral filter alignment via Bernstein polynomials. The approach avoids reliance on pseudo-labels for topology modeling and instead aligns model parameters directly in the spectral domain. The paper presents theoretical insights (including a bound and analysis of Lipschitz continuity) and extensive experiments across multiple benchmark datasets, demonstrating the effectiveness of DGSDA. The rebuttal included further clarifications, new ablation results, and additional supervised comparisons.

**Decision:** All reviewers gave accept recommendations and highlighted the novelty, technical soundness, and practical value of the proposed approach. The method offers a fresh perspective by replacing traditional topology alignment with spectral model alignment, and the use of Bernstein polynomials is well-motivated both theoretically and empirically. While some concerns were raised—such as equation formatting, title clarity, and disentanglement attribution—the authors addressed them comprehensively in the rebuttal. In particular, they provided empirical comparisons to isolate the role of disentanglement and demonstrated that the unsupervised objective is competitive with a supervised variant. Overall, the paper is well-written, the contributions are meaningful, and the empirical validation is thorough. Therefore, I recommend acceptance.
Decision: Accept (poster)