Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
Authors: Shuo Wang
Bokui Wang
Zhixiang Shen
Boyan Deng
zhao kang
Venue: ICML 2025 poster (2025)
Keywords: Graph Foundation Model; Multi-domain Pre-training；Graph Transfer learning; Graph Domain Generalization
Primary area: deep_learning->graph_neural_networks
TL;DR: We propose MDGFM, a foundation model that robustly transfers knowledge across diverse graph domains by aligning their topologies and adapting via prompting.

Abstract: Recent advances in CV and NLP have inspired researchers to develop general-purpose graph foundation models through pre-training across diverse domains. However, a fundamental challenge arises from the substantial differences in graph topologies across domains. Additionally, real-world graphs are often sparse and prone to noisy connections and adversarial attacks. To address these issues, we propose the Multi-Domain Graph Foundation Model (MDGFM), a unified framework that aligns and leverages cross-domain topological information to facilitate robust knowledge transfer. MDGFM bridges different domains by adaptively balancing features and topology while refining original graphs to eliminate noise and align topological structures. To further enhance knowledge transfer, we introduce an efficient prompt-tuning approach. By aligning topologies, MDGFM not only improves multi-domain pre-training but also enables robust knowledge transfer to unseen domains. Theoretical analyses provide guarantees of MDGFM's effectiveness and domain generalization capabilities. Extensive experiments on both homophilic and heterophilic graph datasets validate the robustness and efficacy of our method.

Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
Shuo Wang" Bokui Wang"!
Zhixiang Shen
Boyan Deng'
Zhao Kang +
Abstract
ics (Zhang al., 2021). Inspired by recent advances in CV and NLP (Vidit et al., 2023; Cheng et al., 2024), researchers have sought to develop general-purpose graph foundation models. In particular, multi-domain graph pre-training has gained significant attention for its ability to integrate knowl- edge from various domains and enable effective transfer learning (Yu et al., 2025; Zhang al., 2024b). This ap- proach is viewed critical milestone toward the creation of truly general-purpos. graph models.
Recent advances in CV and NLP have inspired searchers to develop general-purpose graph foun- dation models through pre-training across di- verse domains. However, a fundamental chal- lenge arises from the substantial differences in graph topologies across domains. Additionally, real-world graphs are often sparse and prone to noisy connections and adversarial attacks. To ad- dress these issues, we propose the Multi-Domain Graph Foundation Model (MDGFM), unified framework that aligns and leverages cross-domain topological information to facilitate effective and robust knowledge transfer. MDGFM bridges different domains by adaptively balancing fea- tures and topology while refining original graphs to eliminate noise and align topological struc- tures. To further enhance knowledge transfer, we introduce an efficient prompt-tuning approach. By aligning topologies, MDGFM not only im- proves multi-domain pre-training but also en- ables robust knowledge transfer to unseen do- mains. Theoretical analyses provide guarantees of MDGFM's isss effectiveness and domain generaliza- tion capabilities. Extensive experiments homophilic and heterophilic graph datasets val- idate the robustness and efficacy of our method. Our code is available at https: / tiuh.. com/wbkzww wwekkkwgttw///Dnn ZW/MDGFM.
Despite significant progress, a substantial gap remains in fully understanding the richness and diversity of graph struc- tural knowledge. Existing methods primarily rely on fixed graph topologies and apply uniform encoding mechanisms across all domains (Zhao et al., 2024; Yu et al., 2025), which severely limits their generalizability across diverse domains (Zhang et al., 2024). In this paper, we revisit the multi-domain graph foundation model from a structural perspective and address a critical challenge: topology align- ment across different domains. Achieving effective topology alignment a non-trivial task, presenting two primary chal- lenges that must be overcome advance toward truly Seneral-purpone graph model.
First, structural knowledge across domains often exhibits significant semantic differences. For instance, citation net- works predominantly display highly homophilic patterns, whereas social networks and webpage link graphs frequently contain numerous heterophilic edges (Zheng et al., 2022: Shen and Kang, 2025; Xie al., 2025). These differences necessitate Nomain-speciff adaptations of encoding mech- anisms (Li et al., 2024; Pan and Kang, 2023). As a result, there is an urgent need for a unified framework that can adaptively capture critical information from both features and topology while effectively learning domain-invariant knowledge. Such a framework is essential to ensure robust generalization to downstream domains (Li et al., 2022; Xie et al., 2025).
both
1. Introduction
Graphs, versatile data structure, widely used across various domains, such as citation networks (Ebesu & Fang, 2017), social networks (Traud et al., 2012), and bioinformat-
Equal contribution 'Corresponding author University of Electronic Science and Technology of China, Chengdu, Sichuan Province, China. Correspondence to: Shuo Wang <run- ner21st saane@mmailcom.... Zhao Kang <zkang @uranngessttc.cuu.c.>
Second, real-world graphs are inherently noisy (Jin et al., 2020b), often containing unreliable edges characterized by irrelevant, misleading, or missing connections. Additionally, graph learning algorithms are highly susceptible to adver- sarial attacks (Züggner al., 2020). further exacerbating the challenges of graph representation learning. These issues
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
highlight the limitations of conventional training paradigms that rely on fixed graph topologies. Therefore, developing robust multi-domain pre-training framework capable of effectively handling complex noise distributions is critical for constructing trustworthy and reliable graph foundation models.
2020a).
Pre-training methods can be broadly categorized based on their downstream task-tuning strategies. The first category follows a pretrain-ande -aniii---n--ii--unnn paradigm, emphasizing ef- fective pre-training strategies in the upstream phase through self-supervised learning. For example, DGI (Velickovic et al., 2019) enhances training by maximizing the mutual information between global and local representations. Sim- ilarly, GraphCL (You et al., 2020) and SimGRACE (Xia et al., 2022) focus minimizing the distance between repre- sentations of different augmentations, effectively capturing invariant and robust structural information.
To tackle these challenges, we propose the Multi-Domain Graph Foundation Model (MDGFM), a unified framework designed to effectively align and leverage structural knowl- edge across domains. First, we introduce a decoupled em- bedding mechanism incorporating an adaptive balance token that dynamically weighs feature and topology information. To address inherent noise and achieve topology alignment, we integrate Graph Structure Learning module that learns robust, Nomain-invariaan knowledge, enhancing both gener- alization and transferability. Additionally, we develop an ef- ficient prompt learning strategy to transfer knowledge from multiple source domains to a target domain. By aligning topologies effectively, MDGFM not only improves multi- domain pre- pre-training but also enables efficient and robust knowledge transfer to unseen downstream domains. Theo- retical analyses validate the effectiveness of our topological knowledge alignment and demonstrate superior domain gen- eralization capabilities.
The second type follows a pretrain -errainn-and-peeee -att--anndd--p-tuuuueeaa prompt-tuning paradigm, where pre-trained models are not fine-tuned for downstream tasks. Instead, these methods reformulate the input data to align with the pretext task (Gao et 2020). For instance, GPPT (Sun et al., 2022a) introduces graph prompting function that transforms independent nodes into token pairs, reframing downstream node classification as an edge prediction task. Similarly, GPF (Fang et al., 2024) em- ploys learnable perturbations in the feature space of down- stream graphs, enabling implicit modifications to node fea- tures and graph structures. However, most existing methods are constrained to single-domain pre-training and tuning, significantly limiting their capacity to capture cross- knowledge and generalize to unseen domains.
In summary, our contributions are as follows:
We introduce a novel structural perspective on multi- domain graph foundation models, addressing the com- plexities arising from interconnected nodes and diverse attribute types.
2.2. Multi-domain Generalization
Domain generalization aims to achieve out-of distrribbuionn (OOD) generalization by learning from multiple source do- mains (Zhou et al., 2022). In contrast to domain adapta- which transfers prior knowledge from single source domain specific target domain-domaii generalization focuses leveraging diverse information from multiple source domains to generalize effectively to unseen domains. This approach addresses two critical challenges: domain shift and the absence of target domain data (Blanchard et al., 2011).
We propose MDGFM, a unified multi-domain graph foundation model that employs graph structure learn- ing to reduce inherent noise and align topological struc- tures. Our framework facilitates efficient and robust knowledge transfer to unseen domains, with theoretical guarantees on its effectiveness and domain generaliza- tion capabilities.
We conduct extensive experiments on both homophilic graph datasets, demon- strating the effectiveness, robustness, and versatility of our approach.
Recently, domain generalization on graphs has gained at- tention (Wang et al., 2024). These methods integrate and extract knowledge from multiple source domains during the upstream pre-training phase, enabling the transfer of this knowledge to tackle various graph-related tasks in previ- ously unseen downstream domains (Fang et al., 2025). For example, GCOPE (Zhao et al., 2024) integrates multi-source graph topologies during the pre-training phase by introduc- ing interconnected virtual nodes. Additionally, MDGPT (Yu et al., 2025) incorporates domain-speciftt tokens in the pre-training phase to align node features from different domains, and employs prompt-tuning during downstream tasks for efficient knowledge transfer. Despite these ad- vancements, significant gaps remain in understanding the
and heterophilic benchmark
2. Related work
2.1. Graph Pre-training
Graph pre-training aims to train Graph Neural Networks (GNNs) on large amounts of unlabeled data, enabling the transfer of the learned model to downstream tasks with lim- ited supervision. This approach facilitates the acquisition of general knowledge about real-world graphs while reducing the dependence on labeled data (Hu et al., 2020; Jin al.,
2

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Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
semantic differences and reliability of multi-domain topolo- gies. Consequently, there is urgent need design universal foundation model capable of achieving robust and generalized knowledge transfer across domains.
cies and customize the information propagation process in GNNs, we employ both numerical and semantic alignment to bridge domain gaps. First, we transform the feature ma- trix of each graph into a consistent dimensionality shared across all domains, enabling uniform representation. This transformation is described by:
3. Problem Definition
X; Proj(X(")) E R Vilxa
(1)
A graph is formally represented as G = (V, E, X (A, )) where V is the set of nodes and E denotes the set of edges, X R represents the original feature matrix of nodes. A is the corresponding adjacency matrix. Each graph is associated with a domain D; E D. Without loss of generality, we assume D; * Dj for any pair of distinct graphs.
where d represents the aligned dimensionality, and (roj(.) denotes a specific projection operation. In our approach, the Projection function is implemented using Principal Compo- nent Analysis (PCA) (Abdi & Williams, 2010). While this dimensional alignment ensures uniform feature dimensions across graphs, it primarily addresses numerical consistency and does not resolve the semantic disparities that remain between domains.
Given a set of graphs from different domains, say G2 (A;, ),
1, ..., N. We pretrain our graph founda- tion model on all visible graphs and test the performance on graph T from an unseen target domain Dr & 'D. We summarize the notations in Appendix A.
To achieve semantic alignment, we propose the concept of domain tokens denoted as tp, E Rd, which encode domain- specific characteristics by capturing unique contextual in- formation from each domain. These tokens are applied via element-wise Hadamard multiplication @ with the feature matrices, acting as adaptive filters that modulate features to reflect their respective domain properties within a uni- fied semantic space. This process not only preserves the distinctive traits of individual domains but also facilitates their alignment within a cohesive semantic framework.
4. Methodology
As illustrated in Figure 1, our proposed MDGFM consists of two modules: the Multi-domain pre-training module and the Downstream target-domain adaptation module.
Multi-domain pre-training, For graphs originating from multiple source domains, we begin by unifying their fea- ture matrices to a common dimensionality, ensuring com- patibility across domains. Next, we apply token operators specifically designed achieve semantic alignment between disparate graphs. To address the challenges posed by het- erophily in downstream tasks, we leverage graph structure learning (GSL) to refine each source domain, integrating both feature and topology information during the refine- ment process. Additionally, GSL enables the model to learn domain--nvarian knowledge for better transfer, where we prove the effectiveness in Section 5. Finally, the graph en- coder is trained effectively using self-supervised signals, enabling robust representation learning.
A significant challenge in building a graph foundation model lies in the substantial semantic variations across domains, which can hinder effective knowledge transfer (Hassani, 2022). To address this, we leverage shared, domain- -agnosic information as a critical foundation for robust and efficient transfer learning. Specifically, we introduce the shared to- ken ts Rd which acts common semantic anchor to bridge discrepancies between domains. By focusing on shared patterns, f, captures transferable knowledge broadly applicable to new domains, reducing dependence on domain- specific features that may lack generalizability. This shared representation enables consistent semantic alignment, en- hancing the model's 's adaptability and robustness to domain variations. The overall feature unification procedure can be formalized as:
Downstream target-domain adaptation. Given that graphs often share common intrinsic patterns, pre-trained knowledge can be effectively transferred to unseen graphs (Zhao et al., 2024). To bridge the gap between source and target domains, we propose a dual-stream prompt frame- work. This framework combines meta prompts, which trans- fer broadly learned knowledge, with task-specific prompts designed to align with the unique characteristics of the down- stream domain.
X = ts C otD, @ oX)
(2)
where o-) is non- linearityiy activation function capture complex information.
4.2. Graph Topology-aware Alignment
Previous research on graph prompting (Sun et al., 2023; Dong et al., 2019) has predominantly emphasized feature alignment, often overlooking the critical role of structural discrepancies between graphs from different domains. How- ever, rich semantic information is embedded within rela- tional patterns and topology credibility, as characterized by
4.1. Feature Projection into Unified Semantic Space
Graphs from different domains often exhibit diverse fea- tures and structural patterns. To address these discrepan-
3

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Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
Multi-domain Pre-mriniin
Domale sokonz
Feonser Projection Unified Semantic Space
Gzaph Topologr rwiee Algament
A,X'
token
Graph
Pre-triin loss
X"
+ X
Meta
Ralance rulen
Graph Excodir +
Downstrrom
A
Skared tokea
Unify
Unified Semontic Space
Ainemion
Speeific prowet
Downstream Adaptation
Feemsalle
Hrightdd
Figure The overall framework of the proposed MDGFM.
metrics like homophily and heterophily edge ratios (Zhu et al., 2020). These structural characteristics underscore the importance of aligning graph structure patterns to enable meaningful knowledge transfer (Sun et al., 2022b; Zheng al., 2022).
ization are performed sequentially to produce the final A² Detailed implementation steps are provided in Appendix F. Building on this topology-aware refinement process, we propose general pretraining framework capable of generat- ing high-quality embeddings across multiple domains using contrastive learning. By optimizing contrastive-bast based objective, the model learns generalized representations by leveraging rich comple-to-sape perspectives (Yao et al., 2022). Mathematically, we maxi- mize the mutual information between Gn = (A,, X!) and G2 = (A4, X() follows:
Rather than solely focusing on direct structural unification, our approach seeks to synchromize more reliable topology in- formation through graph structure refinement on both source and unseen downstream graphs. This achieved via the incorporation of GSL, which enhances the robustness and effectiveness of cross-domain graph alignment. By address- ing structural discrepancies, our framework bridges the gap between diverse graph domains, facilitating more compre- hensive knowledge transfer.
relationships from diverse
C = -
G.2 +
(4)
where we use f to indicate positive samples when comput- ing similarities, often utilizing the identity matrix Ie for this purpose. The second term of incorporates the re- fined graph structure A' to increase the number of positive samples. Detailed computations of the loss function provided in Appendix G. This loss formulation reduces the information gap between the original and refined graphs, en- suring that global structural properties are preserved during optimization. By maintaining semantic consistency in the graph structure, it enhances alignment and adaptability for downstream tasks.
Unlike prior GSL approaches (Zhu et al., 2021; Jin et al., 2020b), which primarily optimize graph structures based on feature similarity, our method incorporates both semantic and topological information for more comprehensive re- finement process. Specifically, we utilize A X to capture the original structural information, where represents the or- der of graph aggregation (default set to one). To effectively fuse X and AÇ during graph refinement, we introduce a balance token tB, E 2d which operates in the following manner:
H: Ln. ② [X,, A[X]]
Here, (B, dynamically balances the contributions node features and aggregated structural information.
4.3. Knowledge Transfer Downstream Domain
Similar to in-context learning, downstream adaptation aims to enhance model's ability to learn tasks using only few examples provided as demonstrations (Dong al., 2024). highlighting the need for upstream knowledge transfer. In- spired by the "pre-training & prompting" paradigm, our framework employs a dual-prompt strategy. Specifically, the meta prompt Pm focuses on adjusting the distribution of learned knowledge by modeling the relationships between the target and source domains. This can formalized as a
Finally, consistent with existing GSL methods (Li et al., 2024), we apply post-processing techniques reconstruct the refined adjacency matrix. Specifically, similarity ma- trix is first computed using H, and then sparsified via k- nearest neighbors (kNN). In practice, we employ kNN spar- sification with its hance efficiency (Fatemi et al., 2021). Subsequently, op- erations such as Symmetrization, Activation, and Normal-
approximation to en-

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Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
function defined for feature X:
5. Theoretical Analysis
N
In this section, we provide a theoretical analysis to demon- strate the domain generalization capabilities of MDGFM.
Pm Px(() Is O a OsLD, x)
(5)
Denote Pt as the data distribution feature space in the target domain. t could be changed into i for source domain, while X' could be replaced by label space V With the covariate shift assumption, each domain is characterized by the distribution on X. Thus, we can approximate the target domain distribution P%x within the convex hull of source domain distributions: A == {E> T.Px x E A AM) where Aas is the (M - - -)-dimension simplex so that each π represents a normalized mixing weights. The generalization capability quantified by the following:
where 01,,... CVy are trainable coefficients for the source do- mains. Meanwhile, the learnable specific prompt ps E Rdd aims to directly align the target domain with the unified se- mantic space by learning from the limited available samples, ensuring precise adaptation to the downstream domain.
As discussed, for the downstream graph ['r,, XT) where the feature dimension aligns with the given d, the dual prompts Pm and ps operate on XT to achieve semantic unifi- cation. Since all tokens are optimized during the pretraining phase, only the coefficients and Ps need to be trained in this process. The reduced number of learnable parame- ters enhances transfer performance by compensating for the scarcity of annotated samples. Consistent with upstream operations, we use Hr = l @ [X7,, AG.X, to obtain A/r, where t could be composed of tD, (similar to Pm) or be directly trained. Finally, we obtain the node representations Z as follows, which can be applied to downstream tasks:
Theorem 5.1 (Domain generalization enror bound). Let
Y == minnEDI (Px Ei= T,Px) with minimizer x² be the distance of Pie from the convex hull A, and Px == Σi=1 TH PiPss be the best approximator within A. Let p == su dn(P'x, Pkt) be the diameter of A. Then it holds that
M
+ 7+P + *HH(P()PP)) (8) ((h) S Σ Te (h)
2
Z (pm + (1 B)ps @ Xr; (pr))
(6)
where XH..Ps.. is the ideal joint risk across the target domain and the domain with the best approximator distri- bution Px- e,,.... e represent the source risks and e
where GE denotes the graph encoder and ppe the frozen pa- rameters learned in pre-training phase. To mitigate the risk of harmful transfer from source graphs with significant dis- crepancies, as discussed by (Yan et al., 2024), the prompts are integrated using learnable parameter, 3, instead of re- lying solely on concatenation. For graphs that significantly differ from the source domains, the meta prompt weights are attenuated to minimize the transfer of irrelevant infor- mation, thereby reducing its potential negative impact on downstream tasks.
denotes the target risk (Albuquerque et al., 2019).
To minimize the upper error bound, we introduce the con- cept of invariant graph learning, which is highlighted in work:
Assumption 5.2. Given a graph G, there exists an optimal invariant graph learner 4- (G) satisfying:
Invariance Property: Ve, P (Y = p (G)).
For the downstream node classification task, the loss func- tion Cass follows a universal task template grounded in subgraph similarity. Given a labeled training set S = { .i..... (I;, }, where I; represents a node and yi E Y denotes the class label of x; with Y denot- ing the set of all possible class labels, the loss function is
Sufficient Property: Y = - (g (G))) eeeLG. where g (-) is representation learning function, w" is the classifier, e denotes the environments (i.e., domains), 1 indicates statistical independence and e is the random noise.
4 (G) could generate invariant graphs across different do- mains, which is implemented as GSL procedure on all domains in our work. Moreover, by maximizing the mu- tual information between G,1 (A,,,, X;) and (A2,'', (A), our method retains sufficient telk-relevant information, thereby satisfying the sufficient property. Motivated by Theorem 5. domain invariant representation learning minimizes the risks over all source domains corresponding the first term of the bound, as well as the representation distribution dif- ferences among source and target domains in the hope of reducing y and p. In general, based on the following theo- rem, our model achieves the minimal error bound due to its adherence to the invariant and sufficient properties.
expressed as:
sim( Eer, Zye) Σ In Eyer exp sim(Zz,. ZZzy xn)) (M
Last
Here, 2z, denotes the final embedding of node xi, and By represents the class embedding for class y. computed as the mean embedding of all training instances belonging to y. We practically calculate sim(..) ( by cosine similarity, while T serves as a temperature parameter to control the shape of the output distribution.
5

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Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
Theorem 5.3. Let Ø- be the optimal invariant graph learner and denote the complement as G 1 (G), iee, the corre- sponding variant subgraph. Then we can obtain the optimal predictor under distribution shifts as follows.
parameters of a pre-trained model, unify downstream tasks and tune a single type of prompt accordingly. Represen- tative methods include GPPT (Sun et al., 2022a) and GPF (Fang et al., 2024).
Multi-domain Pre-training Methods: These methods inte- grate multiple source domains during upstream pre-training and transfer knowledge for few-shot learning on unseen tar- get domains. For instance, GCOPE (Zhao al., 2024) con- structs a unified large-scale dataset with inter-dataset con- nections, while MDGPT (Yu et al., 2025) encodes domain- specific characteristics using unique tokens, followed by fine-tuning and prompt adaptation.
arg minwog o g o 40 (G(()) - arg min sup Sup R(fee) wi
(9)
1
if the following conditions hold: (1) (1C))CGV)(() (2) v4 which is an invariant graph learner, Be' such that pe' (G)Y) = Pd (+(G), (GV((G) and pe (O(G) PT)))) (Li al. 2022).
where R(fle) y,y (G) Y) the risk of the pre- dictor/ f on the domain e and ((x-): : Y x Y + IR denotes loss function.
For a fair comparison, we use GCN as the backbone for all methods. To evaluate performance on unseen target do- mains, we focus on scenarios where the model generalizes to domains not encountered during pre-training. For the Cora, Citeseer, Pubmed, Chameleon, Squirrel, and Cornell datasets, we designate one dataset as the downstream target domain while using the remaining five as source domains during pre- pre-training. Additionally, we extend this setup by us- ing all six datasets as source domains and applying Penn94 as the target domain for few-shot learning. More details are provided in Appendix E.
6. Experiments
In this section, we evaluate our proposed MDGFM on few- shot node classification task. Specifically, we aim to answer the following four research questions:
RQ1. How does MDGFM perform on multi-graph tasks in few-shot learning scenarios compared to current baselines? RQ2. How do the key components benefit our model? RQ3. Does our model rely on specific source domains and exhibit sensitivity to the removal of these source domains? RQ4, Does our model demonstrate robustness against attacks and deletion on the source domain?
6.2. Cross-domain Transfer Efficacy under Few-shot Learning Conditions (RQ1)
We compare our proposed MDGFM against all baselines 1-shot and K-shot node classification tasks. Each ex- periment is repeated five times, and we report the average results in Tables I and 2.
6.1. Experimental Sctups
Datasets. To ensure a comprchensive comparison, we con- duct experiments on six primary datasets, including three homophilic graphs-Cora (Sen et al., 2008), Citeseer (Sen al., 2008), and Pubmed (Namata et al., 2012) and three heterophilic graphs- CConeell, Chameleon, and Squirrel (Pei et al., 2020). Additionally, we include Penn94 (Traud et al., 2012), a large-scale graph dataset, as a downstream target domain. Detailed statistical information on these datasets is provided in Appendix C.
In the one-shot setting. MDGFM consistently outperforms all baseline models, demonstrating superior performance on both homophilic and heterophilic graphs. This advantage stems from MDGFM's ability to effectively capture both -speiii information from each source domain and shared patterns across domains. Notably, we observe that supervised methods occasionally surpass graph pre-training approaches, suggesting the presence of negative transfer in certain scenarios.
Baselines. We compare our method against four categories of approaches:
Among multi-domain pre-training methods, MDGPT outper- forms GCOPE on homophilic graphs, as GCOPE's reliance virtual nodes propagates homophilic graph information, inadvertently introducing noise. Conversely, on heterophilic graphs, GCOPE surpasses MDGPT, as MDGPT's simple integration of source domain tokens fails to fully capture cross-domain commonalities. For graph prompting meth- ods, GPPT generally underperforms across most datasets, as it is not specifically designed for few-shot learning. While GPF achieves competitive performance, its high memory requirements pose a significant limitation.
These methods train a GNN on down- stream tasks and directly infer results. We employ two well- known models: GCN (Kipf & Welling, 2016) and GAT (Velickovic et al., 2017).
Supervised methods:
Graph Pre-training Methods: These approaches perform self-supervised pre-training across multiple isolated source domains, such DGI (Velickovic et al., 2019) and GraphCL (You et al., 2020), before fine-tuning on a new downstream task. Notably, source domains are not trained simultane- ously; instead, they are merged into a single batch object, forming an adjacency matrix composed of distinct blocks. Graph Prompting Methods: These methods freeze the
In the few-shot setting, we observe trends similar to those
6

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Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
Table Cross-domain transfer learning performance (mean accuracy + std) of one-shot node classification. The highest result bolded and the runner-up is highlighted with red color. The symbol "OOM" means out of memory.
Methods
Cora
Citeseer
Pubmed
Cornell
Squirrel
Chameleon
Penn94
GCN
28.57=5.07
31.27++4.53
40.55=3.65
31.81+4.71
20.001:.229
24.17::..21
10.45:1..99
GAT
28.4016.225
30.76:15.40
39.99-14.9
18.03:113.19
21.5542..3
23.93:14.11
50.48.: 1.40
DGI
29.3015.82
30.03:14.88
41.85=7.78
15.54
21.15:1.68
21.73:15.47
50.22:10.74
GraphCL
34.94116.49
30.58114.58
40.37=7.81
17.15=
21.42:2..33
22.49:3.02
50.61
GPPT 17.52++5.52
21.45:+3.45
36.56=5.31
25.094:2.92
20.09+
24.53++2.55
48.93++1..99
GPF
37.84.111007
37.61118.87
46.36:17.48
34.54.1..73
21.92:4.500
25.90:18.51
OOM
GCOPE
34.23.18.16
39.05:18.82
44.85-16.22
34.02:111.94
22.46.11.96
24.61:1..99
50.79:10.65
MDGPT
39.54+9.02
39.244+8995
45.39++1..01
33.58+110.38
22.35+3.77
23.68+1.56
50.78+3.005
MDGEM
44.83+7.41
42.18++6.44
46.84-7.31
40.77+55.9
24.20+3.26
28.36+3.65
0..3
Pubmed.
their classification performance against MDGFM. wo-refinedadj: Removes the GSL module.
wo-ssmtoken: Excludes the shared token.
1.00
w/o-topology: Constructs the graph without considering topological information in the GSL process, i.e., merely use X: in (3) rather than [X,, AT XX]] for the i-th graph. w/o- balance: Removes the balance token.
095
0.90
0.80
Comell
0.75
Citeseer
Effectiveness of domain-invariant learning. MDGFM captures shared information across domains through a shared token. As shown Figure 2, the w/o-sumnooken vari- ant exhibits the weakest performance across most datasets, underscoring the importance of capturing common informa- tion for effective domain generalization.
wio-balnnce
wio-topology
wio-refinedaddj
w/o-sumtokeen
Chameleon
Squirrl
MDGFM
Figure? 2. Ablation studies on key components.
Effectiveness of topology structure alignment. MDGFM utilizes graph structure learning to extract information from graph topology structure. The woorrfiiedadj. w/-topoolgg, w/o-balance generally perform much worse than MDGFM, demonstrating the necessity of each component.
in the one-shot scenario. Notably, on heterophilic graphs, certain methods exhibit a decline performance compared to the one-shot case. This degradation is primarily due to the introduction of noise from the few-shot samples. In contrast, our model remains robust and does not experience performance deterioration as the number of training samples increases.
6.4. Domain Sensitivity (RQ3)
In this section, we examine the extent to which our model depends on specific source domains and its performance under limited source-domain conditions. To assess this, systematically remove subsets of source domains and evalu- ate the model using Cornell as the target domain, comparing its performance against multi-domain pre-training baselines. Specifically, we exclude one to three source domains while keeping the target domain and all other parameters fixed.
Quantitatively, across the seven datasets, MDGFM outper- forms the second-bes, model by up to 18.04% in the one- shot scenario and 8.11% in the few-shot scenarios.
We further evaluate large K values on the scale Penn94 dataset. As shown in Table 3, our model's per- formance improvement becomes more pronounced as K increases. In contrast, the performance of GCOPE and MDGPT remains relatively stable, showing minimal change.
According to the one-shot result in Table 4, the performance of GCOPE and MDGPT deteriorates significantly, whereas MDGFM maintains strong performance and stability despite the removal of source domains. Our findings suggest that these methods extract varying degrees of knowledge from specific source domains. A counterintuitive phenomenon is that in some cases, removing some source domains leads to
6.3. Ablation Study (RQ2)
In this section, we analyze the effectiveness of key compo- nents in our model by designing four variants and comparing
7

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Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
Table Cross-domain transfer learning performance (mean accuracy + std) of few-shot node classification, where the values within parentheses represent the K values used in the context of Khotot learning.
Methods
(5ra(5)
Citescer(5)
Pubmed(5)
Cornell(3)
(quirrel33)
Chameleon(5)
Penn94(5)
GCN
60.15+45.33
45.54=4.71
87.82++8.26 57.55:19.37
39.53+
21.61=4.22
22.09:00.99
52.07++1.01
GAT
59.79:13.89
20.48:12.94
34.53:113.01
20.11=3.11
20.83:11.52
10.98:11.23
DGI
56.76:11..29
42.67-8.98
54.04
43.22:15.84
20.23=1.12
27.68:15.21
50.1340.81
GraphCL
61.59:1..1
67.05.16.85
58.5017.38
32.77:16.23
21.18
27.45:12.58
51.8411.58
GPPT
43.67+4.11
47.31+6693
40,47+10.17
34.69+8.54
22.14-1.53
28.25+1.39
51.75++160
GPF
51.21#11.44
56.90=8.84
58.7617.70
38.17:1..15
21.62=3.10
28.09:1:.93
00M
GCOPE
54.63++3.98
53.18:14.47
57.744:2.73
18.21
21.37=4.20
25.50:+1.23
50.69:40.81
MDGPT
59.64:15773
52.71=5.711
58.65:17.54
35.18:1..90
21.42_4.16
26.18:15.18
50.4143.13
MDGFM
64.56.17.29
4..24
63.50
69.56.16.92
30.54
53.58
34
PCJFT
M00
0.29
5.35
0.55
Edige adcitton pahe
Edge deterion cate
Poturbazon FOt2
Aertutbetion cate
(a) Adding edges (Cornell)
(b) Deleting edges (Cornell)
(c) Mcta-attack (Cornell)
(d) Mcta-attack (Cora)
Figure 3. Performance of robustness analysis, where the dataset in parentheses represents the target domain.
Table 3. K-shot node classification accuracy on Penn94.
Table 4, Performance of domain sensitivity analysis.
K
GCOPE
MDGPT
MDGFM
Removed Data
GCOPE
SAMGPT
MDGFM
10
50.30:11.03
50.5914.33
54.43:10.93
-Puhmed
31,4249.53
39.66+5.51
-Squirre
28.88-88.57
33.0248.266
40.78++5.44
50
51.36++0.28
50.73+-3.85
55.69+
100
51.31
51.14:1..88
58.51
-Cora-Citeseer
30.25:18.99
30.04:8.777
39.96:15.99
-ubmeed-Squrrel
30.50:18.41
31.10:77.43
42.68::5.64
500
52.88=+0.40
51.52+3.64
0.825
Citteeer-Corate Corb-u
26.37=+10.85 85
29.38-19.39
40.08=55.74
-Pubmed Citeseer Squirrel
31.50:+:3.11
30.90
42.20:15.02
Original Performance
34.02=+11.94
13.58:10.33
40.77=5.96
improved performance. This occurs because multi-domain graphs may result in conflicts or interferences rather than synergy (Yu et al., 2025). Our model aims to mitigate such negative influences by extracting the shared informative and beneficial aspects of the available knowledge.
6.5. Robustness Analysis (RQ4)
ing data to degrade the model performance after training. Under identical attack conditions, we compare MDGFM against multi-domain methods. As shown in Figure 3, the performance of GCOPE and MDGPT deteriorates as attack intensity increases. Notably, our model is quite stable and consistently outperforms all baselines across all attack see- narios, which can be attributed to our proposed topology alignment. Further analysis, including sensitivity study of hyperparameters, is provided in Appendix D. Time com- plexity analysis is also summarized in Appendix B to show the efficiency of MDGFM.
To evaluate the robustness of our proposed MDGFM, we assess the node classification performance under diverse attacks. Simple modification attacks are implemented by randomly adding removing edges. Considering the vary- ing influence of individual domains on the target domain, we apply these random attacks to all source domains and the target domain under one-shot learning.
Furthermore, we employ Metattack (Zügner& & Günnemann, 2019), an advanced adversarial attack that perturbs the train-
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Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
7. Conclusion
Chien, E., Peng, J., Li, P., and Milenkovic, O. Adaptive universal generalized pagerank graph neural network. In International Conference on Learning Representations, 2021.
In this paper, we introduce MDGFM, novel graph founda- tion model that unifies graphs from diverse domains into universal semantic space. Our approach enhances domain generalization by extracting domain-invarian knowledge through both feature and topology alignment. We validate the effectiveness of MDGFM through theoretical analysis and empirical evaluation. To the best of our knowledge, this is the first work to explicitly address invariant information embedded topology structures across homophilic and het- erophilic domains. Our findings pave the way for advance- ments in graph- based domain generalization, with potential extensions to dynamic graphs and large-scale heterogeneous networks, further enhancing adaptability in real-world ap- plications.
Dong, I.., Yang, N., Wang, W. Wei, F., Liu, X., Wang, Y., Gao, J., Zhou, M., and Hon, H.-W. Unified language model pre-training for natural language understanding and generation. Advances in neural information process- ing systems, 32, 2019.
Dong, Q. Li, L., Dai, D., Zheng, C., Ma, J., Li, R., Xia, H., Xu, J., Wu, Z., Chang, B., et al. A survey on in- context learning. In Proceedings of the 2024 Conference Empirical Methods in Natural Language Processing (EMNLP), pages 1107-1128, 2024.
Ebesu, T. and Fang. Y. Neural citation network for context- aware citation recommendation. In Proceedings of the 40th international ACM SIGIR conference on research and development in information retrieval, pp. 1093-1096, 2017.
Impact Statement
This paper aims advance the field of Machine Learning. While our work has potential societal implications, we do not identify any specific concerns that require particular emphasis at this stage.
Fang, T., Zhang, Y., Yang, Y., Wang. C., and Chen, L. Uni- versal prompt tuning for graph neural networks. Advances in Neural Information Processing Systems, 36, 2024.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 62276053).
Fatemi, B., El Asri, L.,, and Kazemi, S. M. Slaps: Self- supervision improves structure learning for graph neural networks. Advances in Neural Information Processing Systems, 34:22667-2268) 2021.
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Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
A. Notations
Table 5. Frequently used notations.
Notation
Description
G; (Aj, X
The i-th original input graph.
Gn (A; X
Graph after feature projection.
G2 (A/ X
Graph after feature projection topology aware alignment.
T (AT.XT)
The downstream target graph.
X
Features of the original input graph.
X,
Graph features after dimension alignment.
X
Features after feature projection.
Y
The set of all possible class labels.
The order of graph aggregation. Dimension of the original node features.
d'
d
The unified dimension.
V
The set of nodes.
E
The of edges.
Z
Node embeddings. Sct of souree domains.
D
D
The i-th domain from the source domain set.
Dr
Target domain does belong to source domain set.
Proj())
specific projection operation, here we use PCA. Non-linearity activation function.
o())
C
Loss function.
Last
Downstream loss function.
sim(..))
Cosine similarity.
o0)
Element-wise multiplication.
tp,
Domain token for domain D..
is
Shared token.
LB:
Balance token for domain D;-
Pm
Meta prompt.
Ps
Specific prompt.
ai
Trainable coefficients for source domain i.
B
learnable parameter used for integrating prompts.
B. Time complexity analysis
In the pre-training phase, each of the N source graphs is processed independently. For a graph with [V] nodes and [E] edges (here we select [V] = maz(Vx), |E| = max ) n, m E [V, the model first aligns node features via truncated PCA, which reduces the input dimension from d' to d at cost of O([V) d logd'). As for token lightweight element-wise multiplication, the time complexity is VVd. It then applies locality-sensitive hashing kNN for graph structure learning. Denote the batch size of sparse RNN as B, each requiring ((d) operations, resulting in O[V]] B d) time for reconstructing the graph. Next, an L-layer GCN operates on the refined structure contributing an additional (L IV|- + L. EE| d [V] Therefore, the total pre-training complexity across all source graphs is O (N [[V] log d' + IV
+ L [V] de L- L- -d).
Similarly, in the downstream phase, PCA procedure takes O(V] d logd') time. Prompt fusion and token modulation cost ((VVr d). GSL again uses local sensitive ANN, which adds ( B d). The GCN encoder then performs L-layer message passing with cost O(L |V d² + L- |ET| d + V d). Finally, classification is done via prototype matching, where each node compares with C class centroids, yielding ( |- C. d). Summing these terms, the overall downstream complexity is O(V d- log d' + VT B d + L. Er! d - L |V de VTl C-d).
Overall, the model scales linearly with the number of nodes and edges, and benefits from efficient structure refinement and modular design like local sensitive ENN, We will include this complexity analysis in the final version for completeness.
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Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
C. Datasets
Table Statistics of Datasets,
Datasets
Nodes
Edges
Feature dimension
Node classes
Homophily ratio
Cora
2,708
10,556
1,433
0.810
Citeseer
3,327
9,104
3,703
6
0.736
Pubmed
19,717
88,648
500
3
0.802
Squirel
5,201
396,846
2,089
5
0.222
Chameleon
2,277
62,792
2,325
5
0.231
Cornell
183
298
1,703
5
0.305
Penn94
41,554
2,724,458
4,814
2
0,510
We provide an in-depth explanation of datasets, where their statistics details are presented in Table 6.
Citation network: The Cora and Citeseer (Sen et al., 2008) datasets represent a varied collection of scholarly articles in the field of computer science, where each node is associated with bag-of- words features and a categorical label denoting the corresponding topic of the paper. In contrast, the Pubmed (Namata al., 2012) dataset consists of articles focused on diabetes sourced from the PubMed database. Each node in this dataset is characterized by attribute vector, along with label that identifies the specific type of diabetes addressed in the publication.
WebKB: Cornell (Pei et al., 2020) is subset derived from the WebKB dataset. In this dataset, each node corresponds to web page, while the edges illustrate the hyperlinks that connect these pages. The features of nodes are expressed using bag-of-words features derived from the content of the web pages. Nodes are classified into five distinct labels: student, project, course, staff, and faculty.
Wikipedia network:
The Chameleon and Squirrel (Pei al., 2020) datasets comprise two page-to-page networks sourced from Wikipedia, each centered on specific themes. Nodes signify individual web pages, and edges represent their connections. Node attributes are characterized by collections of nouns gathered from the content of the pages. Additionally, each node is categorized according to the average monthly traffic that the corresponding web page receives.
Online social network: The Penn94 (Traud et al., 2012) dataset is a large-scale social network derived from the Facebook 100 networks university students from 2005, where each node corresponds to an individual student. The nodes are annotated with the reported gender of each user, and the objective is to predict this gender. The features associated with the nodes include major, secondary major/minor, dorm/house, year, and high school (Lim al., 2021). The homophily ratio of the Penn94 dataset is 0.51, indicating that its graph topology exhibits a mixture of homophilic and heterophilic patterns. Previous research has shown that such topological information is nearly meaningless, making it difficult for existing topology-fixed GNNs to adapt effectively to this type of graph (Chien et al., 2021; Chen et al., 2024).
D. Additional Experiments
D.1. Expansion Robustness Experiments
We perform robustness experiments on the Cornell dataset before. To investigate whether the MDGFM model retains its robustness under the condition where the downstream target domain consists of homophilic graphs, replicate the robustness experiments previously described the Cora dataset, performance under metaattack is shown in the main text. Specifically, we apply random edge additions and deletions on all domains. We compare our results with those of MDGPT and GCOPE, as summarized in Figure 4. Our observations indicate that our model maintains commendable robustness when evaluated on homophilic graph datasets. Furthermore, we note that in the context of heterophilic graph datasets, GCOPE exhibits greater robustness compared to MDGPT, while the opposite trend is observed for homophilic graphs.
D.2. Sensitivity Analysis
We investigate the impact of the hyperparameter in k-nearest neighbors (kNN) for downstream graph structure learning, which is crucial to our model. The performance of our model varies with changes in the hyperparameter k as illustrated in Figure Overall, our model exhibits low sensitivity to changes in k regardless whether the downstream target domain
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Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
44
42-
Accuracy
Accuracy
40
38
38
36
36
34
32-
Ours
Ours
MIGGPT
32
MDGPT
30
0.0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Edge addition rate
Edge deletion rate
Adding edges
(b) Deleting edges
Figure 4. Performance of robustness analysis, where Cora is the target domain.
50
45
48
40
46
Accuracy (%)
Accuracy (%)
44
35
42
30
40
25
38
Otteeer
Squinel
Pubmed
Chartlon
36
5
10
20
20
5
10
15
20
30
k
(a) Homophilic graphs
(b) Heterophillic graphs
Figure 5. Sensitivity study of k.
homophilic heterophilic. When the target domain homophilic graphs, larger values of k: (such as 20 or 30) yield the best results. In contrast, for heterophilic graphs, relatively smaller values of l can produce satisfactory outcomes.
E. Experimental Details
For both one-shot and few-shot classification tasks, we pretrain the models on five datasets and subsequently perform predictions on the remaining dataset, ensuring that the downstream domain remains unseen during the training phase. The detailed experimental setup is summarized Table where checkmark (v) indicates visibility during pre-train pre-training, while the absence of the mark denotes invisibility. For the large-scale dataset Penn94, it is exclusively used as the target domain evaluate the generalization capability of the model.
All experiments are conducted on a platform equipped with an Intel(R) Xeon(R) Gold 5220 CPU and an NVIDIA A800 80GB GPU, using PyTorch 1.10.1 and DGL 0.9.1. Each experiment is run five times and the average results are reported. We employ the Adam optimizer and set the batch size to 128. During the upstream pre-training phase, we utilize Principal Component Analysis (PCA) to reduce the dimensionality of the initial features to 50 dimensions, thereby unifying the features from multiple source domains. For homogeneous graphs, we set k=30 for Graph Structure Learning (GSL), while for heterophilic graphs, we configure k=15. Additionally, we adjust the upstream and downstream learning rates across different datasets, as detailed in Table 8. We fix the number of graph neural layers 3, with hidden dimension of 256 for the GCN model. When dealing with homophilic graphs, the number of Epochs is set to 60, whereas it is set to 100 for heterophilic graphs. For the downstream node classification tasks, we implement few-shot learning scenarios with 1-shot and 5-shot settings (3-shot for the Cornell and Squirrel datasets). We perform 50 resampling iterations in downstream few-shot classification. The values required for graph construction adhere to the configurations established during the upstream pre-training phase.
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Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
Table Settings of multi-domain transfer.
Target domain
Cora
Citeseer
Pubmed
Squirrel
Chameleon
Cornell
Cora
Citeseer
V
Pubmed
V
Squirrel
v
Chameleon
Cornell
V
Penn94
V
V
V
Table 8. The hyperparameters corresponding each dataset.
Target domain
Pre-training learning rate
Downstream learning rate
Epoch
Unified dimension
Dropout
Downstrearn
Cora
0.0075
0.001
60
50
0.1
30
Citeseer
0.001
0.001
60
S0
0.1
30
Puhmed
0.0001
0.0015
60
50
0.1
Squirrel
0.01
0.0003
100
50
0.1
15
Chameleon
0.02
0,01
100
50
0.1
15
Comell
0.02
0.0003
100
50
0.1
15
Penn94
0.0001
0.003
100
50
0.1
F. Methodology Details
Following the approach proposed in previous works (Liu et al., 2022; Shen et al., 2024), after constructing the cosine similarity matrix of He, we implement post-procsssing techniques to ensure that A exhibits the characteristics of sparsity, non-negativity, symmetry, and normalization. For the convenience of discussion, the subscript is omitted hereinafter.
ANN for sparsity. In most applications, a fully connected adjacency matrix often has limited practical significance and incurs high computational costs. Therefore, we employ the k-Nearest Neighbors (ANN) operation to sparsify the learned graph. For each node, retain the edges with the top-k values and set the others to O, thereby obtaining the sparse adjacency matrix A Note that we employ efficient &NN with lecality-sensiiig hashing (Fatemi et al., 2021) to enhance the model's 's scalability. This approach avoids the resource-intensive computation and storage of explicit similarity matrices, reducing the complexity from (2) to O)) where [V]
the number of nodes and the batch size of the sparse ANN.
Symmetrization and Activation. Since worl-world connections are typically bidirectional, we symmetrize the adjacency matrix. Additionally, the weight of each edge should be non-negative. Given the input AP these operations be expressed follows:
sym a (P)) (A*P)
(10)
where (()) represents non-linear activation implemented by the ReLU function.
Normalization. The normalized adjacency matrix with self-loops can be obtained follows:
A' (D>ym)- -
1/2 /.ymmm (Dsvm)
(11)
where Dagm the degree matrix Ásym with self-loops. Subsequently, for each view, we can acquire the adjacency matrix A/. which possesses the desirable properties of sparsity, non-negativity, symmetry, and normalization.
G. Details of Loss Functions
Note that directly computing mutual information between two graphs is impractical due to the complexity of graph- structured data. Since we focus on node-level tasks, we assume the optimized graph should guarantee that each node's neighborhood substructure contains sufficient task-relevant information. Therefore, this requirement can be transferred to mutual information between node representations (Liu al., 2024), which can be easily computed using a sample-based
15

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Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment
differentiable lower bound. For any view i and j. the lower bound In of the mutual information 1(2;; is (Liang et al., 2023):
xp) log cxpf(2t, =²-) 2/mp(2) EN exps (=', 2d)
Iu(Z'; = EE,
(12)
where .)
-) is score critic approximated by neural network, p(=t, 2d) denotes the joint distribution of node representations from views and j. while p(z') denotes the marginal distribution. 2' and 25+ mutually positive samples, representing the representations of the same node in views and j respectively.
Note that in our work, we redefine : G12 i AD,, where A is the refined adjacency matrix to identify the positive samples. That is to say, z' and 2 are mutually positive samples if and only if node and node j+ are neighbors in matrix A Recall the following loss function in the pre-training phase:
L == - G22 t 4e) - I(G:: G12 t tA)
(13)
where le is the identity matrix. Particularly, the second term incorporates the refined graph structure A to increase the number of positive samples. The minimization objective L in multi-domain pre-training is calculated as follows:
sim, 2e1 9:2) To
IV n= 1 1, mm, A, ni (sim(Z 7th Z \/
Cr. (Zill ) = log
E D22) = loog
(14)
simi Zn ///)/Te
ΣM gorm (2m (2t) 2:2)/To
m=
M
L=
Σ
(L,, (Zil Zi0 E1, (Z Z
1
(Lxy (Zil) 2 ( Z(
(15)
2
where Z denotes the node representations of the original graph G₁1 (A,, X() obtained through the graph encoder, and 712 denotes the node representations of the refined graph G.2 X7)- 2:1 is the non-linear projection of 7 Am[m, n] denotes the element in the m-th row and n-th column of the adjacency matrix A', which corresponds to the edge weights of the learned structure via GSL, siv((::) -) denotes the cosine similarity. is the temperature parameter, which is fixed at 0.2 in this work. During the pre-training phase, we sequentially minimize the objective C for each input graph with different domains.
16


Reviews:
[{"review_id": "dxVu0wgAOt", "reviewer": "Reviewer_XcGn", "summary": "The authors propose a unified approach that aligns graph topologies and features across domains, leveraging Graph Structure Learning (GSL) to refine noisy and adversarial-prone real-world graphs. The framework also introduces an efficient prompt-tuning mechanism to enhance knowledge transfer to unseen domains.", "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": "We sincerely thank the reviewer for the highly encouraging and constructive feedback. We are especially grateful for your recognition of our model’s generalization capability, methodological contributions, and comprehensive evaluations. Below we address your valuable suggestions.\n\n**Weakness: Intuitive explanations of model components**\n\nWe sincerely thank the reviewer for pointing out the potential accessibility issue due to the introduction of multiple novel components.  We agree that intuitive explanations will improve the clarity of our method, especially for broader audiences not specialized in cross-domain graph learning.  Below, we provide a brief intuitive summary of each key component:\n\n**Domain Tokens** ($t_{D_i}$): Each domain token acts like a value vector in a Transformer model, storing domain-specific knowledge during pretraining. During downstream phase, the target domain serves as a query to selectively retrieve and apply relevant knowledge from the source domains.  This design enables flexible and efficient cross-domain transfer through implicit attention-like behavior.\n\n**Shared Token** ($t_S$): This acts like a \"semantic anchor\" shared across all domains. It helps extract and preserve invariant patterns that reside in multiple domains, enabling better cross-domain alignment.\n\n**Balance Token ($t_{B_i}$)**: This component adaptively balances the contribution of node features and graph topology. Intuitively, it acts as a \"tuner\" that decides how much structural information to retain versus how much feature content to emphasize, especially helpful when structural noise or heterophily is present.\n\n**Dual Prompts** (meta-prompt and specific-prompt): These serve as \"adapters\" during downstream transfer.    Meta-prompt transfers generalized knowledge learned from source domains, while the specific prompt fine-tunes the model to the unique structure and features of the target domain.\n\nIn the revised version, we will incorporate these intuitive explanations into the methodology section (Sec. 4) to improve accessibility without sacrificing technical depth.\n\n**Comment 1: \"facilitate robust knowledge transfer.\" → Consider \"facilitate effective and robust knowledge transfer.\" for clarity.**\n\nWe appreciate the suggestion. We will revise the corresponding phrase in the refined version.\n\n**Comment 2: Limitations and assumptions of error bound theorem**\n\nThank you for highlighting this. Our theoretical results (Theorems 5.1 and 5.3) rely on the covariate shift assumption and existence of invariant subgraphs across domains. In the revised paper, we will explicitly enumerate the following limitations:\n\n1. The error bound assumes that the target distribution lies within (or close to) the convex hull of the source domains. In highly diverse or outlier domains, this assumption may be violated.\n2. One potential limitation of our theoretical framework lies in the assumption of the existence of a universal invariant graph learner $\\Phi^*$. This assumption requires that core semantic and structural patterns are preserved across domains after graph structure learning (GSL). However, in real-world scenarios where the relationship between topology and features varies significantly across domains—e.g., in one domain, structure dominates label prediction, while in another, node features are more informative—such shared invariances may not naturally exist. In these cases, identifying a single $\\Phi^*$ that captures consistent and transferable structural knowledge across all domains becomes highly non-trivial. We acknowledge this as a theoretical boundary condition and note that our empirical results suggest MDGFM remains effective even when this assumption is only approximately satisfied.\n\nDespite these theoretical assumptions, we observe in our ablation and robustness experiments (Sections 6.3–6.5) that MDGFM maintains strong performance even under domain removal and adversarial perturbation, which empirically validates the practical soundness of the theoretical setup.\n\nOnce again, we sincerely appreciate your time and effort in reviewing our paper. Your constructive criticism has been invaluable in refining our work, and we are more than happy to add clarifications to address any additional recommendations and reviews from you!", "claims_and_evidence": "Well supported.", "methods_and_evaluation_criteria": "It is convincing.", "theoretical_claims": "Correct.", "experimental_designs_or_analyses": "It is convincing.", "relation_to_broader_scientific_literature": "Current graph models often struggle with generalization due to challenges such as graph heterogeneity and scarcity of domain-specific data. Creating robust and adaptable graph foundation models is the next big thing for practical applications.", "essential_references_not_discussed": "Quite complete.", "other_strengths_and_weaknesses": "1. The paper is well-structured and presents a valuable contribution. It addresses a critical gap in the field of graph representation learning by focusing on topology alignment across diverse domains. \n2. The introduction of domain tokens and shared tokens for semantic alignment is innovative and effectively bridges the gap between domains with varying structural and feature characteristics.\n3. The experimental evaluation is comprehensive. The results demonstrate consistent improvements over state-of-the-art baselines in both one-shot and few-shot learning scenarios.\n4. Some intuitive explanations could be given. For example, the paper introduces several new components (e.g., domain tokens, shared tokens, balance tokens, dual prompts) that may make it not accessible to a broader audience.", "other_comments_or_suggestions": "1.\"facilitate robust knowledge transfer.\" → Consider \"facilitate effective and robust knowledge transfer.\" for clarity.\n2.The error bound theorem is strong, but consider discussing potential limitations or assumptions", "questions_for_authors": "Refer to the 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": "VfDiWqGilh", "reviewer": "Reviewer_6nDL", "summary": "The authors propose MDGFM to solve the graph pre-training issue. The key contributions include: A novel framework that aligns graph topologies across multiple domains using Graph Structure Learning (GSL);An adaptive embedding mechanism that balances features and topologies for improved generalization; A dual-prompt tuning approach that enhances adaptation to unseen domains. Extensive experiments validate the model’s effectiveness.", "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": "We sincerely thank the reviewer for the positive and constructive feedback. We greatly appreciate your recognition of the importance of the problem, the motivation of our proposed components, and the comprehensive empirical evaluation. Below we address your valuable suggestions and questions.\n\n**Weakness 1: Computational complexity**\n\nThanks for your concern. While our method introduces additional modules such as GSL and prompt-tuning, we have carefully designed MDGFM to remain computationally efficient and scalable to real-world graphs. Following your advice, we provide a detailed analysis of the computational complexity of MDGFM for both the pre-training and downstream phases. In the **pre-training phase**, each of the $N$ source graphs is processed independently. For a graph with $|V|$ nodes and $|E|$ edges (here we select $|V|=max\\\\{V_n\\\\},|E|=max\\\\{E_m\\\\},n,m \\in [1,N]$), the model first aligns node features via truncated PCA, which reduces the input dimension from $d'$ to $d$ at a cost of $\\mathcal{O}(|V| \\cdot d \\cdot \\log d')$. As for token lightweight element-wise multiplication, the time complexity is $|V|d$. It then applies locality-sensitive hashing $k$NN for graph structure learning. Denote the batch size of sparse $k$NN as $B$, each requiring $\\mathcal{O}(d)$ operations, resulting in $\\mathcal{O}(|V| \\cdot B \\cdot d)$ time for reconstructing the graph. Next, an $L$-layer GCN operates on the refined structure , contributing an additional $\\mathcal{O}(L\\cdot |V|\\cdot d^2+L \\cdot |E| \\cdot d+|V|d)$. Therefore, the total pre-training complexity across all source graphs is $\\mathcal{O}\\left(N \\cdot \\left[ |V| \\cdot d \\cdot \\log d' + |V| \\cdot B \\cdot d +L\\cdot |V|\\cdot d^2+ L \\cdot |E| \\cdot d \\right]\\right)$.\n\nSimilarly, in the **downstream phase**, PCA procedure takes $\\mathcal{O}(|V_T|\\cdot d \\cdot log d^{\\prime})$ time. Prompt fusion and token modulation cost $\\mathcal{O}(|V_T| \\cdot d)$. GSL again uses local sensitive $k$NN, which adds $\\mathcal{O}(|V_T| \\cdot B \\cdot d)$. The GCN encoder then performs $L$-layer message passing with cost $\\mathcal{O}(L\\cdot |V_T|\\cdot d^2+L \\cdot |E_T| \\cdot d+|V_T|d)$. Finally, classification is done via prototype matching, where each node compares with $C$ class centroids, yielding $\\mathcal{O}(|V_T| \\cdot C \\cdot d)$. Summing these terms, the overall downstream complexity is $\\mathcal{O}(|V_T| \\cdot d \\cdot \\log d'+|V_T| \\cdot B \\cdot d + L \\cdot |E_T| \\cdot d +L\\cdot |V_T|\\cdot d^2+ |V_T| \\cdot C \\cdot d)$.\n\nOverall, the model scales linearly with the number of nodes and edges, and benefits from efficient structure refinement and modular design like local sensitive $k$NN. We will include this complexity analysis in the final version for completeness.\n\n**Weakness 2 & Question 2** \n\nThank you for highlighting this important point. Following your suggestions, we conduct **new ablation studies** and include the results in **https://anonymous.4open.science/r/Ablation-study-on-prompts-FD62**. Specifically, we compare four variants of the model: Full MDGFM (with both meta- and specific prompts), w/o meta-prompt (only using target-specific prompt), w/o specific prompt (only using global meta-prompt), and w/o both prompts (i.e., no prompt tuning at all). The results show that:\n\n1. Removing the specific prompt leads to a noticeable performance drop, especially in domains with strong local structural patterns, confirming its role in target adaptation. Removing the meta-prompt also hurts performance, particularly in low-shot settings, indicating that it captures generalizable cross-domain knowledge. Note that it is normal that meta-prompt causes relatively slight impact than specific prompt. \n2. Obviously, removing both results in degradation, confirming that the two prompts are complementary and essential for effective transfer.\n\nThese findings support our design choice of dual-prompt tuning and clarify their respective impacts. \n\n**Question 1: Hyperparameters**\n\nWe appreciate the reviewer’s attention to experimental details. For feature projection, we apply PCA to reduce the dimensionality of all node features to **$d = 50$**. This value is chosen based on empirical studies and balances expressiveness with computational efficiency. For $k$NN graph construction, we use different $k$ values depending on the structural characteristics of the graph: For homophilic graphs we use $k = 30$ to keep more original relations between nodes. As for heterophilic graphs, we use a smaller $k = 15$ to avoid amplifying noisy connections. We have also provided a sensitivity analysis (Appendix C), which shows that MDGFM remains robust across a range of $k$ values. \n\nTo further demonstrate the robustness of our model to different hyperparameter choices, we additionally conduct a **sensitivity analysis** on the feature dimension $d$, and compare the results against GCOPE and MDGPT. The experimental results are summarized in **https://anonymous.4open.science/r/Sensitivity-on-d-6510**.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "Yes", "experimental_designs_or_analyses": "Yes", "relation_to_broader_scientific_literature": "The problem of multi-domain generalization in graph learning is important, and the paper presents a timely solution inspired by the success of foundation models in NLP and CV.", "essential_references_not_discussed": "The problem of multi-domain generalization in graph learning is important, and the paper presents a timely solution inspired by the success of foundation models in NLP and CV.", "other_strengths_and_weaknesses": "Strengths: \n\nThe problem of domain generalization in graph learning is essential, and the paper presents a timely solution inspired by the success of foundation models in NLP and CV. The topology alignment mechanism and graph structure refinement are well-motivated, addressing key challenges in cross-domain graph learning. The paper evaluates adversarial robustness and domain sensitivity, showing the model’s resilience to noise and distribution shifts.\n\nWeaknesses: The following aspects could further strengthen it.\n\n1. The computation complexity is unclear. Given the increasing size of real-world graphs, a complexity analysis would improve clarity.\n\n2. The paper emphasizes the role of meta-prompts and specific prompts, but it does not extensively analyze their individual contributions.", "other_comments_or_suggestions": "See Strengths And Weaknesses.", "questions_for_authors": "Q1: How are the hyperparameters selected, such as the dimensionality of the feature space and the number of neighbors in kNN?\n\nQ2: How does MDGFM perform with and without prompt tuning?", "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": "wd5psxKqsQ", "reviewer": "Reviewer_QrBx", "summary": "This paper proposes the Multi-Domain Graph Foundation Model (MDGFM) to address the challenge of transferring knowledge across graphs from different domains. MDGFM aligns graph topologies through a decoupled embedding mechanism, a graph structure learning module, and a prompt-tuning approach. This alignment allows MDGFM to effectively transfer knowledge from multiple source domains to a target domain, even for unseen domains. Theoretical analyses and experiments on both homophilic and heterophilic graph datasets validate the robustness and efficacy of MDGFM.", "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": "We appreciate your thoughtful feedback. Your constructive criticism is invaluable in refining our work. Below, we give point-by-point responses to your comments. \n\n**Weakness 1 & Question 1: Further explanations**\n\nThank you for raising this important point. We agree that a clearer explanation of MDGFM’s performance across different datasets enhances the completeness of our work. To address it comprehensively, we now analyze the performance from two perspectives: task setting (one-shot vs. multi-shot transfer) and graph homophily (homophilic vs. heterophilic). \n\n(i) One-shot vs. Multi-shot\n\n**In one-shot settings**, the target domain has few labeled data, so adaptation heavily relies on domain alignment and structural generalization. Thus, methods like GCOPE and MDGPT which pretrain from multiple domains get better results than supervised or graph prompting methods. Specifically, MDGFM outperforms all baselines due to its knowledge transferring capacity. It effectively captures both domain-specific information and shared patterns across domains.\n**In multi-shot settings**, baselines like GraphCL and GPF (even GCN) occasionally catch up or outperform multi-domain pre-training methods. This is because they leverage target supervision more directly, and in high-label regimes, their implicit overfitting can yield short-term gain. However, MDGFM remains competitive and often more stable under cross-domain generalization due to invariant learning.\n\n(ii) Homophilic vs. Heterophilic Graphs\n\n**On homophilic graphs**, traditional GNNs like GCN or GAT may perform decently due to strong neighborhood-label consistency. However, MDGFM still shows advantages in low-label settings due to cross-domain transferability via decoupled embedding and prompt regularization.\n\n**On heterophilic graphs, MDGFM significantly outperforms almost all baselines.** This is because standard message passing methods suffer from abundant noise resided in topology structure, while our framework applies GSL to learn invariant knowledge robustly, learning domain-aligned graphs that reduce harmful interference.\n\n**Weakness 2: Additional experiments on large-scale datasets**\n\nWe sincerely appreciate your suggestion to include large-scale evaluations. In response, we carefully selected and added **three additional large-scale datasets** with 30K+ nodes (i.e., Github, Deezer and T-Finance), covering diverse graph domains and scales, to thoroughly validate the scalability and generalization ability of MDGFM. Experimental settings are same as previous setups. New results along with the original Penn94 evaluation are summarized in **https://anonymous.4open.science/r/Large-scale-datasets-35EE**. It shows that MDGFM still outperform all baseline methods and shows great scalability.\n\n**Question 2: Additional experiments on imbalanced datasets and noisy data**\n\nThanks for your concern. We would like to clarify that our paper includes robustness analysis under multi-type noisy conditions in Section 6.5. Expansion of robustness analysis could be seen in Appendix C.1. Specifically, we simulate noise by randomly adding or deleting edges, even conducting meta-attack. Results demonstrate that MDGFM consistently outperforms baselines under all scenarios, which confirms its robustness to structural noise.\n\nIn response to the reviewer's suggestion, we conduct **new experiments on imbalanced data** using a real-world financial graph dataset T-Finance, which exhibits high class imbalance with skewed label distributions (minority-majority ratio up to 0.048:1). We compare MDGFM with GCOPE and MDGPT under same training setups using ACC/F1/AUC metrics.\n\nThe results, summarized in **https://anonymous.4open.science/r/Imbalanced-Noisy-data-05ED**, show that MDGFM achieves significantly better performance confirming its effectiveness under imbalanced label regimes. We believe these results, together with the original noisy-graph experiments, provide a comprehensive view of its robustness.\n\n**Question 3: Generalization to Dynamic or Temporal Graphs**\n\nCurrently, MDGFM is designed for static graphs. However, due to its modular and decoupled structure, it can be extended to dynamic settings in future work. Specifically the prompt and structure learning modules can be adapted to handle temporal snapshots. We appreciate this insightful suggestion and will discuss temporal extensions in the Conclusion section.\n\nAdditionally, thank you for pointing out the relevant work SAMGPT(WWW 2025), which is the published version of citation [Yu et al., 2024]. We also acknowledge the difference in the font of \"k\", and will correct it in the final version.\n\nOnce again, we thank the reviewer for the constructive suggestions. With the added experiments, domain analyses, and scalability discussion, we believe MDGFM has strong theoretical grounding, broad practical utility, and extensibility toward future challenges. We hope these improvements merit a stronger overall recommendation.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes, the methodology and evaluation make sense to me.", "theoretical_claims": "Yes, I checked them.", "experimental_designs_or_analyses": "Yes, the experiments are sound to me.", "relation_to_broader_scientific_literature": "Existing graph models, such as graph neural networks (GNNs), heavily depend on labeled data, which is often scarce and costly. This paper proposes an effective and robust unified graph foundation model, which performs well on graphs in different domains.", "essential_references_not_discussed": "There is a new publication, which is highly related to this submission. The authors should take a look.\nSAMGPT: Text-free Graph Foundation Model for Multi-domain Pre-training and Cross-domain Adaptation, WWW 2025.", "other_strengths_and_weaknesses": "Strengths\nThe paper is well-structured and easy to follow, with a clear presentation of the methodology and results.\nThe authors provide a solid theoretical foundation for MDGFM, including proofs of its effectiveness and domain generalization capabilities.\nThe proposed MDGFM shows robust performance across various datasets, including both homophilic and heterophilic graphs.\nWeaknesses\nThe authors should provide more detailed explanations of the experimental results, particularly why certain methods outperform others in specific scenarios.\nThe authors should include more large-scale datasets to validate the scalability of the proposed method.", "other_comments_or_suggestions": "The font for k is different in the caption and Fig.5.", "questions_for_authors": "Could you provide more insights into why MDGFM performs better on certain homophilic and heterophilic datasets?\nHow does MDGFM handle imbalanced datasets or noisy data?\nHave you considered applying MDGFM to dynamic graphs or temporal datasets?", "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: All reviewers acknowledged the effectiveness and novelty of the proposed method of graph foundation models and recommended acceptance. The rebuttals addressed some minor issues related to the computational complexity of the proposed algorithm and experiment on larger datasets. In sum, this is a high-quality work and should be accepted.
Decision: Accept (poster)