How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Authors: Haonan Yuan
Qingyun Sun
Junhua Shi
Xingcheng Fu
Bryan Hooi
Jianxin Li
Philip S. Yu
Venue: ICML 2025 poster (2025)
Keywords: graph foundation models
graph multi-domain pre-training
graph prompt learning
cross-domain generalization
Primary area: deep_learning->graph_neural_networks
TL;DR: In this paper, we introduce BRIDGE, a bounded gaph foundation model pre-trained on multi-domains with generalization guarantees.

Abstract: Graph Foundation Models hold significant potential for advancing multi-domain graph learning, yet their full capabilities remain largely untapped. Existing works show promising task performance with the “pretrain-then-prompt” paradigm, which lacks theoretical foundations to understand why it works and how much knowledge can be transferred from source domains to the target. In this paper, we introduce BRIDGE, a bounded graph foundation model pre-trained on multi-domains with Generalization guarantees. To learn discriminative source knowledge, we align multi-domain graph features with domain-invariant aligners during pre-training. Then, a lightweight Mixture of Experts (MoE) network is proposed to facilitate downstream prompting through self-supervised selective knowledge assembly and transfer. Further, to determine the maximum amount of transferable knowledge, we derive an optimizable generalization error upper bound from a graph spectral perspective given the Lipschitz continuity. Extensive experiments demonstrate the superiority of BRIDGE on both node and graph classification compared with 15 state-of-the-art baselines.

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Haonan Yuan
Qingyun Sun
Junhua Shi
Xingcheng Fu? Bryan Hooi Jianxin Li' Philip S. Yu"
Abstract
them a powerful tool for solving real-world problems. As large models like GPTs and DeepSeek have shown remark- able success in handling multi-modal Euclidean data (Chang et al., 2024; Guo et al., 2025a), they struggle with graphs. The inherent non-Euclidean structures, combined with het- erogeneity and task diversity, makes it hard for large models to process effectively (Mao et al., 2024a; Jin et al., 2024).
Graph Foundation Models hold significant poten- tial for advancing multi-domain graph learning, yet their full capabilities remain largely untapped. Existing works show promising task performance with the "pretrain-then-pree poreenppropp" paradigm, which lacks theoretical foundations to understand why it works and how much knowledge can be trans- ferred from source domains to the target. In this paper, we introduce BRIDGE, a Bounded gRaph foundatIon model pre-trained on multi- DoaliinD with Generalization ges.antEes. To learn discrim- inative source knowledge, we align multi-domain graph features with domain-invariant aligners dur- ing pre-training. Then, lightweight Mixture of Experts (MoE) network is proposed to facilitate downstream prompting through self-supervised selective knowledge assembly and transfer. Fur- ther, to determine the maximum amount of trans- ferable knowledge, we derive an optimizable gen- eralization error upper bound from graph spec- tral perspective given the Lipschitz continuity. Ex- tensive experiments demonstrate the superiority of BRIDGE on both node and graph classification compared with 15 state-of-the-ar baselines.
Within this context, Graph Foundation Models (GFMs) have garnered significant attention in very recent studies (Mao et al., 2024b; Shi et al., 2024a: 7i et al., 2024). A GFM is typically defined as a graph learning model pre-trained on extensive multi-domain graph data and adapted for diverse downstream graph tasks (Bommasani al., 2021). Such a "pretrain-then then--inttuee paradigm has validated its feasibility of constructing GFMs (Tang et al., 2024a; He & Hooi, 2024; Lachi et al., 2024). Beyond that, graph prompt learning is promising technique to speed up test-time fine-tuning, demonstrating the updated "pretrain-ther then-theeeeeeeee prompt" paradigm enabling generalization capabilities to any target domains and diverse tasks, showing a potential path to GFMs.
However, despite the initial success, "pretrain pneetrin-tthen-ae thennproomt" frameworks still face notable limitations: (1) Over-reliance on text-attributed graphs. Several methods rely heavily on aligning features with textual descriptions, leveraging lan- guage models to interpret spmaii--peeiffc semantics (Tang et al., 2024a; Li et al., 2024a; Wang et al., 2024; Tan et al., 2024). Yet, many open graph datasets lack explicit text at- tributes, making such approaches impractical for extensive real- world applications. Furthermore, obtaining textual an- notations for non-text graphs can be costly, time-consuming, and prone to domain biases induced by language model hallucinations (Ribeiro et al., 2021), ultimately hindering generalization. (2) Insufficient theoretical analysis. While graph prompts have empirically demonstrated strong perfor- mance in transferring knowledge across graph domains (Sun et al., 2023; Yu et al., 2024b), theoretical insights remain sparse. Beyond discussing learnable parameters and fine- tuning methods, there is limited theoretical understanding of the mechanisms behind effective prompt-based based transfer, es- pecially in few-shot scenarios. Consequently, foundational theoretical bounds guiding graph prompt generalization re- main unclear.
1. Introduction
Graph-structured data is pervasive across wide range of do- mains, including social network analysis (Fan et al., 2019), recommendation systems (Wu et al., 2019), drug discov- cry (Lin et al., 2021), and bioinformatics (Gligorijević et al., 2021), etc. Graphs are capable of representing complex rela- tionships and attributed knowledge between entities, making
'KLCCSE,, School of Computer Science and Engineering, Beihang University Key Lab of Education Blockchain and Intel- ligent Technology. Ministry of Education, Guangxi Normal Uni- versity 'School of Computing. National University of Singapore "Department of Computer Science, University of Illinois, Chicago. Correspondence to: Qingyun Sun <sunqy @ bdaa.cdu.cn>
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
The limitations are non-trivial due to three key challenges,
2. Related Work
(1) Pre-training: How to align multi-domain features? Graph features across different domains show varying distri- butions, often characterized by both distinct dimensions and semantics. Traditional feature alignment techniques like SVD (Stewart, 1993; Yu et al., 2024c) or non-linear map- ping with MLPs (Yin al., 2022: Hou et al., 2024), often fail to capture domain-specifie knowledge. The challenge lies in how to simultaneously and effectively align the het- erogeneous features at both dimension- and semantics-level while preserving shared and domain-invariam knowledge. (2) Down-prompting How to initialize prompts for effi- cient adaptation? Graph prompt learning aims for efficient adaptation with limited target domain data, which implicitly requires fast fine-tuning on restricted learnable parameters within limited epochs instead of re-training. Existing meth- ods suffer from inefficient random initialization of graph prompts, which results in slow convergence in fine-tuning, particularly in few-shot settings where the available labeled data is scarce. The challenge lies in how to initialize graph prompts in seff-super way enabling source knowl- edge to actively and adaptively support target adaptation. (3) Theoretical analysis: How much knowledge can be transferred? The ultimate goal of the GFM is serve as graph intelligence, taking in massive graph knowledge and enabling zero-shot transfer to any target domain. Despite the success of the aforementioned "pretrain- serruiin-the--pe then-prompl paradigm, the challenge lies in the lack of understanding regarding the lower bound of transferable knowledge, i.e., thorough theoretical analysis of the generalization error upper bound is crucial to quantify the limits of knowledge transfer, which can also play as regularizers constraining both the pre-training and prompting optimization.
2.1. Multi-Domain Graph Pre-training (Appendix H.1)
Pre-training on graphs (Liu et al., 2022; Hu et al., 2020a; Xie al., 2022) leverages self-supervised learning to cap- ture intrinsic semantics, but often assumes high similarity between domains, limiting generalization. Cross-domain learning (Zhao et al., 2024; Liu et al., 2024b) aims to transfer knowledge between domains but is often domain-specinf and task-tailored. Multi-domain methods (Liu et al., 2024b; Tang et al., 2024b) seek to align domain features, yet specific graph types constrain many or fail to address conflicts.
2.2. Graph Fine-tuning and Prompting (Appendix H.2)
Fine-tuning pre-trained graph models for downstream tasks is computationally expensive (Zhang et al., 2022; Sun et al., 2024; Zhili et al., 2024). Parameter-efficien fine-tuning (PEFT) (Zhu et al., 2024a; Tian et al., 2024) mitigates these issues but struggles in complex multi-domain scenarios due to the domain biases and task heterogeneity. Prompt learn- ing (Liu et al., 2023a; Zhou et al., 2022b) offers a more efficient way of guiding models with minimal updates but is limited to single-domain tasks and lacks scalability.
2.3. Towards Graph Foundation Models (Appendix H.3)
Graph Foundation Models (GFMs) extend language and vision foundation models to graphs, where pre-training on multi-domain data and transfer to downstream by parameter- efficient fine-tuning and prompt learning provides promis- ing way. A growing line of inquiry is into the existence of scaling laws in GFMs (Ma et al., 2024; Tang et al., 2024c; Chen et al., 2024a; Liu et al., 2024c). which hypothesize increasing parameter scale may yield emergence, such as en- hanced transferability and generalization across tasks. How- ever, its theoretical foundations remain under-developed.
To address these challenges above, we introduce BRIDGE, a novel Bounded gRaph foundatlon model pre-trained on with Generalization guarantEes. To capture discriminative source domain knowledge, we align multi- domain graph features with a set of domain-invariant align- ers. We then introduce lightweight MoE! network to en- hance prompting by enabling self-supervised and selective knowledge transfer. To quantify the maximum transferable knowledge, we derive an optimizable upper bound on the generalization error, grounded in graph spectral theories, leveraging the Lipschitz continuity. Our contributions are,
3. Preliminary
Notation. A graph is denoted as G = (V,E), where V is the node set and È is the edge set. Let A E {0, N be the adjacency matrix and X E R Xxd be the node features, where N = | node number and for input dimension.
Pre-training and Few-shot Fine-tuning, Given n source graphs {G.} E g5 from their labels { E y.S. the pre-training goal is to train graph learner h = (-)) on multi-domain graph datasets, after which the pre-training parameter O' is frozen. During down-promplin given a set of target graphs {GT} E GT from target domain (DT) EDT DT (seen or unseen) with m ac- cessible labels {YT}E) under m-shot setting (m & n), the goal is to fine-tune the pre-trained O' to adapt to predict the labels of the remaining unlabeled samples correctly.
{D$} DS with
We propose a novel multi-domain graph foundation model named BRIDGE with set of domain- invaaiiat feature aligners and efficient graph prompts initializers.
We analyze generalization error for prompt fine-tuning, and derive an optimizable upper bound for the first time.
Extensive experiments have demonstrated the superiority of BRIDGE on downstream node and graph classification compared with 15 of-te-o--hh-arr baselines.
2

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Graph Leamne h go (Graph Neural Networks)
H
H
H
H
H
X
2
X
A
A
Aligner Routing Network
(-))
Semantic-wise Alignment (A,)
Semantic-wise Feature Alignment
(A]]
S
X
Dimension-wise Fealure Alignment
(A,)
Ropc (2; H)
Dimension-wise Feature Alignment (Ai)
X
Graph Signals
X
X(A)
G
G
Academic
E-Commerce
Social
Àg Spectral Regularizer
À1
Transections
Molti-Domain Pre-training
Cross-Domain Down-prompting >>
Figure 1. Framework of the proposed graph features with domain-invariant dimension- and semantic-wise aligners. @ Cross-Domain Down-prompting: facilitates feature adaptation with a lightweight MoE network. @ Spectral Regularizer: determines the maximum amount of transferable knowledge.
-based BRIDGE. Multi-Domain Pre-training: unifics multi-domain
4. Proposed Method: BRIDGE
Assumption 4.1 (Domain Invariance). For each source domain D E {D$), there exist domain-specift invariant aligner A and variant aligner A` that lead to generalized graph representation under any target domains. Domain- invariant features X A C X should sutisfy.
In this section, we elaborate on the proposed "pretrain-then- prompt" -based BRIDGE with its framework shown in Fig- ure 1. During pre-training, we align multi-domain graph features with domain-invarian aligners. Then, a lightweight Mixture of Experts (MoE) network is proposed to facilitate downstream prompting. We derive an upper bound for the generalization error from the graph spectral perspective to determine the maximum amount of transferable knowledge.
Invariance Property: for any target domain D DEE(DSU {DT), the equality IPY:; & D;) Sufficient Condition: given any independent and random noise Y, = =(ff(X A.) + 6, i.e. Y, I AA AAA,, where (.) is the pre-trained graph encoder, g(-) is a discriminator: The logic structure syntaxes 1 and repre- sents label Y: is independent of A", but rely on A,
X holds.
4.1. Pre-training with Domain-Invarian Aligner
The primary goal of multi-domain graph pre-training to learn domain-specific knowledge and transfer the target domain via parameters. However, the multi-domain graph features are often associated with inconsistent dimensions and intrinsic noise, which may hinder knowledge extraction. aligner,
Assumption 4.1 suggests the aligned invariant features X! governed by A contain domain-specife causal correlations to their ground truth label Y,, which is instrumental in fa- cilitating the generalizability of the trained encoder fe) when adapting to unseen target domains. The Invariance Property ensures the aligner A can always filter out dis- criminative features for each domain by eliminating intrinsic noise dimensions and spurious correlations. The Sufficient Condition guarantees the aligned X; is sufficiently robust for forecasting outcomes under the error budget e. We ini- tialize A as random dimension-wise 0-1 mask (A4" is the complementary of A projected by some learnable matrix.
This motivates us to propose the which aligns features both dimensionally and semantically.
We denote Au as the aligner functioned for dimension-wine feature alignment, and A, designed for semantic- wise aligmment. Specifically, given graph G, (V,, E.) (i & n) with its feature matrix X e N.x,d, from the source domain, Au first acts dimension- wise alignment for every graph from each source domain,
Domain-Invariant Aligner.
X E Ax())), for all G, E (G*), (1)
Generalizable Pre-training Objective. Inspired by prior studies (Liu al., 2023b), we adopt link prediction as the pre-training task, as there exist vast number of links on the massive multi-domain graphs without requiring annota- tions, enabling the pre-training phase conducted in self- supervised manner. We leverage a universal task template based on the node similarity, which solves task heterogene- ity by incorporating classification and link prediction tasks.
where di denotes the unified dimension for source graphs. There exist multiple implementations for the As, where we combine SVD and an MLP. consequently. To further align semantic- wise features, encouraging pre-training model re- lying on domain -ppaii- knowledge that better generalizes to unseen domains, we make following assumption w.r.f. In- dependent Causal Mechanism (ICM) (Pearl& & Judea, 2009).
3

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
A series of quadruples (u, U4,,U,, Y.) are consistently and non-redundantly sampled from the source domain graphs {G*), where v4 is the direct neighbor of u, v_ is the nega- tive neighbor does not link to u, and Y, is the binary label indicates the link existence, given by the pair-wise node sim- ilarity discriminator g = MLP(ssi(.;)))) .)). The pre-training
Dimension-wise Adaptation. Similarly, given any sample (node graph) G; = (V., (i & m) with feature X? R-Nixd from target domain Dj E {D"), J. we then utilize the same dimension- wwie aligner A. to unify dimensionality,
X E ERNixdd = =) for all G, E [G"),, (6)
goal is to increase the semantic similarity between the node embedding h and h while decreasing that between h and h We formalize the task objective as,
where dimension d' = dd = d unless otherwise specified.
Selective Knowledge Assembly. However, performing fur- ther semantic-level feature alignment in the target domain proves challenging, primarily as no target domain knowl- edge is available during the pre-training. Building on the assumption that domain experts share common knowledge, we introduce a lightweight MoE network that selectively assembles source domain aligners (experts) to facilitate fea- ture adaptation in the target domain,
explo(h b 1/T) - - Σ explo(hg, h5)/7)'
Cpee ( ( (9;
(2)
(1,04,5-) ) WE (1+.0-)
where T is the temperature, H = stackife(X) A.) the invariant embedding obtained by the graph encoder.
Alignment Risk Regularization. could be expected that pre-training fe via Eq. (2) has limited generalization capa- bilities to target domains as the joint distribution of graphs and their labels being conditioned on domain intrinsic dis- tributions, Le., PP y D)= = P(g D)P('' D). To improve knowledge transferability of the pre-trained model, Eq. (2) is enhanced into a min-max dual optimization strat- egy on the multi-domain semantic alignment risk, Lee,,
X (S A, O X S R" (oftmax(o(XX)) )), (7)
where S is the assignment weight vector, o(-) is a learnable routing network initialized by X Intrinsic source-domain knowledge carried in A, are assembled and transferred to target domains aiding feature adaptation. Assembly MoE is constructed in lightweight scheme as it requires extra training for self-supervised way by minimizing its assembly uncertainty,
experts, and is optimized in a
min max @ DC(D")JLDT77 ((ar))= [D] (3)
[MOB (A,:) = H(S)= = Σ S, log(S.).
(8)
As P(D) is unknown, directly optimize Eq. (3) is infeasible.
Leveraged by the aligner A,, we transform Eq. (3) into,
Fine-tuning with Prompt Learning. The unification of pre- training and downstream tasks enhances knowledge transfer by the universal task template based on subgraph similarity. However, it remains crucial to differentiate domain- domain-specific characteristics. Inspired by prompts in natural language processing (Brown al., 2020), where they traditionally handcrafted instructions that steer downstream learning with task task-specific cues, we propose to recall potential knowledge with learnable prompts as few-shot graph learners, i.e.,
min pre H ('E O H K
(4)
where, H = fe @ X + (4
A). (5)
Note that, a and K are hyperparameters, V denotes variance. Parameter of GNN O* fixed once pre-training converges. Proposition 4.2 (Achievable Assumption). Minimizing Eg. (4) encourages pre-trained fe to satisfy Assumption 4.1. Proposition 4.3 (Equivalent Optimization). Optimizing Eq. (4) is equivalent to minimizing the upper bound of multi- domain semantic alignment risk in Eq. (3).
H. fā (Pn n @ XT Ar),
where P is the graph prompt with initialized tunable vec- tor 22. which is expected to be updated within constrained gradient descent epochs on m-shot supporting samples.
Proposition 4.2 avoids overly strong assumption, and Propo- sition 4.3 guarantees the transformation from Eq. (3) to Eq. (2) is equivalent. This highlights the role of aligners, which not only align dimension- and semantic- wise multi- domain graph features, but also enhance the generalization potential and model robustness during the in-context pre- training. We provide detailed proof in Appendix C.
Overall Down-prompting Objective. As we aim at both the node and graph classification tasks, we formulate the down-prompting objective with the same universal task tem- plate pre- -rriiiiiinn.. Given m (m < n)) supporting (train- ing) samples ((GT, ((GTYY")) Y where H denotes the node (for node classification) or ego-graph embeddings (for graph classification), the downstream classification objective is transformed into determining the highest similarity between the query sample and class prototype embedding, i.e.,
4.2. Down-prompting via Self-Supervised MoE
To bridge the gap between pre-training and downstream tasks (Liu et al., 2023b), our focus shifts towards transfer- ring multi-domain knowledge acquired during pre- prairrining to downstream domains to overcome domain misalignment.
fas (2 H²== E In exp(o(H7 HH,)/T) (GT,T)) yY((T)
(10)
Σ exp(g(HH ,H3))/T)
4

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
where H y; is the mean embedding (prototype) for samples in class Yg. Then, the overall down-prompting objective is,
Bound Implementations. To instantiate Eq. (15), we de- compose the generalization bound into four key components from [a] to [d]. Terms [a] and [b] represent the empirical er- ror on target and source domains, which can be equivalently optimized by Eq. (10) and Eq. (2), respectively.
Laww = Lota )") + 3L (A,;
(11)
where 39 is the trade-off hyperparameter.
In such ways, the lower bound of the amount of transferable knowledge (the upper bound of generalization error) con- strained by both the -ivereedeee (term [c) and the discriminative capability of the graph encoder fe (term [d), both of which are restricted by the Lipschitz constant Lg-- Proposition 4.6 (The Lower Bound of y (Gama et al., 2020; You et al., 2023)). Rewrite the first Wasserstein dis- tance (optimal transport distance) between G. and GJ,
4.3. Optimizing by Minimizing Generalization Error Upper Bound During Knowledge Transfer
To address how much knowledge can be transferred from the source to target domains, we reformulate this problem by minimizing the generalization error, which allows us to maximize transferable knowledge during down-pren pwompprompiing...
Generalization Error Upper Bound. Theoretically, the amount of transferable knowledge depends on the empirical adaptation errors. To this end, we leverage graph spectral theory and Lipschitz continuity to derive an upper bound that explicitly incorporates these factors.
dw(G,, G,)= (t)=-nn inf (IX:: (Xxl- -IAAA (17)
PEll
where P is the optimized permulation matrix by minimizing dw(G,, G2), II is matrix set. Such that, the lower bound of the Lipschitz constant of the encoder f is,
Assumption 4.4 (Lipschitz Continuity). Suppose f and g are Lipschitz continuous with constants y and Lg. satisfies,
Lf 7 LA (1 (1+ Iis / + +0(13)
(18)
Ilf(X, - f(X,, AA)ll
Ly _ *Iflllip = sup G,G,
(12)
-=IIA,-P'A,PE II (y 1,111++))--111199
dw(G., Gj)
where P is the optimal permutaaiion A, = U,A,U is the eigenvalue decomposition for both the adjacency matrices. LA is the spectral Lipschitz constant, defined such that for any eigenvalues X1. x2, the inequality Ir(>a) r(>2)l < Lx 111 Az). where r(-) is the frequency response. No is graph size after padding isolated nodes (Zhu et al., 202
(Hg -9(H, 112
Lg a 1ggllup=
sup H,
(13)
where dw denotes the first Wasserstein distance (Villani et al., 2009), and dg denotes the Euclidean distance.
The Lipschitz Continuity guarantees the smoothness of f and g. laying the groundwork for constraining h = g f with a Lipschitz constant La. which cooperates with the Wasserstein distance to establish a connection between cross-domain distribution shifts and the generalization error.
Eq. (18) bounds L, in a graph spectral perspective by char- acterizing its smoothness. This suggests that changes in the Lipschitz constant L inherently affect the balance between domain divergence (term (c) and discriminability (term [d)).
Proposition 4.5 (Generalization Error Upper Bound (Redko et al., 2017; Li et al., 2021; You et al., 2023)). Let H = th: G + 321h- = fo represent the hypothesis space of bounded real-valued functions, where VCdim (H) = d. Suppose h + hTT with the Lipschitz constant Lt, satisfies,
Bound as Spectral Regularizer. We transform Ls bound to regularization on both spectral and feature signals by evalu- ating its smoothness over consecutive spectral domains (You et al., 2023). Considering our "pretrain -ettraii-thhn--ie promppprpp" set- tings, the constraints should be restricted target domains aiming fine-tuning graph prompt a-- Given transformed signals H! U. H X! = U. XT_ the regularizer is,
||A - ( (G,)lll & &LygTgs (14) G.
L, - 1ip sup
G,)
Repec (Ω: XT )
where D D TT. With probability no less than - 8, the following inequality holds for target domain ervor c (hT)),
1 sum a (A (H.) v (. (20)
TT T(hTT n2
<
[a]
m
E =(hTT)
+
n
E
=(h))
m
where MM)) -M[INN :1 is row- differencing, reflecting signal deviations, v the threshold, and is the non-linear activator implemented by ReLU(-).
2 L (P(GS).P((T)
4
A
d o ( log 1 + 10 log " m
m log
Bounded Down-prompt Fine-tuning. In summary, we update the overall down-prompting objective in Eq. to))
.(55)
d
where à indicates empirical error, and An satisfies,
[Lamn] = Laown + 7 Repec (Ω: 87; H").
(21)
An = min (1e((15) - -(h))) (ns)) - 7 T (16)
Training pipeline in Appendix B, proofs in Appendix C.
5

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
5. Experiments
Multi-Domain Pre-training Settings. We mainly focus the challenging task of generalizing to unseen datasets and domains in downstream applications, incorporating the distinctions between cross-dataset and cross-domain setups,
In this section, we conduct extensive experiments to evaluate BRIDGE'.
We focus on the following research questions,
RQI: How effective for few-shot classification?
Cross-Dataset: the goal is evaluate cross-dataser per- formance when pre-training and down-prompting shares the same multiple domains but different datasets. Specif- ically, we take Academic (Cora, CiteSeer, PubMed) and E-Commerce (P (Photo, Computers) as the shared domains. When each individual dataset is used as the tar- get alternately, the remaining four datasets are collectively used as the pre-training source datasets.
RQ2: Which section contributes most the performance?
RQ3: How time-eflicient during down-prompting phase?
RQ4: How sensitive to hyperparameters fluctuation?
RQ5: Can LLMs enhance knowledge transferrability?
Detailed experimental setups, additional results, and analy- sis are provided in Appendix D and Appendix E.
extends to a more challenging scenario, where the target dataset belongs to an unseen domain than the source domain. Specifically, we take all datasets in Academic and E- E-Commerce domains as the source during pre-training, while Social Network (Reddit) as target.
Cross-Domain:
5.1. Experimental Settings
Datasets. To highlight multi-domain pre-training, we select six benchmark graph datasets from three different domains, which differs from conventional settings that consider any single dataset as one independent domain. Specifically,
Few-shot Down-prompting Settings. We evaluate node and graph classifications as the down-prompting task under the m-shot setting, where the samples (nodes or graphs) from each class are randomly selected supervision. As each dataset consists of single graph that is not directly feasible for graph classification, we construct ego-graphs centered on the target nodes within each dataset (Lu et al., 2021; Yu et al., 2024a;c), which are then applied for graph classification, with each ego-grag ago-graph labeled aligned with its central node. We assess the model performance by accuracy (Acc.) as few-shot is intrinsically balanced classification.
Academic Domain: three citation networks Cora (Mc- Callum al., 2000), CiteSeer (Giles et al., 1998), and PubMed (Sen et al., 2008), where nodes represent publi- cations, and edges denote citations.
E-Commerce Domain: two Amazon product networks Photo (Shchur et al., 2018) and Computers ers (Shchur et al., 2018), where nodes correspond to products, and edges indicate frequent and related co ro-purchases.
Social Network Domain: including Reddit t (Hamilton et al., 2017), which is a social network constructed from its platform, where nodes represent posts, and edges mean interactions, like comments and replies between posts.
5.2. RQI: Few-Shot Node and Graph Classification
One-Shot Classification. We evaluate our BRIDGE against 15 state-of-the-an art baselines under the challenging one-shot settings for node and graph classification of its effectiveness by transferring source knowledge in cross-dataset and cross- domain scenarios. Results reported in Table I and Table 2 illustrate foundational findings. Overall, BRIDGE outper- forms all 15 baselines in both node and graph classification with average improvement of 3.41% and 3.25%, respec- tively, which is relatively promising advancement with the extreme one-shot supervision. @ The most closely related GCOPE and MDGPT fail in both tasks. GCOPE shows subop- timal performance due to its implicit and inadequate domain feature alignment. MDGPT collapses mainly because its shal- low SVD dimension-winn wise feature alignment with the lack of knowledge transferring risk regularization. @ Cross-domain adaptation poses more difficult source knowledge trans- fer condition compared with the cross-dataset settings, and BRIDGE shows satisfying results when not only the source dataset is unseen, but the domains are also not accessible. 4 Additional results for fiye-shot setting in Appendix E. I demonstrate the same trend compared with the above.
Baselines, We compare BRIDGE with four primary cate- gories, 15 state-of-the-art baselines. Specifically,
Vanilla Graph Neural Networks: including GCN (Kipf& & Welling, 2022) and GAT (Velickovic et al., 2017), which are trained in an end-to-end manner without pre- tre-training.
Graph Self-Supervised Pre-training: GCC (Qiu et al., 2020), DGI (Veličković et al., 2019), InfoGraph (Sun et al., 2020), GraphCL (You et al., 2020), DSSL (Xiao al., 2022), and GraphACL (Xiao et al., 2024), which utilize self-supervised objectives pre-train and finetune.
Graph Prompt Fine-tuning: we select pioncering works including GPPT (Sun et al., 2022), GraphP r rompt (Liu et al., 2023b), GraphP rompt+ (Yu et al., 2024b), GPF (Fang et al., 2024), and ProNoG (Yu et al., 2025).
Multi-Domain Graph Pre-training: we select the most closely related baselines, including GCOPE (Zhao et al., 2024) and MDGFT ( (Yu et al., 2024c).
'https:// //g thub. com http///gnnttttccc///ReRe
6

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Table 1. Accuracy (% + standard deviation for five runs) of one-shot node classification. CR=Cora, CS =CiteSeer, PM PM-PubMed,
Ph Ph=Photo, Ccm C =Computers, Rdt =Reddit. The best results are shown in bold and the runner-ups are underlined.
Source
CS
PM
CR
PM
CR
CS
CR CS
CR
CS
CR
CS
PM
(Cross-Dataset Cross-Domuin)
Ph Com
Ph Com
Ph Con
PM Com
PM Ph
Ph Com
Model Target
CR
CS
PM
Ph
Com
Rêt
GCN (bb.) (Kipf& & Welling, 2022)
29.36 = 4.20
30.99 + 4.85
40.94 7.05
40.05 7.25
34.62 + 8.95
52.33 * 7.26 50.70 7.06
GAT (Velickovic et al., 2017)
2
29.46 3.33
40.09 5.221
35.60
33.15 a 6.59
GCC (Qiu et al., 2020)
31.67 2
32.55 2.69
41.66 + 4.58
4.99
35.91 a 5.68
55.67 a 6.01
DCI (Veličković et al., 2019)
40.822:4.44
31.85 0.55::4533 4.36
40.08 +: 6.22
47.23 6:00
37,05 : 6.40
61.95 : 6.22
GraphCI. (You et al., 2020)
33.28 = 6.03
29.12
39.31 + 7.05
42.98 6.554
42.87 + 5.32
60.06 + 5.29
DSST. (Xiao et al., 2022)
30.65 5.24
31.20+
40.89 + 6.94
45.32
38.41 + 6.54
57.23 + 6.61
GraphACL (Xiao et ul., 2024)
2
34.09
43.54 5.58
49.20
41,86 a 6.20
60.97 + 3.39
GPPT (Sun et al., 2022)
=
31.33
41.27 5.60
46,98 4.04
35.18 8.09
55.24 + 6.87
GraphFrompt (Liu et al., 2023b) GræphP rompt+ (Yu et al., 2024b)
37.95
3
45,85 8.58
50,42 8.05
42.58 7.21
64.71 71.71 + 5,21
36.06 6.93
33.85 7.62
45.08 + 7.99
52,28 +751
43.39 + 6.90
64. + 6.25
GPF (Fang et al., 2024)
40.26 8.33
4.10
47.33 6.63
51.48 5.34
40.09 6.19
63.52 = 6.37
ProNoG (Yu et al., 2025)
2.37
7.99
50.48 =: 7.06
63,30
50,29
64.87 5.188
GCOPE (Zhao et al.. 2024)
35.29 a 4.29
40.75
44.55 6.07
51.55
43.74 7.05
64.21 a 4.22
MDCPT (Yu et al., 2024c)
42.29 + 7.75
37.32 7.01
50,89 + 7.74
63.63 +.23
49.78 8:77
63.1 11 + 3.01
BRIDGE (ours)
46.44 8.01
42.18
56.35 7.22
67.87 7.45
54.04 8.52
68.55 3.49
Table Accuracy (% + standard deviation for five runs) of onc-shot graph classification, CR=Cora, CS CS=CiteSee:: PM=PubMed. Ph =h-Photo, C =Computers, Rat =Reddit. The best results are shown in bold and the runner-ups are underlined.
Source
CS
PM
CR PM
CR CS
CR
CS
CR CS
CR
CS
PM
(Cross-Dataset Cross- Dossa-DD
Ph .com
Ph Com
Ph Com
PM Com
PM Ph
Ph Com
Model/ Target
CR
CS
Ph
Com
Rdt
GCN (bb.) (Kipf& & Welling, 2022)
38.59 6.33
29.90 7.20
47.26
56.02 5.28
39.29 6:25
62.77 a 4.22
GAT (Velickovic et al., 2017)
35.64 = 5.09
26.78
41.72
50,23 a 4.98
37.25 a 6.73
60.13 a: 4.95
InfoGraph (Sun et al., 2020)
41.76 4.89
31.03
47.92 6.76
59.88 5.09
41.33 6.67
63.28 5.06
GraphCL (You et al., 2020)
40.23 5.28
3
49.21
58.02+
42.09 5.38
64.01 +: 5.17
DSSL (Xiao et al., 2022)
40.95 e 6.29
32.03 6.80
49.32
58.19 6.10
41.94 6.23
63.90 + 6.
GraphACL (Xiao et al.. 2024)
41.08 5:77
3
49,25 6.22
5,266
43.18 5:99
64.82 1 6.22
Grapherempt (Liu et al., 2023b)
42.18 4.31
38.66
51.28 5.29
60.07 5.70
47.31 5.11
65.56 5.18
CraphPrompt (Yu al., 2024b)
43.87 5.18
40.07
51.83
61.30
47.08 6.32
66.19 5.97
GPF (Fang al., 2024)
41.82
3..00
47.23
59.21
46.03
66.92 9.23
ProNoG (Yu et al., 2025)
50.98 + 7.33
47.37 7.08
55.35 7.90
63.29 6.33
50.21 7.29
73.09 + 4.03
GCCPE (Zhao et al., 2024)
44,29 7.23
39.10
50.82 8.10
61.83 +
$1,12 7.13
70.19
MDGPT (Yu al., 2024c)
49.11 7.63
43.56
53,90 9.83
63.41 6.70
49.56 + 7.63
73.14 6.39
BRIDGE (ours)
53.84 7.52
50.49 9.09
58.41 * 8.52
68.33 + 5.68
54.06 + 7.55
75.76 + 4.04
m-Shot Classification. To additionally evaluate BRIDGE's performance under different few-shot settings varied in m, we selected three competitive baselines in RQ1, including ProNoG, GCOPE and MDGPT for comparison. Results on both the cross-dataset (Cora) and cross-domaie (Reddit) tasks are shown in Figure 2 and Figure 3. (Additional re- sults included in Appendix E.2) Results indicate that, O BRIDGE outperforms the other three baselines in nearly all m-shot settings. However, in very low-shot scenarios, BRIDGE's performance slightly lags behind the baselines. This is due to the initial stabilization of its graph prompt learning, which depends on the generalization error bound and requires a few iterations to adapt with limited supervi- sion. In contrast, baselines like P ProNoG have optimized
parameters for low-shot settings, allowing them to outper- form BRIDGE initially before its regularization takes effect. @ BRIDGE shows a slight advantage when m is small, with its advantage gradually expanding as m grows larger, which we consider as benefiting from the spectral regularizer Rypes that enables more effective fine-tuning on more sufficient supervision through down prompts with a generalization eтror upper bound guarantee. Results in both scenarios show similar trends, highlighting BRIDGE's advantages in knowledge transfer. It demonstrates superior robustness in cross- drmss-n adaptation, particularly when the source and target domains differ, effectively transferring knowledge even without domain-specing data in the target domain, which is crucial for real-world applications.
7

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
ProNoG
GCOPE
MDGPT
BRIDGE (aurs)
PraNoG
GCOPE
MDGPT
BRIDGE (ours)
Node Classification
Graph Classification
Node Classilication
Graph Classificalion
Accuracy (9)
Accuracy (%)
d
80
85
80
60
Accuracy
75
Accuracy
80
50
70
75
50
40
66
10
5
10
10
Number of shats
Number of shots
Number of shots
Number of shots
Figure 2. m-shhot evaluation on Cora (cross-dataset)
Figure 3. m-shot evaluation on Reddit (cross-domails
BRIDGE
BRIDGE (wi A,)
BRIDGE (Wla MoE)
BRIDGE (wla R.pe)
PraNoG
GCOPE
MDGPT
BRIDGE (ours)
Node Classification
Graph Classifiation
Loss Curvafure
Accuracy Cuivelure
1.2
60
60
25
80-
Acquracy (%)
Accurscy (1)
Accuracy (5%)
1.1
50
50
60
Lea AT2
40
14sh
1100
40-
0
100
200
300
100
200
300
one-shot
fivesshot
one-shot
five-shot
Epech
Epoch
Figure Ablation studies on CiteSeer.
Figure 5. Down-prompt Fine-tuning on Photo (m = 5).
5.3. RQ2: Ablation Study
5.4. RQ3:
Efficiency Evaluation for Down-prompting
We conduct ablation studies concerning three key sections,
Graph prompt learning necessitates efficient training with a limited number of tunable parameters and epochs. To assess the down-proopting efficiency of BRIDGE, we analyze the loss and accuracy training curves for node classification task the Photo dataset with m shots. Results (Fig- ure 5) reveal that, O BRIDGE outperforms the competitive baselines in both loss reduction and accuracy improvement within fewer epochs. Specifically, BRIDGE converges to a lower loss (0.996) and achieves a higher accuracy (81.95%) much faster compared to other methods. This demonstrates that BRIDGE's training process is more efficient, reach- ing optimal performance without requiring as many itera- tions, which is crucial when computational resources and time are limited. @ In contrast, other baselines show rel- atively higher loss values, ranging from 1.0 to 1.1, and achieve weaker performance (accuracies between 68% and 75%). These methods take longer to converge, which in- creases the overall training time and computational cost. This inefficiency could be problematic in real-world settings where resources constrained. ? These results highlight BRIDGE's efficiency as graph adapter. The model lever- ages fewer training iterations to reach higher accuracy and lower loss, which is a significant advantage when perform- ing fine-tuning with limited labeled data or computational power, The efficiency validation demonstrated here rein- forces that BRIDGE is well-suited for real-world applica- tions, especially in scenarios where training time and com- putational cost are critical factors. This makes BRIDGE a viable option for environments where rapid deployment and scalability are essential.
BRIDGE (w/o A.): remove the semantic-wise aligner Ag. and only align multi-domain features with A..
BRIDGE (w/o MoE): remove the lightweight MoE, and replace the routing network by simple average routing.
BRIDGE (w/o Repec): remove the spectral regularizer Rspec in the bounded down-prompp fine-tuning objective.
Results (Figure 4) show that, O All three sections demon- strate effectiveness in boosting BRIDGE's s knowledge trans- ferability in classification performance. The lightweight MoE has a particularly pronounced impact on the one-shot learning. selectively assembles expert knowledge from pre-training, which helps compensate for the limited super- vision and makes it essential component for effective adaptation in low-shot scenarios. ② The spectral regular- izer Ryee becomes increasingly beneficial in the five-shot learning. It helps constrain the generalization error during cross-dataset and cross-domain tasks by leveraging more target supervision. Removing the semantic-wise aligner As results in a noticeable drop in performance, highlighting the importance of semantic alignment in ensuring effective knowledge transfer across domains, especially when feature semantics vary. O The performance improvement observed when including all components suggests that each plays a complementary role enhancing BRIDGE's robustness. The MoE and spectral regularizer are especially valuable in scenarios requiring efficient adaptation to unseen data.
8

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Aoquracy (%)
66.0
03.1
65.0
65.0
Table 3. Accuracy (%) of L.-Meehhanced few-shot performance targeted on Reddit under the cross doos--omain setting.
60.0-14
60.0-14e03
80.3
Source Target
CR
CS
PM
Fh
Com
Rd
3
0.01
0.05
0.1
0.5
K
0
Task
Node Classification
Graph Classification
Accurasy (%)
85.0
048
54.4
65.0
Model
0
5
0
5
64
61.3
MDGPT
49.69
63.11
75.29
64.80
73.14
80.29
60.0
64.0
MDGPT + LLM
51.08
65.56
76.34
66.07
75.35
82.02
0.01
0.05
0.1
0.
0.01
0.05
0.1
0.5
BRIDGE
55.72
68.55
80.12
67.18
75.76
84,42
BRIDGE LLM
57.16
70.70
82.31
69.14
77.31
86.87
Figure 6. Hyperparameter studies on Computers (m = 5).
5.5. RQ4: Hyperparamter Sensitivity Investigation
noticeable improvements across all few-shot settings. For example, BRIDGE improves from 55.72% to 57.16% in zero-shot node classification, and from 84.42% to 86.87% in five-shot graph classification. This demonstrates that LLM- generated text effectively enhances representation quality even in graphs originally lacking textual attributes. @ Com- pared with MDGPT, BRIDGE benefits more consistently from LLM enhancement. This observation suggests that the architectural components of BRIDGE, such as its semantic aligner and mixture-of-expers routing, are more capable of leveraging the additional semantic cues introduced by LLMs. @ The results support the view that integrating LLM- generated features can significantly reduce domain shifts in cross-domain scenarios. Even when textual data is not available natively, generating pseudo-description through language models provides scalable and generalizable solu- tion to improve knowledge transfer.
We evaluate the sensitvity of important hyperparameters in graph classification on Computers in Figure 6, where,
K: controls variant embedding augment times in Eq. (4).
Q: trades off between expectation and variation in Eq. (4).
3: balances between Lets and LM in Eq. (11).
7: acts between Ldown and Rapse in Eq. (21).
Results (Figure 6) show that, O BRIDGE is sensitive K, with performance peaking at K = 3 (65.0%). A moderate number of embedding augmentations improves robustness, while excessive augmentation slightly degrades. @ Increas- ing a leads to steady performance gains, reaching 64.9% at a = 0.5. This suggests that emphasizing the variation term in alignment risk enhances generalization. @ BRIDGE per- forms best when 3 0.1 (64.9%), indicating that moderate MoE regularization benefits expert selection. Higher values may over-constrain the adaptation process. 2 shows clear effect on performance, with the best result (65. (656%) at y = 0.1. The spectral regularizer helps reduce general- ization error, though overly strong constraints may hinder flexibility. All hyperparameters exhibit stable behavior within reasonable ranges, with identifiable optimal points, demonstrating the robustness and tunability of BRIDGE.
We can conclude that, enhancing graph models with L.M-- generated features proves to be a promising strategy for improving few-shot performance, particularly under chal- lenging settings where the target domain lacks labeled data.
6. Conclusion
We introduce a novel multi-domain graph pre-training and prompting framework BRIDGE with domain-invariant fea- ture aligners and efficient prompt initializers. We analyze knowledge transfer generalization error and derive an op- timizable bound as a spectral regularizer. Extensive exper- iments demonstrate its superiority in classifications com- pared with 15 tfate-of-the-art baselines. We also demon- strate compatibility by integrating with LLMs to enhance domain understanding and knowledge transferability.
5.6. RQS: LLM-Enhanced Multi-Domain Alignment
To explore the potential of language models in bridging cross-domain gaps, we leverage pre-trained large language models (LLMs) to generate textual descriptions for nodes based on raw graph data (Liu et al., 2024b; Li et al., 2024b). These generated descriptions serve as auxiliary node fea- tures and provide an implicit form of cross-domain align- ment, especially for graphs without inherent textual at- tributes. Results (Table 3) demonstrate that, O Both MDGPT and our BRIDGE perform relatively poorly in the zero-shot setting, which reflects the inherent difficulty of transferring knowledge across domains in the absence of any supervi- sion. This finding is consistent with prior research, where the availability of semantic-rich text information signifi- cantly enhances model generalization. @ After incorpo- rating L...--eeerated descriptions, both models exhibit
A Acknovledgments
The corresponding author is Qingyun Sun M. This work is supported in part by NSFC under grants No.623B2010 and No.62225202, by NSF under grants III-2106758 and POSE-23361558 and by the Fundamental Research Funds for the Central Universities. We extend our sincere thanks to all authors for their valuable contributions.

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Impact Statement
and new insights. arXiv preprint arXiv:2406.10727 2024b.
This paper advances the development of Graph Foundation Models (GFMs) by introducing BRIDGE, a novel GFM pre-trained on multi-domains with bounded generalization guarantees. As GFMs emerge cornerstone for leverag- ing graph-structured data across diverse domains, this work provides critical step toward robust and scalable knowl- edge transfer. With applications spanning social networks, e-commerce, and bioinformatics, this framework demon- strates the potential to significantly enhance multi-domain graph learning while maintaining computational efficiency. This work primarily aims to advance the field of machine learning, and we do not identify any specific ethical issues or societal risks requiring further attention.
Du, Y., Wei, F., Zhang, Z., Shi. M., Gao, Y., and Li, G. Learning to prompt for ppen-vocabulary object detection with langugle model. CVPR, pp. 14084-14093, 2022.
Fan, W. Ma, Y., Li, Q., He, Y., Zhao, E., Tang, J., and Yin, D. Graph neural networks for social recommendation. In www. pp. 417-426, 2019.
Fang, T., Zhang, Y., Yang, Y., Wang, C., and Chen, Uni- versal prompt tuning for graph neural networks. NeurlPS, 36, 2024.
Federici, M., Tomioka, R., and Forré, P. An information- theoretic approach to distribution shifts. NeurIPS, 34: 17628-17641, 2021.
References
Bai, Y. Ying, J., Cao, Y., Lv, X., He, Y., Wang, X., Yu, J. Zeng, K., Xiao, Y., Lyu, H., et al. Benchmarking foundation models with inguugg-moodellas-----cceo NeurIPS, 36, 2024.
Gama, F., Bruma, J., and Ribeiro, A. Stability properties of graph neural networks. IEEE TSP, 68:5680-5695, 2020.
Giles, C. L.,, Bollacker, K. D., and Lawrence, S. CiteSeer:
Bommasani, R., Hudson, D. A., Adeli, E., Altman, R., Arora, S., von Arx, S., Bernstein, M. S., Bohg, J., Bosse- lut, A., Brunskill, E., et al. On the opportunities and risks of foundation models. arXiv preprint 0rXiv:2108.07256 2021.
An automatic citation indexing system. In Proceedings of the Third ACM Conference on Digital libraries, pp. 89 1998.
Gligorijević, V., Renfrew, P. D., Kosciolek, T., Leman, J. K., Berenberg, D., Vatanen, T., Chandler, C., Taylor, B. C., Fisk, I. M., Vlamakis, H., et al. Structure- based protein function prediction using graph convolutional networks. Nature Communications, 12(1):3168, 2021.
Brown, T., Mann, B., Ryder, N., Subbiah, M., Kaplan, J. D., Dhariwal, P., Neelakantan, A., Shyam, P., Sastry, G., Askell, A., et al. Language models are few-shot learners. NeurlPS, 13:1877-1901, 2020.
Gong, C., Li, X., Yu, J., Yao, C., Tan, J., Yu, C., and Yin, D. Prompt tuning for multi-view graph contrastive learning, arXiv preprint arXiv:2310. 10362, 2023.
Cao, Y., Xu, J., Yang. C., Wang, J., Zhang, Y., Wang, C., Chen, L., and Yang, Y. When to pre-train graph neural networks? from data generation perspective! In KDD, pp. 142-153, 2023.
Gui, A., Ye, J., and Xiao, H. G-Adapter: Towards structure- aware parameter-efficient transfer learning for graph transformer networks. AAAI, 38(1 38(1)))122222612224 2024.
Chang, Y., Wang, X., Wang, J., Wu, Y., Yang. L., Zhu, K., Chen, H., Yi, X., Wang, C., Wang, Y., et al. A survey evaluation of large language models. ACM TIST, 15(3):
Guo, D., Yang, D., Zhang, H., Song, J., Zhang, R., Xu, R., Zhu, Q., Ma, S., Wang, P., Bi, X., et al. DeepSeek-RL In- centivizing reasoning capability in llms via reinforcement learning. arXiv preprint arXiv:2501.12948. 2025a.
1-45, 2024.
Chen, K.-., Zhang, J., Jiang, L., Wang, Y., and Dai, Y. Pre-training on dynamic graph neural networks. Neuro- computing, 500:679-66 687, 2022.
Guo, Z., Sun, Q., Yuan, H., Fu, X., Zhou, M., Gao, Y., and Li, J. GraphMoRE: Mitigating topological heterogeneity via mixture of riemannian experts. In AAAI, volume 39, pp. 11754-11762. 2025b.
Chen, Z., Mao, H., Li, H., Jin, W., Wen, H., Wei, X., Wang, S., Yin, D., Fan, W., Liu, H., et al. Exploring the po- tential of large language models (Ilms) in learning on graphs. ACM SIGKDD Explorations Newsletter, 25(2):
Hamilton, W., Ying, Z., and Leskovec, J. Inductive repre- sentation learning large graphs. NeurIPS, 30, 2017.
42 61, 2024a.
Chen, Z., Mao, H., Liu, J., Song, Y., Li, B., Jin, W., Fatemi, B., Tsitsulin, A., Perozzi, B., Liu, H., et al. Text-space graph foundation models: Comprehensive benchmarks
He, Y. and Hooi, B. UniGraph: Learning a cross-domain graph foundation model from natural language. arXiv preprint arXiv:2402. 13630, 2024.
10

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Hou, Z.,, Li, H., Cen, Y., Tang, J., and Dong, Y. GraphAlign: Pretraining one graph neural network multiple graphs via feature alignment. arXiv preprint wwxxi:2406.029555 2024.
Lachi, D., Azabou, M., Arora, V., and Dyer, E. GraphFM: A scalable framework for multi-graph pretraining. arXiv preprint arXiv:2407.1 2407. 2024.
Li, B., Wang. Y., Zhang. S., Li, D., Keutzer, K., Darrell, T., and Zhao, H. Learning invariant representations and risks for semi-supervised domain adaptation. In CVPR, pp. 1104-1113, 2021.
Hu, W., Liu, B., Gomes, J., Zitnik, M., Liang, P., Pande, V., and Leskovec, J. Strategies for pre-training graph neural networks. In ICLR, 2020a.
Hu, Z., Dong, Y., Wang, K., Chang, K.-W., and Sun, Y. GPT- GNN; Generative pre-training of graph neural networks. In KDD, pp. 1857-1867, 2020b.
Li, J., Sun, X., Li, Y., Li, Cheng, H., and Yu, J. X. Graph intelligence with large language models and prompt learn- 2024a.
ing. In KDD, pp.
Huang, Q., Ren, H., and Leskovec, J. Few-shot relational reasoning via connection subgraph pretraining. NewrIPS, 35:6397-6409, 2022.
Li, Y., Wang, P., Li, Z., Yu, J. x., and Li, J. ZeroG: Inves- tigating cross-dataset zero-shot transferability in graphs. In KDD, pp. 1725-1735, 2024b.
Huang, Q., Ren, H., Chen, P., Kržmanc, G., Zeng, D., Liang, P. S., and Leskovec, J. PRODIGY: Enabling in-context learning over graphs. NeurIPS, 36, 2024a.
Lin, M., Li, W., Hong, X., and Lu, S. Scalable multi-source pre-training for graph neural networks. ACM MM, pp. 1292-1301, 2024.
Huang, R., Xu, J., Jiang. X., Pan, C., Yang, Z., Wang. C., and Yang, Y. Measuring task similarity and its implica- tion in fine-tuning graph neural networks. AAAI, 38(11): 12617-12625, 2024b.
Lin, X., Quan, Z.,, Wang, Z.-J., Ma, T., and Zeng, X. KGNN: knowledge graph neural network for drug-drug interac- tion prediction. In IJCAI, pp. 2739-2745, 2021.
Jensen, W. V. Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta mathematica, 30(1):175-193, 1906.
Liu, H., Feng, J., Kong, L... Liang, N., Tao, D., Chen, Y., and Zhang, M. One for all: Towards training one graph model for all classification tasks. In ICLR, 2024a.
Jiang, B., Wu, H., Wang, B., Tang, J., and Luo, B. Reliable and compact graph fine-tuning via graphsparse prompting. arXiv preprint ar arXiv:2410.21749, 2024.
Liu, J., Mao, H., Chen, Z., Fan, W., Ju, M., Zhao, T., Shah, N., and Tang. J. One model for one graph: A new per- spective for pretraining with cross-domain graphs. arXiv preprint arXh::2412.00315, 2024b.
Jang, X., Lu, Y., Fang, Y., and Shi, C. Contrastive pre- training of gnns on heterogencous graphs. In CIKM, pp. 803-812, 2021.
Liu, J., Mao, H., Chen, Z., Zhao, T., Shah, N., and Tang. J. Neural scaling laws on graphs. arXiv preprint arXiv:2402.020 02055, 2024c.
Jin, B., Liu, G., Han, C., Jiang, M. Ji, H., and Han, J. Large language models on graphs: A comprehensive survey. IEEE TKDE, 2024.
Liu, P., Yuan, W., Fu, J., Jiang, Z., Hayashi, H., and Neubig, G. Pre-train, prompt, and predict: A systematic survey of prompting methods in natural language processing, ACM Computing Surveys, 55(9):1-35, 2023a.
Joyce, J. M. Kullback-lebblrr divergence. In Interna- tional Encyclopedia of Statistical Science, pp. 720-722. Springer, 2011.
Liu, Y., Jin, M., Pan, S., Zhou, C., Zheng, Y., Xia, F,, and Philip, S. Y. Graph self-supervised learning: survey. IEEE TKDE, 35(6):5879-5900. 2022.
Kaplan, J., McCandlish, S., Henighan, T., Brown, T. B., Chess, B., Child, R., Gray, S., Radford, A., Wu, J., and Amodei, D. Scaling laws for neural language models. arXiv preprint arXiv:2001.0883
Liu, Z., Yu, X., Fang, Y., and Zhang, X. GraphPrompt: Unifying pre-training and downstream tasks for graph neural networks. In WWW, pp. 417-428, 2023b.
2020.
Kingma, D. P. and Ba, J. Adam: A method for stochastic optimization. In ICLR, 2015.
Lu, Y., Jiang, X., Fang. Y., and Shi, C. Learning pre-train graph neural networks. AAAI, 35(5):4276442844 2021.
with graph convolutional networks. ICLR, 2022. Kipf, T. N. and Welling, M. Semi-supervised classification
Ma, Q., Mao, H. Liu, J., Zhang, Z., Feng, C., Song, Y., Shao,
Kraskov, A., Stögbauer, H., and Grassberger, Estimat- ing mutual information. Physical Review E Statistical, Nonlinear, and Soft Matter Physics, 6966106613,, 2004.
Y., and Ma, Y. Do neural scaling laws exist on graph self- supervised learning? arXiv preprint arXiv:2408.1 11243, 2024.

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Mao, H., Chen, Z., Tang, W., Zhao, J., Ma, Y., Zhao, T., Shah, N., Galkin, M., and Tang. J. Graph foundation models. arXiv preprint arXiv:2402.02216, 2024a.
Shen, J., Qu, Y., Zhang, W., and Yu, Y. Wasserstein dis- tance guided representation learning for domain adapta- tion. AAAI, 32(1), 2018.
Mao, H., Chen, Z., Tang, W. Zhao, J., Ma, Y., Zhao, T., Shah, N., Galkin, M., and Tang, Position: Graph foundation models are already here. In ICML, 2024b.
Shi, C., Chen, J., Liu, J., and Yang. C. Graph foundation model. Frontiers of Computer Science, 18(6), 2024a.
Shi, C., Yang, C.,, Fang, Y., Sun, L., and Yu, P. Lecture- style tutorial: Towards graph foundation models. In wwW, pp. 1264-1267, 2024b.
McCallum, A. K., Nigam, K., Rennie. J., and Seymore, K. Automating the construction of internet portals with ma- chine learning. Information Retrieval, 3:127-163, 2000.
Shirkavand, R. and Huang, H. Deep prompt tuning for graph transformers. arXiv preprint 2301:22309.1131,, 2023.
Mohri, M. Foundations of machine learning, 2018.
Myers, D., Mohawesh, R., Chellaboina, V. I., Sathvik, A. L., Venkatesh, P., Ho, Y.-H., Henshaw, H., Alhawawreh, M., Berdik, D., and Jararweh, Y. Foundation and large language models: fundamentals, challenges, opportuni- ties, and social impacts. Cluster Computing, 27(1):1-26, 2024.
Singh, A., Hu, R., Goswami, V., Couairon, G., Galuba, W., Rohrbach, M., and Kiela, D. FLAVA: A foundational language and vision alignment model. In CVPR, pp. 15638-15650, 2022.
Stewart, G. W. On the early history of the singular value decomposition. SIAM Review, 35(4):551-560 1993.
Pearl and Judea. Causal inference in statistics: An overview. Statistics Surveys, 3:96-146, 2009.
Sun, F.-Y., Hoffman, J., Verma, V., and Tang, InfoGraph: Unsupervised and semi-supervised graph-level represen- tation learning via mutual information maximization. In ICLR, 2020.
Pearl, Causal inference. Causality: objectives and assess- ment, pp. 39-33--55 39-58.220 2010.
Peters, J., Janzing, D., and Schölkopf, B. Elements of causal inference: foundations and learning algorithms. The MIT Press, 2017.
Sun, M., Zhou, K., He, X., Wang, Y., and Wang, X. GPPT: Graph pre-training and prompt tuning to generalize graph neural networks. In KDD, pp. 1717-1727, 2022.
Qiu, J., Chen, Q. Dong, Y., Zhang, J., Yang, H., Ding, M., Wang, K., and Tang, J. GCC: Graph contrastive coding for graph neural network pre-training In KDD, pp. 1150-1160, 2020.
Sun, X., Cheng, H., Li, J., Liu, B., and Guan, J. All in One: Multi-task prompting for graph neural networks. In KDD, pp. 1130-2131, 2023.
Redko, I., Habrard, A., and Sebban, M. Theoretical analysis of domain adaptation with optimal transport. In Machine Learning and Knowledge Discovery in Databases: Euro- pean Conference, pp. 737-753. Springer, 2017.
Sun, Y., Zhu, Q., Yang. Y., Wang, C., Fan, T., Zhu, J., and Chen, L. Fine-tuning graph neural networks by preserv- ing graph generative patterns. AAAI, 38(8):0053-90) 9061, 2024.
Reimers, N. Sentence-BERT: Sentence embeddings using siamese bert-networks. arXiv preprint arXiv: 10084, 2019.
Tan, Y., Zhou, Z., Lv, H., Liu, W., and Yang, C. WalkLM: A uniform language model fine-tuning framework for attributed graph embedding. NeurlPS, 36, 2024.
Ribeiro, L. F., Schmitt, M., Schütze, H., and Gurevych, Investigating pretrained language models for graph-to- text generation. In Proceedings of the 3rd Workshop on Natural Language Processing for Conversational AI, pp. 211-227, 2021.
Tang, J., Yang, Y., Wei, W., Shi, L., Su, L., Cheng, S., Yin, D., and Huang, C. GraphGPT: Graph instruction tuning for large language models. In SIGIR, pp. 491-500, 2024a.
Tang, J., Yang, Y., Wei, W., Shi, L., Xia, L., Yin, D., and Huang, C. HiGPT: Heterogeneous graph language model. KDD, pp. 2842-2853, 2024b.
Sen, P.. Namata, G., Bilgic, M., Getoor, L., Galligher, B., and Eliassi-Rad, T. Collective classification in network data. Al magazine, 29(3):93-93, 2008.
Tang, W. Mao, H., Dervovic, D.,, Brugere, I., Mishra, S., Xie, Y., and Tang, J. Cross-domain graph data seal- ing: showcase with diffusion models. arXiv preprint arXiv:2406.0 09899 2024c.
Shchur, O., Mumme, M., Bojchevski, A., and Günnemann, S. Pitfalls of graph neural network evaluation. arXiv preprint arXiv: /8/ 10/.1//1108000 05868, 2018.
12

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Tian, S., Luo, Y., Xu, T., Yuan, C., Jiang, H., Wei, C., and Wang. X. KG-Adapter: Enabling knowledge graph integration in large language models through parameter- efficient fine-tuning, In ACL Findings, pp. 3813-3828, 2024.
Yang, Y., Guan, Z., Wang. Z.., Zhao, W., Xu, C., Lu, W., and Huang. J. Self-supervised heterogeneous graph pre- training based on structural clustering. NeurIPS, 35: 16962-16974, 2022.
Yang. Y., Xia, L., Luo, D., Lin, K., and Huang, C. Graph pre-training and prompt learning for recommendation. arXiv preprint arXiv:231 1. 2023.
Velickovic, P., Cucurull, G., Casanova, A., Romero, A., Lio, P. Bengio, Y., al. Graph attention networks, stat, 1050 (20):10 12)550048550, 2017.
Yang, Y., Xia, L., Luo, D., Lin, K., and Huang. C. GraphPro: Graph pre-training and prompt learning for recommenda- tion. In WWW, pp. 3690-3699, 2024.
Veličković, P., Fedus, W., Hamilton, W. L., Liò, P., Bengio, Y., and Hjelm, R. D. Deep graph infomax. In ICLR, 2019.
Villani, C. et al. Optimal transport: old and new, volume 338. Springer, 2009.
Yi, Z., Ounis, I., and Macdonald, C. Contrastive graph prompt-tuning for cross-domain recommendation. ACM TOIS, 42(2):1 2023.
Wang, H., Feng, S., He. T., Tan, Z., Han, X., and Tsvetkov, Y. Can language models solve graph problems in natural language? NeurlPS, 36, 2024.
Yin, J., Li, C., Yan, H., Lian, J., and Wang, S. Train once and explain everywhere: Pre-training interpretable graph neural networks. NeurlPS, 36:35277-35299 2023.
Wang, W., Dai, J., Chen, Z., Huang, Z., Li, Z., Zhu, X., Hu, X., Lu, T., Lu, I.., Li, H., et al. InternImage: Exploring large-scale vision foundation models with deformable convolutions. In CVPR, pp. 14408-14419, 2023.
Yin, N., Shen, L., Li, B., Wang, M., Luo, X., Chen, C., Luo, Z., and Hua, X.-S. DEAL: An unsupervised domain adaptive framework for graph-level classification. In ACM MM, pp. 3470-3479, 2022.
Wu, Q., Zhang, H., Yan, J., and Wipf. D. Handling distribu- tion shifts graphs: An invariance perspective. In ICIR, 2022.
Ying, W., Wang, D., Hu, X., Zhou, Y., Aggarwal, C. C., and Fu, Y. Unsupervised generative feature transformation via graph contrastive pre-training and multi-objective fine- tuning. In KDD, pp. 3966-396699 09766,9766400 2024.
Wu, S. Tang, Y., Zhu, Y., Wang, L., Xie, X., and Tan, T. Session- based recommendation with graph neural networks. AAAI, 33(01):346-33) 2019.
You, Y., Chen, T., Sui, Y., Chen, T., Wang, Z., and Shen, Y. Graph contrastive learning with augmentations. NeurlPS, $3:5812-5823, 2020.
Xiao, T., Chen, Z., Guo, Z., Zhuang. Z., and Wang, S. Decoupled self-supervised learning for graphs. NeurlPS, 35:620-665 634, 2022.
You, Y., Chen, T., Wang, 7., and Shen, Y. Graph domain adaptation via theory-grounded spectral regularization. In ICLR, 2023.
Xiao, T., Zhu, H., Chen, Z., and Wang, S. Simple and asym- metric graph contrastive learning without augmentations. NeurlPS, 36, 2024.
Yu, X., Fang, Y., Liu, Z., and Zhang, X. HGPrompt: Bridg- ing homogeneous and heterogeneous graphs for few-shot prompt learning, AAAI, 38(15) 2024a.
Xie, Y.. Xu, Z., Zhang, J., Wang, Z., and Ji, S. Self- supervised learning of graph neural networks: A unified review. IEEE TPAMI, 45(2):2412-24299 2022.
Yu, X., Liu, Z., Fang, Y., Liu, Z., Chen, S., and Zhang, X. Generalized graph prompt: Toward a unification of pre- training and downstream tasks on graphs. IEEE TKDE, 2024b.
Xu, J., Huang, R., Jiang, X., Cao, Y., Yang, C., Wang, C., and Yang, Y. Better with less: A data-active perspec- tive on pre-training graph neural networks. NeurIPS, 36: 56946-56978, 2023.
Yu, X., Zhou, C., Fang, Y., and Zhang, X. Text-free multi- domain graph pre-training: Toward graph foundation models. arXiv preprint arXiv:2400 13934, 2024c.
Xue, R., Shen, X., Yu, R., and Liu, X. Efficient large language models fine-tuning tne-tuning on graphs. arXiv preprint arXiv:2312.04737. 2023.
Yu, X., Zhang, J., Fang, Y., and Jiang, R. Non- Non-homophilic graph pre-training and prompt learning. KDD, 2025.
Yan, Y., Zhang, P., Fang, Z., and Long, Q. Inductive graph alignment prompt: Bridging the gap between graph pre- training and inductive fine-tuning toeiitunng from spectral perspec- tive. In wwW, pp. 4328 4339, 2024.
Yuan, H., Sun, Q., Fu, X., Zhang, Ji, C., Peng, H., and Li, J. Environment- dynamic graph learning for out-of- distribution generalization. NeurIPS, 36:49715-49747, 2023.
13

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Zeng, H., Zhou, H., Srivastava, A., Kannan, R., and Prasanna, V. GraphSAINT: Graph sampling based in- ductive learning method. In ICLR, 2020.
Zhang, J., Zhang, H., Xia, C., and Sun, L. Graph-BERT: Only attention is needed for learning graph representa- tions. arXiv preprint arXiv:2001.00 100000..55440 4000,00000 2020.
Zhang, J., Xiao, X., Huang, L.-K., Rong, Y., and Bian, Y. Fine-tuning graph neural networks via graph topology induced optimal transport. In IJCAI, pp. 3730-3736, 2022.
Zhang, P.. Yan, Y., Zhang, X., Kang, L., Li, C., Huang, F,, Wang, S., and Kim, S. 4PT4Rec: Graph prompt tuning for streaming recommendation. In SIGIR, pp. 1774-1784, 2024.
Zhang, Z., Liu, Q., Wang, H., Lu, C., and Lee, C.-K. Motif- based graph self-supervised learning for molecular prop- crty prediction. NeurlPS, 34:15870-15882, 2021.
Zhao, H.. Chen, A., Sun, X., Cheng, H., and Li, J. All in One and One for All: simple yet effective method towards cross-domain graph pretraining. In KDD, pp- 4443-4454, 2024.
Zhili, W., Shimin, D., Lei, C., and Xiaofang, Z. Search to fine-tum pre-trained graph neural networks for graph- level tasks. In ICDE, pp. 2805-2819. IEEE, 2024.
Zhou, K., Yang, Loy, C. C., and Liu, Z. Conditional prompt learning for vision- vision-language language models. In CVPR, pp. 16816-16822 2022a.
Zhou, K., Yang. J., Loy, C. C., and Liu, Z. Learning to prompt for vision-language models. IJC,, 130(7):2337- 2348, 2022b.
Zhu, Q. Yang, C., Xu, Y.. Wang, H., Zhang, C., and Han, Transfer learning of graph neural networks with ego- graph information maximization. NeurlPS, 34:1766. 1779, 2021.
Zhu, Q., Zheng, D., Song, X., Zhang, S., Jin, B., Sun, Y., and Karypis, G. Parameter-efficient tuning large language models for graph representation learning. arXiv preprint arXiv:2404.18277 2024a.
Zhu, Y., Guo, J., and Tang, S. SGL-PT: A strong graph learner with graph prompt tuning. arXiv preprint 12444 2023.
Zhu, Y., Wang, Y., Shi, H., and Tang, S. Efficient tuning and inference for large language models on textual graphs. arXiv preprint arXhv:2401.15569 2024b.
Zi, C., Zhao, H., Sun, X., Lin, Y., Cheng, H., and Li, ProG: A graph prompt learning benchmark. NerulPS, 2024.
14

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
A. Notations
Notations
Descriptions
G (V,8)
Graph G with node V and edge &.
A E {0,1) 1
Adjacency matrix A, where N is the number of nodes.
X Nxd
Node feature matrix X, with N nodes and d feature dimensions.
{G*) egs, {GT) e gr
Graphs sampled from source domain (S) and target domains (7).
(Yé) ys, (YT e EVT
Corresponding graph labels in source domain (S) and target domains (7).
[D$) E DE,
DT
Source domains and targeted domains.
P(G").P(GT) fe(): :R' H
Probability distributions of the source graph and target graph. Graph encoder with learnable parameter O
g()) Rh 1K" " R2
Graph discriminator composed of sim(::) :) and binary ML.. The graph learner compound of the encoder f() and the binary discriminator g(-) Graph signal frequency response function.
h= gof
r(-)
6())
Learnable routing network in the MoE.
a()
Non-linear activation function (ReLU). The number of training samples in the source domain.
The number of supporting samples (m-shot setting) in the target domain.
d
The aligned feature dimensions.
Aa RR + Rd
Dimension-win feature aligner (implemented by composing SVD and MLP). Semantic-wiss invariant and variant feature aligner (implemented by pair of masks). Source domain graph features, dimension-aligned features, and semantic-
A. A" 10
X X X1
aligned domain-invariant features.
H. h
Encoded source domain graph features and node features. Target domain graph features, dimension aliigned features, and adapted target domain features assembled with the lightweight MoE. Encoded target domain graph features and node features.
xT, "T,T,"" R
H
S
Assignment weight vector for the MoE network.
W
Learnable weight matrix for projection.
Pa
Graph prompt with learnable vector f.
H
Hypothesis space, defined as a mapping from graphs to labels.
VCdim(-)
The VC (Vapmik-Cheeronenkiin Chervonenkis) dimension of the hypothesis space.
Ly
Lipschitz constant for the graph encoder f.
Lg
Lipschitz constant for graph discriminator g. Lipschitz constant for graph spectral.
Lx d dw
Eigenvalues of the graph spectral signal.
Wasserstein distances.
dg
Euclidean distances.
e, E,
етror and empirical error for generalization in the source domains. еттог and empirical error for generalization in the target domains. Discrepancy between source and target domain errors.
TTeT
T, T.
Terms related the smoothness of graph features in Eq. (19). Permutation matrix and the optimized permutation matrix. Pre-training loss function.
P. p*
Cue
EMoE
Loss function for the MoE network.
Lets
Loss function for the down-prompting classification.
Laown
Overall Loss function for the down-prompling phase few-shot training. Bounded overall Loss function for the down- bown-prompting phase few-shot training. Graph spectral regularizer for multi-domain generalization. Hyperparameter that controls variant embedding augmenting times in Eq. (4). Hyperparameter trades-off between expectation and variation term in Eq. (4). Hyperparameter that balances between Las LMBE in Eq. (11). Hyperparameter that between Cdown and Rypc Eq, (21).
Lsown]
spec
K
o
B
?
Temperature parameter.
15

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
B. Algorithm and Complexity Analysis
The pre-training pipeline of BRIDGE is illustrated in Algorithm 1, and the down-prompting (fine-tuning) pipeline shown in Algorithm 2. Next, we theoretically analyze their computational complexity, respectively.
Algorithm Pre-training pipeline of BRIDGE.
Input: A set of source graphs (G$) E g from multi-domains (D$) E DD, where each of the graph G; is associate with its feature matrix X, E xR and adjacency matrix A invariant aligners Aa and A,; Learnable weight matrix for projection W; Number of pre-training epochs E;; Temperature parameter т; Number of variation times K; Hyperparameter a for trading-off expectation and variation terms. Output: Pre-trained graph encoder and discriminator g (-)).)).
E IR N; N:;
Initialize all parameters randomly; for e 1, Ej do
# Di Dimenst i on-wise feature alig gnment
Aligned graph feature: X + Eq. (1) for each G; E {G*);
Semanti c-wise feature a alignment
Aligned domain-invariant feature: X A o X with domain-invariant aligner A
# General i izable pre wee-training objective
Encode and stack source domain graph embeddings: H stack fo (I 1 Calculate pre-training loss: (0; t- Eq. (2);
# Ali Alignment Risk Regulari zation
for k = 2, ..., K do
Calculate intervened embedding: H + Eq. (5); end for Calculate alignment risk regularization by variations: fie risk + V([RCppe ( {()))
# Opt zation
Calculate the overall pre-training loss: Eprcoverrall Eper + aCrro.riki
Update e by minimizing Epreooeerll and back-propagation end for
Complexity Analysis of Algorithm We analyze the computational complexity of each part in Algorithm as follows.
fimension-wise feature alignment: We implement A, by a two-layer MLP (R& + Rd ). Its computational complexity takes O N,d,dS), where N; represents node numbers in source graph G,, and de is its initial feature dimension. We omit the computational complexity of the preceding SVD computation here.
Semantic-wise feature alignment: We implement A,, A..A! A E {0, 13mxd by a pair of 0-1 mask. The mask operations on x takes the computational complexity of O(E",))
Feature encoding: We encode n source graphs by a two-layer GNN, which takes the complexity of O (IE + dn)) + N,d))), where |& is the number of edges, d$ is the aligned feature dimension, and d, the hidden dimension. Pre-training loss calculation: Suppose there n& direct neighbors, n22 indirect neighbors (nodes that are connected) averaged for each node in the source domain graphs, we could sample n x x n2 quadruples, which is utilized to formalize the self-supervised pre-training contrastive loss. Its computational complexity takes ON + +12) ))))
Causal intervention variant dimensions: We perform K times weighted projection as means of causal intervention, which takes the computational complexity of
(2)))))))
Alignment risk regularization: K times variant feature encoding by the same two-layer GNN takes this computational complexity of O(K) Σn (IE + da) + Nzd,)).
The overall computational complexity for the pre-training phase takes,
((E + da) + +Nddt)) + < 2 N.dS) +
Nid,d)) + +
(M(
1=1
0 K
N((d813)++ +0(K Σ (18](ds dr.) + N.dk))
(B.1)
16

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Suppose every source graph has average of N nodes, E edges, with do initial feature dimension, the overall computational complexity is simplified to (dS d===0
= d),
(n((E(d+ d.) + Nd))) + OnNNd(do + d m2),
(B.2)
where the additional complexity beyond that of regular two-layer GCN does not lead to perceptible increase in pre-training time, n, R1, < N in most cases. Such that, the pre-training computational complexity of BRIDGE is on par with
other state-of ah--thr-rtt graph multi-domain pre-training models.
Algorithm& 2 Down-prompting (fine-tuiig)) pipeline of BRIDGE.
Input: m nodes class-balanced sampled from a target graphs {G") } E G" (node classification) or m target graphs {GT) E GJ from target domain (DT} E DJ (graph classification), where each of the graph G3 is associate with feature matrix X, E R R.d,dd, and adjacency matrix A E IR N:x N:; The pre-trained graph encoder and discriminator ((f) Domain- invariant aligners A, and A,; Routing network (b:); Assignment weight vector S; Tuna- ble graph prompts Pa; Number of pre-training epochs Ez; Temperature parameter т; Hyperparameter 3 and y.
Output: Fine-tuned graph encoder and discriminator g(f (-)).
Initialize all parameters randomly;
for -== 1,2, Eg do
# Dimension-wise feature alignment Aligned graph feature: X + Eq. (6) for each G; E {GT]:
Semant ic-wise feature alignment Initialize the assignment weight vector: S Softmax(e(X7))
Assemble source domain aligners (experts) to facilitate semantic alignment: X7 <- Eq. (7);
Optimize the MoE network: LMaE( (A,; S) +- Eq. (8);
# Down-prompt ing classi f cation objective
Encode target domain node or graph embeddings with graph prompts: H. + Eq. (9);
Calculate wwwn-ppromptigggggoo classification loss: Las (S2; + Eq. (10);
Generalizen pnteraalizatiin ion bound as spectral reqularizer
Calculate the spectral regularize: Rxpec (S2; 3T; + Eq. (20);
# Optim imi zation
Calculate overall loss: [ < fosss (2:H7) + 8 ML (A,;S) + R (P; ;T;
Update Q by minimizing Ldown] and back- back-propagation. end for
Complexity Analysis of Algorithm 2. We analyze the computational complexity of each part in Algorithm 2 as follows.
We apply the same A₁ in pre-training, which is implemented by two-layer MLP "d ). Its computational complexity takes O(N,,adT) ) for node classification and O N,d,d") for graph classification, where N; represents node numbers in target graph G,, and d, is its initial feature dimension.
Dimension-wise feature alignment:
(R:4 -
Routing network initialization: The learnable routing network <b-) is implemented by single-layer MLP, which is initialized by X The initialization takes the computation complexity of O(N,dT m) for node classification and O NddTn, n) for graph classification.
Selective knowledge assembly: The assignment weight vector takes the Softmax operation output of d(+), and then assemble A from source domains to guide dimension selection on XT, which takes the computational complexity of Omymd dd)) for node classification and O(nd d' Σm No) for graph classification.
MoE optimization by uncertainty: We applied the self superviedd optimization by minimizing the uncertainty of the assignment weight vector, where the uncertainty calculation takes the computational complexity of O(n) for both node and graph classification.
Graph prompt insertion: We initialize graph prompt with learnable vector Ω over the aligned features X which takes the computational complexity of O(N,d") for node classification and O N₂d") ) for graph classification.
17

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Feature encoding: We encode target graph(s) by the pre-trained two-layer GNN, which takes the complexity of + +z) (d" [E]] is the number of edges, d' is the aligned feature dimension, and h is the hidden dimension.
N,d)) for node classification and (
(I" de) + N;d )) for graph classification, where
Down-prompting classification loss: Suppose there are ne classes, the contrastive loss takes the computational complexity of (mmndd") ) for both node and graph classification.
Spectral regularizer: Extracting spectral signals with eigenvalue decomposition (EVD) requires ( for both node and graph classification.
The overall computational complexity for the Nown-prompting (fine-tuning) phase of node classification takes, da) +
O(Nd,"")+ONN,"" n)+ +00mm +0000
(B.3)
Suppose every target graph has an average of N nodes, E edges, with do initial feature dimension, the overall computational complexity of node classification is simplified to (dS d" = dd),
O + + +
+ O(Ndido + n)) + Omm(ndd + +nd + NN))+)N)>> + On))
(B.4)
As m, n, Nes d, do are all small constants compared with N, beyond that of regular two-layer GCN, the complexity primarily increases due to the ((N3) complexity required for eigenvalue decomposition.
The overall computational complexity for the down- poonpprmppingggge (fine-tuning) phase of graph classification takes,
Nid,d + + 0 Σ Ned n) + +0 (n(d
EN:) nat d" Σ No) + +O(n) + +0 + N,d")
Σ ((E + +dh) + N.dh)) + ((mmdd))
(B.5)
Similarly, the overall computational complexity of graph classification can also be simplified to (d = dd = d).
O(m(E(d+ + d.) + Nd.)) + O(mNd((d + + +d)) + O(mdn.) + ((n)+0(N3))
(B.6)
In the same way, since m, n, ne. d, and dg are all small constants relative to N, the additional complexity compared to a standard two-layer GCN is mainly attributed to the (NN)) complexity introduced by the eigenvalue decomposition.
To mitigate the O(N3) complexity of eigenvalue decomposition, several effective strategies are implemented (You et al., 2023). Mini-batch training is utilized to reduce computational overhead by processing only small subset of nodes in each batch rather than the entire graph, ensuring that each batch is completed in tolerable time (Hamilton et al., 2017 10:7: Zeng et al., 2020). Additionally, when EVD is indispensable, is performed only once for the entire graph, and the results are reused across subsequent training steps to avoid unnecessary computations. These optimizations keep the computational burden manageable, as efficiency experiments demonstrate that the total woom-prompting time remains in line with other competitive baselines, striking a balance between computational efficiency and scalability for large-scale graphs.
It is worth noting that we temporarily set aside the theoretical analysis of space complexity and instead focus on evaluating the actual computational time and space requirements through efficiency analysis experiments. Experimental results indicate that, under the hardware and software configurations detailed in Appendix F.3, the pre-training time and space complexity of our proposed BRIDGE are comparable to those of other competitive baselines. While the fine-tuning phase incurs slightly higher time and space complexity than other baselines, this can be effectively mitigated by adopting optimized training strategies achieve a balance between performance and efficiency.
In summary, the computational complexity of the BRIDGE framework consists of two phases: pre-training and fine-tuning. During the pre-training phase, the overall complexity is on par with other state-of-the-arr pre-training frameworks, as the additional computations for feature alignment and risk regularization remain managcable. In the fine-tuning phase, although both node classification and graph classification are influenced by the ((N)) complexity introduced by eigenvalue decomposition (EVD), the total down-pronpting time remains in line with other competitive baselines, striking a balance between computational efficiency and scalability for large-scale graphs.
18

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
C. Proofs
C.1. Proof of Proposition 4.2: Achievable Assumption
We first restate Assumption 4.1 and Proposition 4.2 for reference.
Assumption 4.1 (Domain Invariance). For each source domain D, {D$}. there exist a domain-specitte invariant aligner A, and variant aligner A", that lead to generalized graph representation under any target domains. Domain- invariant features X;- A @ X should satisfy,
Invariance Property: for any target domain D E {D3} ( }. the equality P(Y, | -
{ holds. Sufficient Condition: given any independent and random noise c. Y; =, g(fe ( A.)) + i.e., Y; A," AT, where fe(-) is the pre-trained graph encoder, g(-) discriminator. The logic structure syntaxes 1 and represents label Y; is independent of A but rely A
Proposition 4.2 (Achievable Assumption). Minimizing Eq. (4) encourages pre-trained fe to satisfy Assumption 4.1.
Proof. We begin by introducing the following lemma reformulate the Invariance Property and the Sufficient Condition in Assumption 4.1 via the mutual information theory (Kraskov et al., 2004).
Lemma C.1 (Mutual Information Equivalence). The Mutual Information I(;-) provides an equivalent representation of the Invariance Property and the Sufficient Condition in Assumption 4.1, which takes the following form,
Invariance Property: P(Y; X D2) = PPG X + I(Y; D3 X = 0;
Sufficient Condition: Y; = g(fe ( A.)) + X+) maximized.
Proof. We establish Lemma C.1 by independently proving the validity of both the Invariance Property and the Sufficient Condition (Yuan et al., 2023).
(1) Proofs of Invariance Property. Given three random variables x, y and the mutual information between and y, conditioned on is defined by,
T(x y/z) = Ep(2) (P(x, P(x(2))P() 2))]
(C.1)
The joint conditional distribution P(x. y (z) can be written as,
F(x.ylz) = P(xlzzy))P(yla)
(C.2)
Substituting Eq. (C.2) into Eq. (C.1),
F(x
1(x;y/z) = EF(())
log
(C.3)
Ix(1z)
Eq. (C.3) is exactly the Beilbackk-Leibles divergence (Joyce, 2011) between the distributions P(x|z, y) and P(x|2), weighted by y). Hence, we can conclude,
[(x;)(z) DKI,(P(x|z,, F(x|2)).
(C.4)
Then, substitute Y, into x, Dj into y, and X into in Eq (C.4), we readily obtain the following equation,
I(Y;: D2 X, DxL (P( D,) P(G X()).
(C.5)
When P(Y. Dj) P(Y. X)), the Bullback-Leible divergence between the distributions P(Y: (J, and P(Y; X equals 0, such that [Y;;; D, x = 0. Similarly, it is also straightforward to derive the left-hand side from the right-hand side of the equivalent condition of the Invariance Property. This will be elaborated further here.
(2) Proofs of Sufficient Condition. We establish the sufficiency and necessity through the following steps.
Step 1: We first prove for Y, XJ, and satisfying Y, = =f (X.. Ad) + condition X = arg maxx: I(Y;; X]) holds.
We proceed by contradiction. Assume that X # arg max 7(;;; ;;; XF), and suppose that there exists another X satisfies
19

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
X arg maxx; &: I(Y;; X)), such that X + X We always construct permutatioo-iivaaian mapping function IT
such that X = [I(X;, T., where и a random variable. This leads to the following sequence of equalities,
I(Y; x I( σ) = (lo(So A:)); 7 I X
(C.6)
Since X and x yield the same mutual information with Y,, this contradicts the assumption that X + X7 Therefore, the condition X arr maxx: x: I(Y;; X must hold.
Step Next, we prove that for Y, X and satisfying X arg maxx; T(Y;; X the condition Y; = g(Se(( A.)) te must also hold. Again, we use proof by contradiction. Assume Y; # (X; A,)) + c, and instead suppose Y, = gSS X A,)) + where X o X Under this assumption, we have the following inequality,
T(g(Se "X,", (d)) { < [(([J A.)): X
(C.7)
which implies that X," arg max x I(Y; X This contradicts the original assumption X arg max I(2;; X Hence, we conclude that Y; = g( (X) A.)) + must hold.
Next, we establish Proposition 4.2 from two perspectives. Step 1 proves minimizing the expectation term in Eq. (4) corresponds to maximizing the lower bound of [(;;; XY); thereby encouraging the pre-trained model to satisfy the Sufficient Condition. Step proves minimizing the variance term in Eq. (4) minimizes the upper bound of Z(Y;; D, X;). which ultimately leads the model satisfies the Invariance Property. The respective proof' are as follows.
Step I: To begin, we show that minimizing the expectation ferm in Eq. (4) (E.C (O; H ensures that the pre-trained model satisfies the Sufficient Condition stated in Assumption 4.1.
Referring to the Structural Causal Model (SCM) (Pearl & Judea, 2009; Pearl, 2010; Peters et al., 2017), given the dependency relationship X + G. + Y,, we can analyze that maxo( (Y;;; X;) is equivalent to (ino();(c.)) I(Y;; as we remove the spurious correlations embedded in the dependencies between Y, and G; by the set of comainn-nvarrian aligners. By considering Q[{} G:) as the variational distribution, the following upper bound can be derived, ie..
X;),
I(Y;;; { (P(, G;, || P(Y D,)
(C.8)
= (P(, D,) || P(Y, X,) - (P(y X,, QYY, X())
(C.9)
< (P(, G,, || [|{ {())
(C.10)
min (PP, G., D2) || QQY ((X))))). ((1)))
(C.11)
Additionally, the following expression holds,
P(Y G. Dÿ)
(P(, G., DI QY X()) = ED, E.,.,..K)~7G.11,,
log
(C.12)
Q(1; X
P(Y: Gi, D2) log
< Ep, Bo,,
(C.13)
E -x--0(X110)) Q(Y, X::)
Here, Eq. (C.13) relies on Jensen's Inequality (Jensen, 1906; Wu et al., 2022), and requires the GNN encoder fe(:) is Dirac delta distribution (f-distribution). Therefore, we reach,
min DKL (P(6 G,, D2) Q(Y X()) + min mun. Q(Y: X))
(C.14)
e
Thus, we have demonstrated that minimizing the expectation term in Eq. (4) corresponds to minimizing the upper bound of I(Y; G. X,) (or equivalently maximizing the lower bound of I(Y)) thereby encouraging the pre-traint pre-trained model to satisfy the Sufficient Condition.
20

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
We now demonstrate that minimizing the variance term in Eq. (4) (V( (e, guides the model to satisfy the Invariance Property in Assumption 4.1.
Step 2:
By considering Q QX}
G₃) as the variational distribution, the following upper bound can be derived, i.e.,
I(Y; Dj
x (PY, X || P(Y: x
(C.15)
(P(, X Dj) || Bn, D;))
(C.16)
(RL( X BD, Q(Y) X()) (QL( X P(Y, X,)
(C.17)
(ED, P(Y; { [D, (x:))
(C.18)
PKI( X || En, Q( (X])
(C.19)
min DRL (QY x Ep, QYY x)).. (07188))
(C.20)
Additionally, we have,
Q( X))
QQLY x ED, Q(Y x :. P2
log Ep, Q(Y; D2)
(C.21)
Derived from Jensen's Inequality (Jensen, 1906; Wu et al., 2022), have,
(Q X || Ed, Q(Y, x & ED, () - LD, Spce C (O;
(C.22)
where denotes embeddings bound with misaligned features under any domain D, E (D$). Finally, minimizing,
min DKL (Q(Y x ED, Q(Y X + [o..eerrpppp.a.a............aa.a i=1 ((X(X))
(C.23)
Thus, we have proven that minimizing the variance term in Eq. (4) minimizes the upper bound of 7(Y; Dj Xt), which ultimately leads the model satisfies the Invariance Property.
We conclude the proof for Proposition 4.2.
Proposition 4.2 avoids making overly stringent assumptions regarding the Sufficient Condition and Invariance Property.
C.2. Proof of Proposition 4.3: Equivalent Optimization
We first restate Eq. (3), Eq. (4), and Proposition 4.3 for reference.
Eq. (3):
min о
max B(G.))-)) [ee (O;H))
D DE(DT)
Eq. (4):
min ECpe ( EC
HS; (V((E where H. fe(A X + O &))...)).
Proposition 4.3 (Equivalent Optimization). Optimizing Eq. (4) is equivalent to minimizing the upper bound of
multi-domain semantic alignment risk in Eq. (3).
Proof. As stated in Eq. (3), the intrinsic generation process of graph (G,, from any domain D, E {D$} is determined by the distribution PP D2), where the semantic misalignment difference D, ~ P(D) between domains introduces
21

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
graph pre-training variations. Let Q(Y, G₂) denote the inferred variational distribution of the true posterior Q(Y; G2 D₁), then the multi-domain semantic alignment risk can be expressed as the KI. divergence between these distributions, i.e.
PPY: G D5) (TP, G2, D2) QVV G,)) = ED, B(G.X))+(GG31D))) log Q(Y G.)
(C.24)
Following Federici et al. (2021); Wu et al. (2022), we adopt an information-thheortii perspective to analyze this problem. To simplify the multi-doman semantic alignment risk in Eq. (C.24), we propose the following lemma.
Lemma C.2 (Multi-Domain Semantic Alignment Risk). The multi-domain semantic alignment risk satisfies the following upper bound.
(P(,, Gi Dy) QV, G.)) & (P(, Gi, DJ) QI &()).
(C.25)
where can be interpreted as the variational distribution for the label predictor (if two nodes are connected in Eq. (2)).
Proof. We start from Eq. (C.24).
P(Y: G., Dj) Q(, (..)
Dx (P(Y G. D,) I| Q(Y, GJ)) = Ep, ((1
log
(Eq. (C.24))
(C.26)
P(Y, G,,
= ED (.D.)).I..))))))))D))
log
(C.27)
E Rx(w0(2:1a)) Q(Y, | {{:)
log PP(Y G. Dj)
< ED, (.G...)..))).....D.
E Q G.)
(C.28)
Q(Y; X;)
(P(Y G. D2) Q(Y, x (Jensen's Inequality),
(C.29)
where the upper bound (Eq. (C.29)) emerges from the assumption that Q(} (i.e., Dirac delta function), as the GNN encoder fe(-) holds the assumption. This completes the proof for Lemma C.2.
G.) represents a deterministic distribution
From Lemma C.I, we can reframe Eq. (4) as,
min
(PL G D,, |I QY x()) + I(Y; (
(C.30)
Q(.
Consequently, using Lemma C.2, minimizing Eq. (4) reduces to minimizing the upper bound of multi-domain semantic alignment risk in Eq. (3), i.e.,
min prc ( H (+)++aV(BBB) )) ( (O: (H +
DKL(P(Y; G., D))IQQ( x)) + IIY;; X
min 0(X;)))).0))
negalive
(C.31) (C.3))
min Q ()).0..))
DPL( Q)) XF))
(C.32)
min (PL( G Q,) G.).
(C.33)
Q & (.).
is evident that as K increases (to positive infinity), it approaches the theoretical upper bound more closely. formal proof omitted for brevity. This concludes the proof for Proposition 4.3.
Proposition 4.3 declares optimizing the pre- pre-training optimization goal is equivalent minimizing the upper bound of multi-domain semantic alignment risk in Eq. (3), which is intractable. It also highlights the role that the set of feature aligners play during pre-training, which not only aligns multi-domain graph features dimension-wind wise and semantic-wise but also guarantees the learned knowledge is domain -spacffc and invariant, which enhances the generalization potential and model robustness during the in-context pre-training phase, laying a better and solid foundation for multi-domain knowledge transfer by -rompproo..
22

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
C.3. Proof of Eq. (8): Why Decrease Uncertainty Lead to Optimization?
Let the routing vector be S Softmax(((X")) where o(-) is a learnable routing network initialized by the target feature X", and S R" represents the assignment weight distribution over source domain experts (aligners). The uncertainty of S is quantified by its entropy.
H(S) Σ S, log(S.),
(C.34)
i=1
where S, is the i-th component of S, representing the weight of the i-th expert. The goal is to minimize the entropy (S) reduce the uncertainty of the routing vector.
We answer why decreasing uncertainty leads to optimization from four perspectives.
(1) The relationship between uncertainty and optimization. The entropy (S)) measures the uncertainty of the distribution S. High entropy indicates that the distribution more uniform, implying that the model does not prioritize any specific expert. This leads to an averaging effect across experts, which dilutes the influence of the most relevant ones. Conversely, low entropy corresponds
sharper distribution, where the model assigns higher weights to few critical experts, improving its ability adapt to target domain features. Thus, minimizing (S)) aligns with the goal of encouraging the model to focus on selecting high-quality experts while ignoring irrelevant ones, thereby improving feature adaptation.
(2) Lower entropy implies sharper distribution. We give the following lemma without proof.
Lemma C.3 (Relation between Distribution and Entropy).
I is uniform distribution (i.e., S; for all i), the entropy reaches its maximum,
H(S) = - Σ log = log(n).
(C.35)
1=1
this case, all components are equally weighted, indicating no preference for any expert. If concentrated on a single component (i.e., S > 1 for some j. and S, - ffr i + j), the entropy reaches its minimum value 0f
H(S) (1- + - log 0) = 0.
(C.36)
This indicates a fully deterministic selection, where only one expert is prioritized.
By analyzing the behavior of the entropy function, we can quantitatively demonstrate that, by minimizing entropy, the routing vector S transitions from a uniform distribution sharper one, emphasizing a few key experts while suppressing others. This sharpness ensures the model focuses on relevant experts, improving feature adaptation efficiency.
(3) Entropy minimization implicit regularization. Minimizing HS) introduces an implicit regularization that encourages S to focus on the most relevant experts. The mathematical intuition behind this process is as follows.
Entropy gradient: The gradient of (S) encourages larger S,, thereby concentrating the routing distribution: #H(S)
88,
-(1 +og S,).
Routing distribution convergence: Minimizing HS) ensures that S, 1 for few relevant experts, while S, - 0 for irrelevant ones. This leads to effective feature selection, where the model emphasizes only the most critical experts. Explicit objective: By reducing LMOE, the model suppresses noise from irrelevant experts, improving feature adaptation the target domain.
(4) Self-supervised mechanism. Minimizing (S) operates as a self-supervised mechanism. As the routing vector is derived entirely from the target feature X", implicitly encoding the relationship between the target domain and source domain experts. This process does not require additional labeled data, relying instead on the internal structure of S guide optimization. As result, the model automatically learns to select the most relevant experts, achieving effective domain adaptation without external supervision.
In summary, minimizing the uncertainty of the routing vector S by reducing its entropy H (S) leads to optimization because it concentrates the distribution on a few key experts, reducing noise and improving feature adaptation. Lower entropy implies a sharper distribution, which aligns the model's 's focus on the most relevant source domain experts. This sulf-super mechanism ensures efficient domain adaptation and enhanced performance in downstream tasks.
23

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
C.4. Proof of Proposition 4.5: Generalization Error Upper Bound
We first restate Proposition 4.5 for reference.
Proposition 4.5 (Generalization Error Upper Bound (Redko et al., 2017; Li et al., 2021; You et al., 2023)) Let H {(h + y g o f} represent the hypothesis space of bounded real-valued functions, where VCdim(H)
Suppose h * h with the Lipschitz constant Ln satisfies,
(G,(( hP(G))l
L - 1hllpp sup G.G,,
dw(G;, G3) & I.gT.
where D (..
With probability no less than m 2T 2'hh")
6, the following inequality holds for target domain error e (h?).
e (hh) <
[0]
b
n
2 SS()))
+
(4 14 Ly )(GG)))((GG)))
[4] A
n n+m m
n m
+ (
1 +
1
log n
m
log
+
log
8
d
d
where indicates empirical error, and An satisfies.
A = min - -
Proof. We first introduce the following lemma that introduces Wasserstein distance to relate the source and target domain generalization errors.
Lemma C.4 (Wasserstein Distance to Relate Errors (Shen et al., 2018)). Let pt,, Het E P(X) be two probability measures. Assume the hypotheses H are all L.p-Lipschizz continuous for some K. Then the following holds,
e (he (h,l') < e (h, + 2Lxdw(n , ,,²),
(C.37)
for every hypothesis h, h' E H.
Proof. We first prove that for every L -ip-hiisc continuous hypotheses, h, 1/ E H. 1h - N'| Lip-Lipschitz continuous.
Using the triangle inequality, we have,
[M(x) [(xx) [h(x)) h(v) + (k(y) 11(x))
(C.38)
1h(x) h(v)) (h(v) n1(v)) + [h((x) h'(v))
(C.39)
and thus for every E X 1h(x) - (x)) - (h(y) 11 (v))
[h(x) h(y) + 11(xx)) - k(vil
(C.40)
p(x,y)
p(x, y)
2L
(C.41)
where p(..)) is some distance measurement. Then, for every hypothesis h. h., we have,
ET {(h, h²) e (h,hh') = T [1(x)) N'(x))) - F [1(x) - (x)1
(C.42)
& sup [,r(f(x)] E,, [f()]]
(C.43)
11
2Lx (HTTTT)
(C.44)
Derived by Lemma C.4 and You et al. (2023) with the Lipsshit continuous assumption,
[IA - ~P(G,)) < Lg dw dw G3)
L A |
sup G,G,,
(C.45)
24

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
we reach the following inequality,
- -
-()) < 160
(C.46)
< < -
(C.47)
+ 11 P(G"))
(C.48)
Similarly,
- - (h")) < (h0) -c , P(G"))
(C.49)
We then combine Eq. (C.48) and Eq. (C.49) into,
|e (1)) e (hT)) < 2L,,, AA.
(C.50)
where,
A min -1,,) Such that, the following upper bound of target domain generalization error holds,
(C.51)
((hT) & ( + LT, P(GT) + An.
(C.52)
We next present the following corollary with proof given in Mohri (2018) Theorem 11.8.
Corollary C.5 (VC-Dimension Generalization Bounds (Mohri, 2018) Corollary 3.19). Let H. be a family of functions with (H) = d. Then, for any >> O, with probability at least 1 - 6, the following holds for all h E H.
2d
em
( log 2m
1 1 5)
e(h) (h)
log
(C.53)
777
d
Thus, the form of this generalization bound is,
( d
1/
<(h) (
log m
m d
(C.54)
which emphasizes the importance of the ratio m/ d for generalization. The theorem provides another instance of Occam's razor principle where simplicity measured in terms of a smaller VC-dimension.
((TT) in Corollary C.5, with probability at least 1 - 6, the following holds, By replacing e(h) with eT "(hT), ē (h) with @T
$
T ((hT) < < (1T)+ +
2d
log
cm
+ 2m
log
(C.55)
m
d
Upon the reference of Corollary C.5, we transform Eq. (C.52) into, with probability at least the following holds,
$ +
2d (2
1 8
((h))< (L)
+ +A +
log
log
(C.56)
n2
2n
Inspired by Li et al. (2021); You et al. (2023), we apply union bound to combine Eq. (C.55) and Eq. (C.56) with coefficients m/(n (n + m) and n/ (n + m), respectively, we reach.
T T(hT) <
m
ēT (T((T)) +
(
2d m
log
em
1
log
1
1/2
+
d
2m
n1 + m
n
(1()) + 2b, L (P(CS),P((T) + A +
2d
log
en
d
1/
+
2n
log
3
1
$ )
(C.57)
(TT(T")+ +
(2 log em
2d
m)) + (
$ ) 1og 1
'2d
en d
7/
1
1
1
log
+
log
71 + m
m
12 + m
2n
71
(P(G))
+ An).
(C.58)
11 + m
25

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Definition C.6 (Simplfied Cauchy-Schwarz Inequality). For any real numbers x and y, the Cauchy- inequality in its alternative additive form states that,
x x < +2)]..
(C.59)
This inequality is derived directly from the standard form of the Cauchy-Schwarz inequality and can be used to simplify expressions involving sums or to bound the magnitude of quadratic terms.
We applied the simplified Cauchy-Schwart Inequality to the two square root terms in Eq. (C.58), then we reach,
e ((h))
(1)) +
4d m
log
em + m
1 log 8) 3)
n n a++ m
+
[d
log
en + log
1 1 3 ))²²
n + m
n
d
n
(2L (2 ( )) + Am)
(C.60)
m
? (ThTT)
2 ( ) +
An)
n + m
n n m
n n m
m²
8d
em + m d
2
1 !)
n²
( 88d
log en + log 8
2
! )))
log
log 8)
+ (n + m)2 n
(C.61)
(n m)2
m
d
As m & n. we have,
((h)) < n +++ m
m
?Th²")
(5 2L, L.g (P(GS), + Sb.
n + m
1 ( 1 + n
d n + log
d
m
+
log log
(C.62)
d
m
d
We conclude the proof of Proposition 4.5.
C.5. Proof of Proposition 4.6: The Lower Bound of L We first restate Proposition 4.6 here.
Proposition 4.6 (The Lower Bound of Ly (Gama et al., 2020; You et al., 2023)) Rewrite the first Wasserstein distance (optimal transport distance) between G2 and Gj,
dw(G., inf (IX, - PX, AA, PA,PT(F) Penl
where P is the optimized permutation matrix by minimizing dw Gj), II is matrix set. Such that, the lower bound of the Lipschitz constant of the encoder f is,
Lf > LA 1 [ + O(T))
T= - - P'A,P' Di = (|Jl,
where P is the optimal permutation. A, U.A.U, is the eigenvalue decomposition for both the adjacency matrices.
Lx is the spectral Lipschitz constant, defined such that for any eigenvalues X1, X2, the inequality |r(>1) r r(>a)) <
La A - X2), where r(-) is the frequency response. N& is graph size after padding isolated nodes (Zhu et al., 2021).
Proof: Derived by Theorem in Gama et al. (2020), Lemma and Lemma 2 in You et al. (2023), we have the inequality,
I.)
& LAI' (I+ToVNs) NA + +
(C.63)
where I = AA; P'A, || is the structure perturbations, and F'sj is the eigenvector misalignment. Since,
[l A A2)
Lj A lffllup
sup G,G,
(C.64)
dw (w G3)
26

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees for any G2 and G3 E (GGS (IG",,I/(X; A.)- ,,,, & Lydw Gj). Then, Eq. (C.63) is transformed to,
LAT (1 (1+TayVNa) + O((T) + XT & ( G2) (Ly is the tight supremum)
(C.65)
< LyI.
(C.66)
As X and I' are both non-negative, then reach,
L,T LAT (1 (1+ToVNs) + +(T3)
(C.67)
Dividing both sides of the inequality by we obtain,
L > LA (1+ + +TT).
(C.68)
Since elements in adjacency matrics A, and A, belongs [0, 1/, such that I -A-PPA,P" ||F [0, 1]. Consequently, ((T2) & OT).. Then, reach,
L 2 Ln (1 (1+ + 0((3)
(C.69)
We conclude the proof of Proposition 4.6.
D. Experiment Details
In this section, we provide additional experiment details. Statistic of datasets is illustrated in Table D.1.
Table D. Statistic of the multi-domain graph dataset.
# Feature Dimensions
Dataset
Domain
# Nodes
# Edges
# Classes
Avg. # Deg.
Cora
Academic
2,708
10,556
1,433
7
3.90
CiteSeer
Academic
3,327
9,104
3,703
6
2.77
PubMed
Academic
19,717
88,648
500
3
4.50
Photo
E-Commerce
7,650
238,162
745
8
31.13
Computers
E-Commerce
13,752
491,722
767
10
35.76
Reddit
Social Network
232,965
114,615,892
602
41
492.00
D.1. Datasets Details
To highlight multi-domain pre-training, select six benchmark graph datasets from three different domains, which differs from conventional settings that consider any single dataset one independent domain. These datasets span varying structures, scales, and feature distributions, offering diverse set of challenges for pre-trained graph models. Specifically.
(1) Academic Domain. This domain includes three prominent citation networks, which are widely used as benchmarks graph representation learning. Each dataset represents a single graph where nodes correspond to documents, edges correspond to citation links, and node labels indicate the category or topic of the corresponding paper.
Cora (McCallum et al., 2000): consists 2,708 machine learning-related papers organized into research categories, including "Neural Networks" and "Probabilistic Methods". Each paper is represented by 1.433-dimensional sparse bag-of-words feature vector, where each dimension corresponds the presence of specific word in the document, The graph contains 10,556 citation links, making it moderately sparse. The dataset frequently used for semi-supervised node classification tasks due to its small size and well-defined class boundaries.
CiteSeer (Giles et al., 1998): comprising 3,327 scientific publications grouped into 6 predefined categories, including "Artificial Intelligence" and "Robotics". Nodes are represented by 703-diimensional sparse feature vectors, capturing word occurrences the documents. The graph has 9,104 citation edges, forming a sparse structure with fewer connections per node compared to Cora. CiteSeer is challenging for its noisy class labels and relatively low connectivity, presenting obstacles in capturing structural information.
27

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
PubMed (Sen al., 2008): a biomedical citation network includes 19,717 research articles diabetes, categorized into 3 classes. Each node is represented by dense 500-dimensional feature vector derived from Term Frequency- Inverss Document Frequency (TF-IDF) statistics of medical terms. The graph contains 88,648 citation edges, resulting in denser structure compared to the other two citation datasets. Due to its size and real-world biomedical context, P PubMed is often considered a benchmark for scalable graph learning algorithms.
(2) E-Commerce Domain. This domain includes two product networks derived from Amazon, capturing relationships between items frequently bought together. The two datasets are widely used for evaluating recommendation and product- related tasks. Nodes denote products, edges for co-purchase relationships, and node labels correspond to product categories.
Photo (Shchur et al., 2018): constructed from the "Electronics" category on Amazon, focusing specifically on camera and photo products. It consists of 7,650 products (nodes) and 238,16, 162 co-purchase relationships (edges). Each node is represented by 745-dimensional feature vector capturing product reviews and descriptions word embeddings. Nodes are categorized into classes based on product types. The graph exhibits moderately sparse structure and serves as benchmark for semi-supervised node classification.
Computers (Shchur et al., 2018): is also derived from the Amazon "Electronics" category but focuses on computer- related products. It comprises 13,752 products (nodes) and 491,722 co-purchase relationships (edges), forming denser graph compared to Photo. Each node is characterized by the same 767-dimensional feature vector format, based on product reviews and descriptions. The products are categorized into 10 classes. Computers is often used to test the scalability and robustness of graph learning models on larger and denser graphs.
(3) Social Network Domain. This domain includes Reddit (Hamilton et al., 2017), a large-scale social network constructed from the Reddit platform. this dataset, nodes represent individual posts, and edges signify interactions between posts, such comments, replies, votes. The graph comprises 232,965 nodes and 1 114,15,892 edges, making it one of the largest datasets commonly used in graph representation learning, Each node is associated with 602-dimensional feature vector generated from the textual content of the posts using pre-trained text embeddings. Posts are grouped into 41 distinct categories, corresponding to specific subreddits or discussion communities. Due to its size and dynamic nature, Reddit t is frequently used as a benchmark for evaluating models on inductive learning tasks, particularly to test their scalability and generalization ability on unseen nodes and large-scale graphs.
D.2. Baseline Details
We compare the proposed BRIDGE with four primary categories, 15 state-of-the-art baselines, including traditional graph neural networks, self-supervised pre-training methods, prompt-based fine-tuning approaches, and multi-domain pre-training frameworks. These baselines provide a comprehensive benchmark to assess the effectiveness and versatility of our approach.
(1) Vanilla Graph Neural Networks. This group includes traditional graph neural networks trained directly on downstream tasks without any pre-training. These models serve as foundational baselines for graph representation learning.
GCN (Kipf& & Welling, 2022) (backbone): The Graph Convolutional Network (GCN) ppectral-based approach that aggregates neighborhood information using a layer-wise propagation rule, effectively capturing local graph structure. It is applied to both node and graph classifications.
GAT (Velickovic et al., 2017): The Graph Attention Network (GAT) incorporates attention mechanisms to assign learnable weights to neighbors, allowing the model to focus on the most relevant nodes in graph during the aggregation process. It is applied in both node and graph classifications.
(2) Graph Self-Supervised Pre-training. These methods leverage self-supervised learning objectives to pre-train graph representations, aiming at enhancing performance across various downstream tasks.
GCC (Qiu et al., 2020): This framework focuses on learning transferable structural representations by using subgraph instance discrimination across multiple networks. It employs contrastive lcarning to capture universal topological properties, enabling better performance on diverse downstream tasks and out-of-domain generalization. It is applied to node classifications.
DGI (Veličković et al., 2019): DGI maximizes mutual information between local patch representations and global graph summaries, capturing meaningful node-level representations. Unlike prior approaches, it does not rely on random walks
28

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
and works well in both transductive and inductive settings, showing strong performance node classification tasks. It is applied to node classifications.
In foGraph (Sun et al., 2020): InfoGraph learns graph-level representations by maximizing mutual information between graph-level and multi-scale substructure representations (e.g. nodes, edges, triangles). This approach encodes shared structural and semantic information, achieving superior performance on graph classification and molecular property prediction tasks. It is applied to graph classifications.
GraphCL (You et al., 2020): This method introduces graph contrastive learning with various graph augmentations, enabling the generation of robust and generalizable graph representations. By systematically exploring augmentation strategies, GraphCL achieves strong results in semi-supervised, unsupervised, and transfer learning scenarios. It is applied to both node and graph classifications.
DSSL (Xiao et al., 2022): DSST, addresses the limitations of homophily assumptions in existing methods by decoupling semantic information in self-supervised learning. Through latent variable modeling framework, it captures diverse neighborhood semantics and achieves improved performance across homophilic and heterophilic graph benchmarks. It is applied to both node and graph classifications.
GraphACL (Xiao et al., 2024): GraphACI introduces asymmetric contrastive learning to better handle heterophilic graphs. It eliminates the need for prefabricated augmentations and homophily assumptions, effectively capturing local and monophily similarities. This simple yet powerful approach outperforms existing contrastive and self-supervised methods in both homophilic and heterophilic graph scenarios. It is applied to both node and graph classifications.
(3) Graph Prompt Fine-tuning. Prompt-based methods introduce learnable task-specific prompts to guide the fine-tuning of pre-trained models. These approaches are designed to improve transferability and adaptability to downstream tasks.
GPPT (Sun et al., 2022): GPPT introduces novel paradigm combining pre-training and prompt tuning. It uses masked edge prediction as the pre-training task and reformulates downstream tasks (e.g., node classification) into edge prediction tasks via token-pair prompts. This approach minimizes the objective gap between pre-training and downstream tasks, enabling efficient knowledge elicitation from pre-trained models without extensive fine-tuning. It is applied to node classifications.
GraphProme (Liu et al., 2023b): GraphPrompt unifies pre-training and downstream tasks into a shared template using learnable prompt. By narrowing the gap between pre-training and downstream objectives, it improves task-specific knowledge extraction and reduces labeling requirements. It provides generalizable framework suitable for diverse downstream tasks. It is applied to both node and graph classifications.
GraphPrompt (Yu et al., 2024b): An enhanced version of GraphP rompt, this method expands compatibility with pre-training tasks and introduces hierarchical prompt vectors across graph encoder layers. These advancements improve representation learning, enabling better utilization of multi-scale structural information and enhancing performance across tasks. It is applied to both node and graph classifications.
GPF (Fang et al., 2024): GPF proposes a universal prompt-based tuning method applicable to pre-trained GNNs under any pre-training strategy. Operating in the input feature space, it adapts pre-trained models to downstream tasks flexibly and effectively, outperforming traditional fine-tuning and specialized prompt-tuning methods across diverse scenarios. is applied to both node and graph classifications.
P ProNoG (Yu al., 2025): ProNoG focuses on non-homophilic graphs, recognizing their mixed structural patterns. incorporates conditional network to adaptively model node-specific characteristics during downstream tasks. This approach generalizes well to both homophilic and heterophilic settings, providing robust performance on real-world non-homophilic graphs. It is applied to both node and graph classifications.
(4) Multi-Domain Graph Pre-training. This category focuses on learning graph representations that generalize across different domains, which are the most relative baselines with ours. By leveraging shared structural or semantic patterns, these methods aim to improve performance in multi-domain scenarios.
GCOPE (Zhao et al., 2024): GCOPE introduces a unification framework to aggregate diverse graph datasets during pre-training, effectively distilling shared structural knowledge. By addressing the negative transfer issue often encountered in cross-domain learning, GCCPE enhances few-shot learning performance on target tasks. It leverages the synergistic potential of multiple graph datasets, pioneering the development of graph foundational models for multi-d multi-domain pre- training. It is applied to both node and graph classifications.
29

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
MDGPT (Yu al., 2024c): MDGPT proposes text-free multi-domain graph pre-training framework designed to align and integrate knowledge from diverse graph domains without relying on textual attributes. It introduces domain tokens to unify features across source domains and employs dual prompts (unifying and mixing prompts) to adapt the model to target domains. This approach effectively mitigates domain conflicts and enhances the relevance of source domain knowledge, achieving significant performance improvements across various datasets. It is applied to both node and graph classifications.
D.3. Experimental Setting Details
In this section, we introduce detailed experimental settings.
Detailed Settings for Section 5.2 (RQT). This section evaluates BRIDGE's ability to transfer knowledge from source domains to unseen target datasets domains under few-shot learning settings, specifically
five-shot, and mn-shot (1 & m & 10), which goal is to evaluate the model's generalization ability by pre-training BRIDGE on source domain graphs and then fine-tuning it with only a few labeled target samples. 1 m-shot settings. In all cases, only the m randomly selected labeled samples per class are used for training (supporting set), and the remaining samples in the dataset are used for testing (query set). @ Generalization settings. Two types of generalization settings are considered, i.e., cross-dataset setting (the target dataset belongs to the same domain as the source datasets but was unseen during pre-training) and cross-domain setting (the target dataset belongs to different domain from the source datasets). It is noteworthy that, some prior cross-domain graph pre-training and fine-tuning works (Zhao et al., 2024; Yu et al., 2024c) consider different datasets as different domains. However, in this paper, argue that treating datasets as domains is too coarse- grained, and instead, the distinction between domains should be based on the broader fields to which the datasets belong. Therefore, the "cross-dataset" setting in this section aligns with what other works refer to as "cross-domain , while the "cross-domain setting in this paper refers to a higher-level distinction between broader domains. @ Downstream Task settings. We evaluate both the node classification and graph classification. For graph classification, we sample 2-hop ego- gro-gaapss for each node to synthesize the corresponding datasets for graph-level tasks. Note that, GCC, DGT, and GPPT are only for node-level tasks comparison, while InfoGraph only for graph-level tasks comparison (as an extension of DGI).
Detailed Settings for Section 5.3 (RQ2). To analyze the effectiveness of BRIDGE's s key components, we further elaborate on the following three variants, each with specific module removed. The full BRIDGE is also included as a baseline to compare against the variants. O BRIDGE (w/o A.). Removes the semantic-wise aligner A, responsible for feature aligmment across domains. This variant only applies fimension-wiss alignment (A.,) but does not ensure feature consistency across domains at the semantic level. Specifically, directly apply X in Eq. (1) with adjacencies A to encode and stack HS, which is then be integrated in Liee O H in Eq. (2) as the ultimate goal of pre-training phase. Consequently, the alignment risk regularization objectives in Eq. (3), Eq. (4), and Eq. (5) are abandoned. This variant tests the necessity of ensuring semantic consistency between different graph domains. @ BRIDGE (w/o MoE). Removes the lightweight Mixture of Experts (MoE) network, which is responsible for selectively assembling source domain knowledge during prompting. Instead of selectively ussembling knowledge from different source domains, this variant uses fixed, ion-adaptive knowledge transfer mechanism (treats all pre-trained features as equally relevant for the target task with averaged routing). Specifically, the constant routing weight vector S = [I/n]" is applied on A!)) to obtain T in Eq. (7). This variant tests whether a static prompt learning strategy can be as effective as an adaptive expert routing mechanism. BRIDGE (w/o Repec): Removes the spectral regularizer Rspec- which constrains the upper bound of generalization error. Without the spectral regularizer, the model still follows the "pretrain-then- theroomptthn-ppomppt paradigm but lacks an explicit regularization mechanism to guarantee the transferability of knowledge. Specifically, we directly omit Repec term in Eq. (21), which only leaves Ldown in Eq. (11) as the prompt objectives. This variant tests the role of theoretical constraints on knowledge transfer and whether they significantly impact downstream performance.
Detailed Settings for Section 5.4 (RQ3). This section aims to analyze how efficiently BRIDGE performs downstream adaptation (prompt-based fine-tuning) compared to other baselines, which represents a challenging real-world scenario where labeled samples are limited, making efficient adaptation crucial. Datasets and Baselines. The evaluation is conducted on the Photo dataset under five-shot node classification setting (cross-dataset). Results are compared to ProNoG, GCOPF, F, and MDGPT, which represent sfate-of-the-art graph pre-training and adaptation approaches. @ Efficiency Metrics. We report results with the loss curvature, and accuracy curvature, where the reduction in training loss over epochs is tracked to measure how quickly each model stabilizes, the accuracy is recorded at different epochs to show how rapidly the model reaches its peak performance, and the total number of fine-tuning epochs needed for convergence is measured and compared.
30

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Detailed Settings for Section 5.5 (RQ4). Hyperparameter sensitivity analysis aims to analyze how different key hyperparam- eters in BRIDGE affect its performance. We track the fluctuation led by four hyperparameters. K. Controls the number of domain-specific variations (times of causal interventions) generated during the multi-domain semantic alignment process in Eq. (4). It determines how diverse the source domain embeddings become before knowledge transfer. Higher K simulates more varied domain diversity, which potentially brings better generalization performance, but heavier computational burdens. @ a. Balances the expectation term and variation term in the multi-domain semantic alignment objective in Eq. (4). influences how strictly domain-specific features are aligned across different datasets. 3.3 Controls the weight between the classification loss and the self-supervised MoE loss Eq. (11). Higher values give more importance to expert routing in knowledge selection. 4 y. Regulates the effect of spectral regularization controlling generalization error in Eq. (21). It
determines the extent to which the model constrains feature transferabilityy.
Detailed Settings for Section 5.6 (RQ5). This section investigates whether Large Language Models (LLMs) can improve the alignment of graph features across different domains. Specifically, this experiment evaluates whether leveraging LLM--enerated textual descriptions enhances the cross-domale generalization of BRIDGE. Since node features in the benchmark dataset are considered as seconnday data" derived from raw text through feature engineering, we follow Liu et al. (2024a); Li et all (2024b); Liu et al. (2024b); Chen et al. (2024b) and establish a correspondence between the raw text and the processed features. Specifically, for given raw sentence text s, we encode it using SentenceBERT (Reimers, 2019) to obtain a textual feature representation X, with the same dimension as the original node feature X. The two types of features are then concatenated as [X] X to form the new initial node feature, enabling the enhancement of node representations by leveraging the semantic understanding capabilities of LLMs.
E. Additional Results
In this section, we provide additional experiment results.
E.1. Additional Results: Five-Shot Node and Graph Classification
We evaluate our BRIDGE against 15 state-of-the-art baselines under the challenging one-shot setting for both node and graph classification, assessing its effectiveness in transferring source knowledge across cross-dataset and cross-domain scenarios. The results, presented in Table 1.1 and Table E.2, highlight the following key insights. O Results of the five-shot node classification graph classification demonstrate significant performance improvements for all models, including BRIDGE and baselines, compared to the one-shot results in Table 1 and Table 2. For example, BRIDGE achieves notable results such as 63.85% on Cora and 71.14% on Computers in node classification, significantly higher than the runner-up methods. In graph classification, BRIDGE attains accuracies such as 70.53% on PubMed and 84.42% on Reddit, showcasing its robustness. @ Compared to the one-shot setting, BRIDGE's performance improvement in one-shot tasks more substantial, highlighting its capability to effectively leverage additional labeled samples for fine-tuning. For example, BRIDGE achieves an accuracy gain of nearly 4% on Reddit compared to the best baseline in the five-shot setting, whereas the gain was less pronounced in the one-shot setting. O We also witnessed the improved generalization in cross-domain scenarios, i.e. BRIDGE's superior performance is particularly evident in cross-doman tasks, where domain shifts are more significant. For instance, BRIDGE achieves 80. 12% (node classification) and 84.42% (graph classification) on Reddi t, far exceeding the performance of baselines such as MDGPT and ProNoG.
E.2. Additional Results: m-Shot Classification on Diverse Datasets
To further evaluate BRIDGE's performance under different few-shot settings varied in m. we selected three competitive basclines in RQI, including ProNoo, GCOPE and MDGPT for comparison. Results on more diverse dataset are shown in Figure 1, Figure E.2, Figure E.3, and Figure E.4. The results demonstrate how accuracy evolves as the number of labeled samples per class increases, key observations include, 1 BRIDGE consistently outperforms the other three baselines across nearly all m-shot settings. Its superiority is particularly evident in both node and graph classification tasks, where it achieves state-of-the-an performance at higher shot levels. @ While BRIDGE shows only a slight advantage over the baselines when m is small, this advantage grows significantly as m increases. This trend can be attributed to the spectral regularizer, which enables more effective fine-tuning under increased supervision by leveraging down prompts with a theoretical generalization error upper bound guarantee. For example, in the graph classification task on PubMed dataset, BRIDGE steadily outperforms baselines such as MDGPT and GCOPE as m increases, showcasing its ability to scale with additional labeled data. @ Similar trends are observed across both cross datseet and cross-domain scenarios, suggesting that
31

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Table E.I. Accuracy (% = standard deviation for five runs) of five-shot node classification. CR=Cora, CS =CiteSeer, PM PM-PubMed,
Ph Ph=Photo, Com =Computers, Rdt =Reddit. The best results are shown in bold and the runner-ups are underlined.
Source
CS
PM
CR
PM
CR
CS
CR CS
CR
CS
CR
CS
PM
(Cross-Dataset Cross-Domuin)
Com
Ph
Com
Ph Con
PM Com
PM Ph
Ph Com
Model Target
CR
CS
PM
Ph
Com
Rêt
GCN (bb.) (Kipf& & Welling, 2022)
46.53 5.36
46.59 6.31
51.07 5.89
56.17 5.75
51.82 + 6,93
66.65 + 6.89
GAT (Velickovic et al., 2017)
46,47 a 7.10
46.09:55.9
50.47 7.26
53.11 7,80
49.78 a 5.68
66.59 a 5.07
GCC (Qiu et al., 2020)
47.05 7.88
6.78
$2.45
7.17
51.96 5.31
68.10 5.12
DCI (Veličković et al., 2019)
47.77 = 6.85
47.22
52,43
54.66 + 7.03
67. 67.64 5,52
GraphCI. (You et al., 2020)
49.61 + 6.67
6..7
52.81 5.27
59.55 7.50
57.88 + 7.36
69.94 5.43
DSST. (Xiao et al., 2022)
46,28 + 7.61
4.224
7.54
62.35 6,99
54.15 + 7.58
68.22 + 5.40
GraphACL (Xiao et ul., 2024)
51.38 7.06
4.121
54.48 6.01
65.87 6.54
59.72 a
70.93 a 7.41
GPPT (Sun et al., 2022)
47.51 7.54
52.86 6.98
6.78
52.11 5.16
71.05 5.59
GraphFrompt (Liu et al., 2023b) GraphP rompt+ (Yu et al., 2024b)
52.48 7.67
52.01 5.54
56.85 1 5.65
68.19
60.05 7,30
72.79 6.23
52,16 7.84
52.76 7.09
55.82 + 6.16
66.31 5,90
61.07 5.37
72.63 + 5.77
GPF (Fang et al., 2024)
53.97=
55. 88 5.77
67.65 6.558
55.69 + 6.05
74.00 + 5.46
ProNoG (Yu et al., 2025)
58.06 2.006
7.55
56.53 7.29
74.23 5,90
63,54 6.12
76.48 + 6.82
GCOPE (Zhao et al.. 2024)
55.93
53.05 a 7,86
68.80 6.63
60.46 a 5.56
73.61 a 5.27
MDCPT (Yu et al., 2024c)
59.51 4.54
55,28 4:88::44 4.47
56.25 + 6.76
75.73 +.554
64.38 + 4.13
75.29 + 4,25
BRIDGE (ours)
63.85 5.87
60.20 7.72
62.35 5.92
81.95 5.38
71.14 4.86
80.12 2.66
Table E.2. Accuracy (% + standard deviation for five runs) of five-shot graph classification. CR=Cora, =S=iiteeeer. PM PM-PubMed, Ph =Phhooo Com =Computers, Rat =Reddit. The best results are shown in bold and the runner-ups are underlined.
Source
CS PM
CR
PM
CR
CS
CR
CS
CR
CS
CR
CS
PM
(ooss-Dataset Coos-Domain))
Ph Com
Ph Com
Ph Com
PM Com
PM Pa
Ph Com
Model / Target
CR
CS
PM
Ph
Com
Rdt
GCN (bb.) (Kipf& Welling, 2022)
51.93:4.55
44. 15
56.88
59.29+4.78
51.67+5.38
75.70
GAT (Velickovic et al., 2017)
48.95=6.08
43.65+5.66
55.8945.86
57.18+492
50.9146.22
74.82+6.1 16
InfoGraph (Sun et al.. 2020)
52.8
45.51
60.4726.25
12...25.24
53.40a5.49
76.3424497
GraphCL (You et al., 2020)
53.92s4.65
48.47=4.91
61.8446.48
62.7425.19
57.73a6.27
78.5 78.5925.40
DSSL (Xiao et al., 2022)
55.03+5.57
45.58=6.20
62.45x44.5
63,04s6.13
57.2646.33
78.8244
GraphACI. (Xiao et al., 2024)
55.52+5.83
47.26=5.24
61.44+5.90
63.6946.33
56.54x6.46
79.75+6.20
GraphPrompt (Liu et al., 2023b)
55.2446.39
52.3125.04
63.0916.22
64.27a5.86
58.1245.19
79.48sa4.56
GraphPrompt: + (Yu et al., 2024b)
59.53+5.07
51.2245.88
1
65.73a4.60
58.6146.24
78.7746.15
GPF (Fang et al., 2024)
56.19+5.28
53.4745.33
61.39+5.93
64,08+5.24
57.47
79.6615.26
ProNoG (Yu et al., 2025)
61.07=5.31
51.25=5.98
64.73+6.12
66, 60s5..4.14
60,55+5.17
80,26+5.05
GCOPE (Zhao al., 2024)
57.29+4.77
53.97=6.13
62.61+4.94
65.7745.35
58.2044.65
79.87+6.22
MDGPT (Yu et al., 2024c)
61.77+4.10
55.37+6.33
62.70+6.20
67. 67.06+3.72
59.76+8.56
80,29+5.46
BRIDGE (ours)
67.21s3.78
59,80+5.04
67.1743.77
70.5326.53
64.9116.24
84.42+1.43
BRIDGE not only excels in domain-specific tasks but also demonstrates strong generalization in transferring knowledge across diverse domains. This robustness is further validated in challenging scenarios such the Reddit t dataset, where BRIDGE achieves significant accuracy gains compared to its competitors.
E.3. Additional Results: Ablation Studies
The additional ablation results in Table E.3 and Table E.4 provide detailed analysis of the impact of three key components in BRIDGE on node classification and graph classification across various datasets and shot settings (m 1 and 5). We find that, 1 Removing the semantic- wiie aligner (A.) results in consistent accuracy drops across all datasets and tasks, particularly in node classification. For example, in Cora (m 5), the accuracy decreases by 4.9%, demonstrating the importance of semantic alignment for effective knowledge transfer. The removal of the Mixture of Experts (MoE) module has a significant impact, especially in more challenging cross-domain datasets like Reddit. For instance, in node classification on Reddi t (m = 5), accuracy drops by 10.1 %, underscoring the role of selective knowledge assembly in
32

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
ProNoG
GCOPE
MDGPT
BRIDGE (aurs)
PraNoG
GCOPE
MDGPT
BRIDGE (ours)
Node Classification
Graph Classification
Node Classilication
Graph Classificalion
70
70
60
Accuracy (%)
Accuracy (%)
Accuracy (%)
70
60
80
50
Accuracy
50
80
50
40
50
10
5
10
10
Number of shats
Number of shots
Number of shots
Number of shots
Figure E.I. m-shot evaluation on CiteSeer.
Figure E.2. m-shot evaluation on FubMed.
ProNoG
GCOPE
MDGPT
BRIDGE (ouns)
ProNoG
GCOPE
MDGPT
BRIDGE (ours)
Node Classification
Graph Classification
Nocie Classffication
Graph Classification
80
80
Accuracy (86
Acsuracy (%)
75
00
70-
70
70
Acturacy
80
Accuracy
60
60
65
500
1
2
5
6
7
8
10
1
2
5
6
7
8
9
10
1
4
5
6
7
9
10
1
2
4
5
6
7
10
Number shols
Number shots
Number of shots
Number of shols
Figure E.3. m--hoot evaluation on Photo.
Figure E.4. m--shot evaluation on Computers.
Table E.3. Accuracy (%) with relative decrease (A%) of ablation on node classification. =Corora, CS =Ci teSeer, PM= Ph =Photo, Com =Computers, Rdi =Reddit t. The biggest decrease is shown in bold and the runner-ups are underlined.
CS
PM
PM
CR PM
Source
Ph Com
Ph Coo
Ph Com
PM Com
PM Ph
Ph Com
Target
CR
PM
Ph
Com
Variants
m
YT 5
5
5
TUL
5
YIL 1
5
BRIDGE
46.4
63.9
A
42.2
A
60.2
56.4
A
62.4
A
67.9
A
62,4
A
54.0
A
71.1
68.5
80.1
A
(wwo A.)
44.2
61.2
4.2
40.2
4.8
57.3
55.0
2.5
60.1
3.7
65.7
3.3
61.9
0.8
53.4
1.2
70.7
0.6
68.3
0.4
78.9
1.6
(w/o MoE)
44.3
4.6
61.0
4.5
39,9
5.3
58.3
52.2
74
57.6
7.6
65.7
3,2
58.3
6.6
53.1
1.8
68.4
39
64.3
6.2
72.1
10.1
(w/o )
45.6
59.9
6.2
41.3
2.0
54.6
9.4
55.3
1.8
55.9
10.3
66.5
2.0
56.0
10.2
53.6
0.9
66.7
6.2
67.7
1.2
72.4
9.7
addressing complex cross-domain tasks. ① The spectral regularizer (Repee) has the most critical impact on performance. Its removal causes the largest accuracy drops, particularly in tasks with higher supervision (m 5). For example, in P PubMed for node classification, accuracy decreases by 10.3%, highlighting Rye 's effectiveness in enhancing fine-tuning with robust generalization guarantees. O Overall, the effects of ablation are more pronounced in the five-shot setting, as the increased supervision enables the full BRIDGE to leverage its components more effectively. This highlights the importance of these modules in scenarios with more labeled data. Among the three key components, the spectral regularizer (Ropec) consistently demonstrates the most critical role in both node and graph classification tasks. Its ability to constrain generalization error provides substantial benefits for cross-dataset and cross-domain learning. The Mixture of Experts (MoE) is essential for selectively transferring domain-specific knowledge, particularly in challenging cross-domain scenarios. The semantic-wise aligner (A,), while less impactful than Rapoe and MoE, still contributes to stable improvements across tasks.
E.4. Additional Results: Embedding Visualizations
We visualize node embeddings after pre-training and down-prompting (fine-tuning) in Figure E.5 and Figure E.6, respectively, During pre-training, simultaneously take six datasets (belonging to three different domains) the source knowledge. Note that, "BRIDGE Ablated" of the left subfigure in Figure E.6 refers to the removal of both the lightweight MoE. network and the spectral regularize, as the comparison with the right subfigure. We find that, The pre-trained embeddings for nodes from six datasets, spanning three distinct domains (academic, e-commerce, and social networks), exhibit well- clustered structures. Nodes belonging to the same dataset are grouped tightly, demonstrating the ability of BRIDGE's s pre-training phase to align domain-specific features effectively. Different domains remain distinguishable (forming into three
33

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
Table E.4. Accuracy (%) with relative decrease (A%) of ablation on graph classification. CR-Cora, CS aB5=-Ctteeee PM PM-PubMed,
Ph Ph=Photo, Ccm =Computers, Rdt =Reddit. The biggest decrease is shown in bold and the runner-ups are underlined.
CS
PM
CR PM
CR Ca
CR CS
CR CS
CR CS PM
Source
Ph Com
Ph Com
Ph Com
Com
PM Ph
Ph Coo
Target
CR
C8
PM
Ph
Com
Rdt
Variants
m
m
172
m
m
5
m
=
m
m
m
m
BRIDGE
53.8
A
67.2
A
50.5 A
59.8
A
58.4
A
67.2
A
68.3
A
70.5 A
54.1
4
64.9
A
75.8
A
84,4
A
-(w/ A.)
51.3
4.7
64.9
3,5
49.2
2.6
58.8
1.6
57.3
1.9
66.6
0.8
68.0
0.5
68.8
2.4
53.9
0.4
63.8
1.7
75.0
1.
83.0
1.7
MoE)
51.3
4.8
63.8
57.0
4.6
55.3
5.3
4.3
66.4
2.8
65.2
7.5
3.6
61.7
4.9
4.1
80.6
4.5
(w/ )
53.0
1.6
61.2
47.3
6.4
55.2
7.7
57.1
2.2
62.2
7.4
67.5
12
66.3
6.0
53.9
0.3
60.1
7.4
74.9
11
78.6
7.0
Com
CanSoer
Pabttedd
Photo
Comp don
Redd
Fine-luned Embedding (BRIDGE Atlided)
Fine-lared Embaiding (BRIDGE)
-90
Silhouette Score: 0.3029
Silhouette Score: 0.7488
Figure E.S, Pre-trained Embeddings.
Figure E.6. Fine-tuned Embeddings on Cora (m 5). Colors denote class labels.
obvious domain clusters), reflecting the preservation of -nter-domain characteristies while ensuring intra-domain consistency. This well-separated clustering highlights the success of BRIDGE's domain-invarian feature alignment mechanism in the pre-training stage, which is critical for effective transfer learning. @ With ablation (left), the silhouette score of 0.3029 indicates relatively poor class separability, reflecting the challenges posed by the absence of MoE and Repec- For full BRIDGE (right), the embeddings show significantly improved clustering, with clear boundaries between nodes different classes. The silhouette score increases to 0.7488, demonstrating enhanced class separability. Overall, the additional visualization results provide strong evidence of the synergy between BRIDGE's pre- pre-training and fine-tuning mechanisms, further validating the effectiveness of its key components.
F. Implementation Details
In this section, we provide implementation details.
F1. Implementation Details of BRIDGE
Pre-training. The number of pre-training epochs for optimizing our proposed BRIDGE and all baselines is set to 10,000. An early stopping strategy is employed, where training is terminated if the pre-training loss does not decrease for 50 consecutive epochs. This ensures computational efficiency without sacrificing performance. The aligned feature dimension is set to 50, and the encoded hidden dimension is set 256. For BRIDGE, the hyperparameters a, 3, and y are chosen from the range of le-3 to 5e-1. The optimal values are determined using Weights & Biases (wandb)², framework for hyperparameter optimization and experiment tracking. For the optimization process, we utilize the Adam optimizer (Kingma & Ba, 2015), carefully tuning the learning rate and weight decay. The learning rate is chosen from the range of le-4 to 1e-2, and weight decay is selected from le-5 to le-2 through grid search on the validation set. All parameters are initialized randomly to avoid any bias from pre-trained weights.
F.2. Implementation Details of Baselines
We provide the baseline methods implementations with their respective licenses as follows.
pertps://wandb.ce
34

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
GCN (Kipf & Welling, 2022): [MIT License] http://///...i.i..ooooooooo/oorr
GAT (Velickovic al., 2017): [MIT License] http:////saaaaaaaoo/////////////
GCC (Qiu al., 2020): [MIT License] http:s///gthhhh.co/////////CCC
DGI (Veličković et al., 2019); [MIT License] | https////aattbboo////a/aaa////11
InfoGraph (Sun et al., 2020): [Unspeci fied] http::///gtttttioooo////////////
GraphCL (You et al., 2020): [MIT License] httprr//aathua...cmm///aaaaaaaaa
DSSL (Xiao et al., 2022): [Apache-2.0 License]
https://github. http:///www...p/////////////////
GraphACL (Xiao et al., 2024): httppp/.....aaaaee///....//////a
GPPT (Sun et al., 2022): [Unspecified] http:://gggtuuuoocoom//hhhaaaaas
GraphPrompt (Liu et al., http:///aaaaaa//aa///aaaaaaaaaaa
GraphPrompt (Yu al., al. 2024b): [Unspecified]
GPF (Pang al., 2024): [MIT License] http:://ggtthubb.c/////////////r
ProNoG (Yu al., 2025): cuspecified) http:://ggthiocccmm//g/gggg////
GCOPE (Zhao al., 2024): [Unspecified] httpss//gghhabb...cm/ccchhh/////
MDGPT (Yu al., 2024c): [Unspecified]
For all baselines, we follow the recommended byperparameter settings from the original papers or official implementations to ensure their optimal performance. cases where recommended configurations are unavailable or suboptimal, we carefully fine-tune the hyperparameters to achieve the best possible performances. To ensure a fair comparison, we align other experimental settings such as hidden dimensions, number of model layers, etc., with those of our proposed method. This consistent setup guarantees that performance differences arise solely from the model designs and not from external factors.
F.3 Hardware and Software Configurations
We conduct the experiments with the following hardware and software configurations.
Operating System: Ubuntu 20.04 LTS.
CPU: Intel(R) Xeon(R) Platinum 8358 CPU@2.60GHz with ITB DDR4 of Memory,
GPU: NVIDIA Tesla A100 SMX4 with 80GB of Memory.
Software: CUDA 10.1, Python 3.8.12, PyToorch² 1.9.1, Py PoTch Geometric 2.0.1.
G. Discussions
To make BRIDGE easy-to-follow to readers, we discuss through the following "How to Understand?" questions.
G.1. How to Understand Domain-Invariant Aligner?
The Nomain-invarian aligner in BRIDGE facilitates robust cross-domain knowledge transfer by combining dimension- wise and semantic-wise alignment. Dimension-wise alignment ensures basic dimensionality-Leed consistency by aligning statistical distributions of features across domains, addressing simple discrepancies such as scale and distribution differences. However, this alone is insufficient for handling deeper semantic mismatches that arise in complex domain shiftss Semantic-wise alignment complements this by focusing on aligning features with similar semantic meanings across domains, even when their original representations differ significantly.
Semantic- wise alignment disentangles domain-invarian semantics from domain-specific noise by isolating shared factors that are consistent across domains. This process helps BRIDGE avoid relying on spurious correlations or domain-specific
hhttps: httrps://gti thub. cww wwt./pyytrcc/yyttttttooo hhtpps: pettps:/gtte thub. httm:/ppg-eeam/pptorrccoooooo
35

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
confounders, but causal relations. Additionally, it enforces consistency through principles of invariant learning, ensuring that the predictive relationships between features and lahels remain stable regardless of the domain. By integrating these two alignment strategies, BRIDGE creates representations that are not only robust to domain shifts but also capable of preserving meaningful semantic relationships, enabling effective knowledge transfer and strong generalization across diverse domains.
G.2. How to Understand the Objective of Pre-training?
The objective of pre-training in BRIDGE is to create a robust and transferable representation that bridges the gap between heterogeneous tasks encountered during pre pre-training and fine-tuning. To achieve this, a self-supervised loss is employed, which serves as a unifying mechanism for multi-task learning on graphs.
The self-supervised loss focuses on aligning tasks by framing them as similarity computations (Liu et al., 2023b), Regardless of the specific downstream task, whether maybe node classification, link prediction, or graph classification, these tasks can be reformulated into the shared objective of learning embeddings through similarity maximization. By doing so, the pre-training process effectively reduces the task gap between pre-training and fine-tuning, ensuring that the representations learned during pre-training are directly applicable to a wide variety of graph tasks. This unified approach not only enhances knowledge transfer across domains and tasks but also provides a flexible framework for handling the diverse objectives inherent in graph-based learning.
G.3. How to Understand the Alignment Risk Regularization?
The alignment risk regularization in BRIDGE is designed to mitigate spurious correlations and enhance the learning of invariant dimensions across domains. This is achieved by leveraging a causal interveniion-inspreed mechanism that systematically targets and reduces the influence of domain-specific variations in the embeddings.
Specifically, the regularization introduces stochasticity into the embeddings by randomly replacing the variant dimensions K times, achieved by applying different projection weight matrices. For each replacement, the embeddings are perturbed to simulate domain-specific variations, and the model computes the downstream loss for each perturbed version. The regularization objective minimizes the variance of these K times loss values. By forcing the model to remain robust under these perturbations, it effectively learns to focus on the invariant dimensions that are consistent across domains, while reducing reliance on domain-specific, spurious correlations. This approach ensures that the learned representations capture the essential causal factors shared across domains, improving generalization and transferability.
G.4. How to Understand Proposition 4.2 and Proposition 4.3?
Proposition 4.2 establishes that minimizing Eq. (4) encourages the pre-trained model få to satisfy Assumption 4.1, which defines the conditions for effective multi-domain alignment. Essentially, it avoids relying on overly strong assumptions by explicitly integrating domain alignment into the optimization process. This ensures that the model is capable of capturing domain-invariant features without requiring unrealistic assumptions about perfect alignment between source and target.
Proposition 4.3 formalizes that optimizing Eq. (4) is equivalent to minimizing the upper bound of the multi-domain semantic alignment risk defined in Eq. (2). This guarantees that the optimization process is not only aligning features dimension- and -emantic-wise but also systematically reducing alignment risks. By transforming the problem into an equivalent optimization of alignment risk, it provides a more direct way to enhance generalization and robustness.
Proposition 4.2 highlights the practicality of the alignment process by avoiding overly strong assumptions, while Propo- sition 4.3 bridges the gap between theoretical alignment risk and its empirical optimization. Together, they demonstrate how the aligner in BRIDGE contributes to both feature alignment across domains and robustness during pre-training. The propositions solidify the theoretical foundation for BRIDGE's ability to generalize effectively in multi-domain scenarios.
G.5. How to Understand the Selective Knowledge Assembly?
Selective knowledge assembly is a mechanism designed bridge the gap between pre-training and downstream tasks by dynamically integrating domain-specific knowledge. During pre-training, BRIDGE aligns graph features across domains to extract invariant representations, ensuring these representations retain core knowledge while filtering out -smaiic noise. This aligmment lays the foundation for selective transfer.
In downstream tasks, the lightweight Mixture of Experts (MoE) network facilitates this assembly process by assigning
36

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
weights to the pre-trained aligners based on the input graph's characteristics. Instead of uniformly applying all pre-trained knowledge, the MoE selectively activates the most relevant components, tailoring the knowledge transfer the target domain. This targeted assembly improves generalization to unseen tasks while maintaining computational efficiency, especially in few-shot settings. By focusing only on what is essential, BRIDGE ensures the effective adaptation of multi multi-domain graph knowledge to diverse tasks.
G.6. How to Understand the Downstream Classification Objective?
The downstream classification objective unifies the pre-training and Nown-prompting stages by adopting a universal task template based on subgraph similarity. This objective ensures consistency across tasks such as node and graph classification, enabling efficient knowledge transfer from pre-trained embeddings to downstream tasks.
Formally, the classification loss aims to maximize the similarity between embeddings of correctly matched subgraphs while minimizing the similarity with mismatched ones. Specifically, the model utilizes the pre-trained graph encoder and learnable prompt adapt the embeddings of target graphs. The classification loss is formulated over pairwise embedding similarities, where few-shot labeled samples serve as the supervision to guide the model in distinguishing classes. This design not only aligns with the sulf-vupe pre- tre-train task but also enables effective fine-tuning under limited labeled data, ensuring robust performance even in few-shot settings.
cross-entropy
G.7. How to Understand the Derivation Process of Generalization Error Upper Bound?
The derivation of the generalization error upper bound quantifies the extent to which knowledge can be transferred between domains. It decomposes the target domain error into four components: (term [a] and term [b]) empirical errors on source and target domains, which can be optimized through classification objectives. (term [c]) domain divergence, measured using the Wasserstein distance between source and target distributions. (term [d) hypothesis mismatch, capturing the gap between source and target models' predictions. (Term O()) residual terms related to sample complexity and model capacity. To ensure smooth generalization, Lipschitz continuity is introduced for the graph encoder and classifier, bounding how much the output can change with respect to input variations.
To further constrain domain divergence. the bound is refined using graph spectral theory. The Wasserstein distance is related to spectral differences in adjacency matrices, leading to spectral regularization term that stabilizes knowledge transfer. This regularizer ensures that down-prompting effectively preserves task-relevant features while mitigating overfitting to domain-specific noise. By incorporating this bound into the optimization process, BRIDGE explicitly balances knowledge transfer and generalization, making it theoretically sound and practically efficient for adapting to new domains.
G.8. How to Understand Proposition 4.6?
Proposition 4.6 provides a lower bound for the Lipschitz constant Ly, which measures how much the graph encoder's output changes in response variations in input graphs. A smaller L, suggests smoother model that generalizes better across domains, while larger I,y implies that small input differences can lead to large output variations, potentially reducing transferability. The bound is derived by linking L5 to the spectral properties of graphs, specifically the eigenvalues of their adjacency matrices. Since graphs from different domains have different structures, their adjacency matrices vary, and the optimal transport (Wasserstein) distance between them helps quantify these differences.
Mathematically,, the bound shows that L depends on both the spectral gap (differences in eigenvalues) and the structural distance between source and target graphs. If two graphs have highly similar adjacency matrices (i.e., small spectral differences), the bound indicates that the knowledge transferability is high, leading to better generalization. Conversely, if the graphs are structurally different, the transfcrability is limited, and adaptation becomes harder. This result is crucial for BRIDGE because it provides a theoretical guarantee that minimizing spectral differences during training enhances the model's ability to generalize across domains.
G.9. How to Understand How the Spectral Regularizer Takes Effectt
The spectral regularizer in BRIDGE ensures that knowledge transfer remains stable by controlling how much the graph representations change when adapting from source to target domains. It works by enforcing smoothness in the spectral space, meaning that graph embeddings should not fluctuate too much between adjacent frequencies. This is done by minimizing deviations in spectral components of node features and their transformed embeddings, reducing the risk of
37

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
overfitting to domain-specifie noise. The regularizer acts as a constraint during fine-tuning, preventing drastic shifts in feature distributions while ensuring the model retains transferable knowledge.
Mathematically, the regularizer is formulated based on spectral differences in graph signals, ensuring that variations in embeddings align with the inherent structure of the target domain. By penalizing excessive spectral changes, it helps maintain consistency in feature representations, leading to better generalization. In practice, this means that BRIDGE can adapt to new domains while preserving key structural properties, ensuring efficient and stable knowledge transfer even with limited labeled data.
Notably, if pre-training and fine-tuning are jointly optimized rather than conducted separately, the spectral regularizer is expected to have even greater impact. In this setting, the model can dynamically adjust feature alignment while simultaneously learning more generalizable representation space. This synergy allows the regularizer to directly influence both phases, reinforcing domain-invarisnn structures early in training and further stabilizing adaptation.
G.10. How to Understand Why LLM Can Enhance Knowledge Transfer?
LLMs enhance knowledge transfer by improving feature alignment across different domains through semantic enrichment. In BRIDGE, node features are often derived from raw text via feature engineering, which may lead to information loss. By leveraging LLMs to generate textual descriptions and encode them into numerical representations, BRIDGE integrates richer semantic information into its graph embeddings. This augmented representation bridges the gap between structurally different graphs by introducing a shared semantic space, making it casier for the model generalize across domains.
Specifically, the experiment constructs node representations by concatenating original graph features with LLM--enerated textual embeddings. This hybrid approach allows the model to retain domain-specific graph structures while benefiting from the broader contextual understanding provided by LLMs. As a result, the cross-domain adaptation process becomes more effective, as semantically similar nodes from different domains are better aligned. This method is particularly useful in scenarios where domain shifts are significant, enabling BRIDGE to achieve stronger generalization with minimal supervision.
H. Additional Related Work
H.1. Multi-Domain Graph Pre-training
Pre-training methods on graphs (Liu et al., 2022; Hu et al., 2020a; Xie et al., 2022) capitalize on the intrinsic properties of graph structures and feature semantics via self-supervised learming. These approaches leverage optimization objectives such as node masking, edge prediction, and contrastive learning to learn transferable representations from massive graph data, such as GCC (Qiu et al., 2020), GPT-GNN (Hu et al., 2020b), GPPT (Sun et al., 2022), L2P-GNN (Lu et al., 2021), APT (Xu et al., 2023), CPT-HG (Jiang et al., 2021), Aut-GNNN (Yang et al., 2022), PT-DGNN (Chen et al., 2022), W2PGNN (Cao et al., 2023), =-NNN3 (Yin al., 2023), etc. Pre-trained knowledge can be transferred to downstream domains and tasks through parameter fine-tuning or prompting. However, these methods typically assume that the pre- pre-training and downstream graphs originate from the same very similar domains. For instance, they may involve different subgraphs of large graph (Huang et al., 2022) or collection of similar graphs within the same field (Zhang et al., 2021). Consequently, these methods struggle to generalize across multi-domain graphs due to domain-specific biases inherent in the pre-training process.
Another line of research in cross-domain learning addresses the challenge of transferring knowledge from a single source domain to a different target domain, such as GCOPE (Zhao et al., 2024), OMOG (Liu et al., 2024b), PGPRee (Yi et al., 2023), UniAug (Tang et al., 2024c), UniGraph (He & Hooi, 2024), etc. These studies focus on exploiting domain-invariant representations to bridge the gap between disparate domains. While effective for nogle-source to single-target scenarios, they fall short in utilizing comprehensive multi-domain knowledge. Additionally, many of these methods are tailored to specific tasks or domains and often require domain expertise, which limits their applicability to diverse domains and tasks. Towards addressing multi-domain graph pre-training, methods such as OFA (Liu et al., 2024b) and HiGPT (Tang et al., 2024b) employ large language models to derive and align node features from different domains through the medium of natural language. While innovative, these methods are constrained to aext-attributed graphs (Wang et al., 2024; Tan et al., 2024), thereby limiting their applicability
broader range of more general graph types. For nonttext-attributed graphs, GCOPE (Zhao et al., 2024) introduces domain-specific, interconnecting virtual nodes to link across domains. Though it facilitates some level of connection, they do not explicitly align node features, thereby failing to overcome domain conflicts effectively. MDGPT (Yu et al., 2024c) proposes a set of domain tokens to explicitly align multi-domail features for
38

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
synergistic pre-training. However, it simply utilizes SVD to align features dimension-wise without considering semantics.
Unlike existing multi-domain graph pre-training methods that often struggle with domain-specific biases or lack explicit feature alignment, BRIDGE introduces a principled approach by aligning multi-domain features with a set of domain- invariant aligners. Additionally, BRIDGE offers a theoretical framework that quantifies transferable knowledge through an optimizable upper bound on the generalization error. This combination of alignment, selective transfer, and theoretical grounding leads to improved performance across various tasks, setting it apart from prior work in the domain.
H.2. Graph Fine-tuning and Prompt Learning
Fine-tuning pre-trained graph models for specific downstream tasks has become a widely adopted approach to adapting general representations to particular applications. Traditional fine-tuning techniques involve modifying the entire graph model or a subset of its parameters to optimize performance on the target task, such as GTOT (Zhang et al., 2022). G- ootimiize
Tuning (Sun al., 2024), S2PGNN (Zhili et al., 2024), IGAP (Yan et al., 2024), WalkLM (Tan et al., 2024), NEAT (Ying et al., 2024), Bridge-Tune (Huang al., 2024b), GSP (Jiang et al., 2024), GRAPH-BERT (Zhang al., 2020), etc. While effective, these methods can be computationallyy expensive and prone to overfitting, especially when labeled data is limited. To mitigate these issues, parameter-efficieee fine-tuning (PEFT) methods have emerged, adjusting only a small portion of the model's parameters (Zhu et al., 2024a; Tian et al., 2024; Gui et al., 2024; Xue et al., 2023) or introducing lightweight modules (Lin et al., 2024; Zhu et al., 2024b; Yang et al., 2023; 2024) to adapt the model to new tasks without extensive retraining. PEFT-based methods significantly reduce computational cost and the risk of overfitting while maintaining strong performance on downstream tasks. This allows for more efficient adaptation to new domains, especially in scenarios with limited labeled data. However, these methods still face challenges in handling complex multi-domain scenarios, where domain-specific biases and task heterogeneity can affect their effectiveness.
Prompt learning has emerged as an alternative approach, initially popularized in natural language processing (Liu et al., 2023a; Zhou et al., 2022b;a: Du et al., 2022). For graph learning, prompt-based methods aim to guide the behavior of pre-trained models with minimal parameter updates. Techniques such as graph prompt tuning (Sun et al., 2022: Zhu et al., 2023; Shirkavand & Huang. 2023; Zhang et al., 2024) involve designing task-specific prompts that interact with pre-trained embeddings to achieve the desired outcomes. These methods provide a more parameter-efficient adaptation strategy, reducing the risk of overfitting and improving generalization across diverse tasks. Recent advancements in graph prompt learning have introduced various strategies to improve adaptability and performance, such as GPPT (Sun et al., 2022), All in One (Sun et al., 2023), GraphPrompt (Liu et al., 2023b), Self-Pro (Gong et al., 2023), SGL-PT (Zhu et al., 2023), GPF-Plus (Fang et al., 2024), PRODIGY (Huang et al., 2024a), MDGPT (Yu et al., 2024c), HGPROMPT (Yu et al., 2024a), ProG (Zi et al., 2024), etc. These methods integrate learnable prompts into the graph neural networks, enabling the model to focus on task-relevant features or structural semantics, which captures key patterns in the downstream graph data, facilitating transfer learning across different graph tasks, enhancing the model's ability handle complex graph structures.
Despite these advancements, existing prompt learning methods for graphs are often limited in their ability to handle multi- domain scenarios effectively. Many approaches are designed with a focus on single-domain tasks, requiring domain- sspeiffc prompt engineering that restricts their scalability and generalization. Furthermore. there is lack of theoretical insights into how and why prompt learning works in the graph domain, which constrains the development of more robust and principled methods. BRIDGE, however, not only integrates multi-domain graph feature alignment but provides a theoretical foundation for knowledge transfer, addressing these challenges and offering a more scalable solution for multi-domain graph learning.
H.3. Towards Graph Foundation Models
Graph Foundation Models (GFMs) extend the idea of foundation models, which have shown remarkable success in fields, like natural language processing (Myers et al., 2024; Bai et al., 2024) and computer vision (Singh et al., 2022; Wang et al., 2023), to graphs (Mao et al., 2024b; Shi et al., 2024a; Zi al., 2024; Guo et al., 2025b). The concept of GFMs revolves around pre-training models on large-scale, diverse graphs to learn generalizable representations that can then be fine-tuned for a wide variety of downstream tasks (Shi et al., 2024b; Bommasani et al., 2021). These models aim to combine the flexibility and generalization ability seen in large language models with structural precision required for traph-based tasks.
While traditional graph deep models have made significant progress in tasks like node classification, link prediction, and graph classification, their generalization across different domains remains limited. Pre-training GNNs on domain- graphs has been a common approach, but such methods often face challenges in transferring learned representations across diverse domains due to domain-specifie biases and the heterogeneous nature of graph data. This limitation motivates the
39

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
exploration of GFMs, which aim to learn domain-invariant representations through multi-domain pre-training. Existing multi-domain graph pre-training methods, such as GCOPE (Zhao et al., 2024) and MDGPT (Liu et al., 2024b), have attempted to bridge the gap by aligning features from different domains. These approaches focus on aligning node and edge features across domains, but they are typically restricted to text-attributed graphs or rely on shallow domain alignment techniques. Methods like OFA (Liu et al., 2024b) and TSGFM (Chen et al., 2024b) leverage language models to align features via language descriptions, yet they still struggle with non-textual graphs and fail to fully capture semantic similarities.
GFMs, however, aim to provide a more unified approach by leveraging large-scale pre-training on graph data, followed by task-specific adaptation. These models are expected to exhibit both emergence and homogenization (Shi al., 2024a), similar to LLMs, enabling them to generalize across tasks and domains while maintaining their ability to perform graph- specific reasoning. By aligning features across domains in principled way and using techniques like contrastive pre-training or prompt learning, GFMs have the potential to handle diverse graph data types and tasks more effectively. A growing line of inquiry is into the theoretical underpinnings of GFMs. Some works have begun to explore the potential existence of scaling laws in graph models (Ma et al., 2024; Tang et al., 2024c: Chen et al., 2024a; Liu et al., 2024c), analogous to those found in large language models (Kaplan et al., 2020). These studies hypothesize that increasing the scale of graph models may yield emergent properties, such as enhanced transferability and generalization across tasks. However, empirical validation is still limited, and the theoretical foundation for scaling GFMs remains underdeveloped.
Several key challenges remain for GFMs. First, the heterogeneous nature of graph data ranging from node features to edge types poses significant difficulties for creating unified representations. Second, the scalability of GFMs remains a concern, as large-scale graph data requires substantial computational resources for pre-training. Third, while prompt learning techniques show promise for fine-tuning pre-trained graph models, their application across multiple domains and tasks is still an open challenge. Overall, GFMs present an exciting opportunity to revolutionize graph learning by combining the strengths of pre-trained models with the precision of graph-based reasoning.
Our work, BRIDGE, addresses these gaps by aligning multi-domain graph features during pre- vre-training and using Mixture of Experts (MoE) network for efficient prompting. Moreover, we provide a theoretical upper bound on the generalization error, offering new insights into the amount of knowledge that can be transferred across domains. This foundational work paves the way for more robust and scalable GFMs.
I. Limitations
While BRIDGE demonstrates notable advancements in multi-domain graph pre-training and prompt learning, certain areas remain open for further exploration to maximize its utility and versatility towards Graph Foundation Models (GFMs).
Evaluation on Wider Domains. The current evaluation focuses on well-known benchmark datasets across three domains (academic, e-commerce, and social networks). However, niche domains, such as chemical reaction networks (where nodes represent molecules and edges denote reactions) or financial fraud detection graphs (with complex temporal and hierarchical patterns), may pose unique challenges due their specialized data distributions. For example, highly sparse or asymmetric adjacency matrices prevalent in these graphs may affect feature alignment or generalization capabilities. Additionally, multimodal graphs, such those combining image features with graph structure (e.g., protein folding graphs with 3D spatial data), remain unexplored.
Assumptions on Feature Alignment. BRIDGE relies on domain-invariant aligners to unify graph features across domains. However, this assumes that such alignments can adequately capture shared characteristics across all graphs. In graphs with extreme heterogeneity, such as heterogeneous information networks (HINs) where nodes and edges have diverse types (e.g., author-paper-venut graphs), this assumption may not always hold. Similarly, in hypergraphs, where edges connect multiple nodes, or dynamic graphs with time-evolving structures, fixed aligners may struggle to generalize effectively. Developing adaptive context-awand atteexttwwaeeaaeee aligners could mitigate these issues by dynamically reconfiguring based on the data.
Hyperparameter Exploration. The sensitivity analysis indicates that BRIDGE's performance depends on hyperparameters to some extent, such as trade-off weights for the spectral regularizer and MoE components. Extending this exploration to cover interactions between hyperparameters and task-specific factors (e.g., dataset size, graph sparsity) could yield practical guidelines. Automated hyperparameter optimization techniques, such as Bayesian optimization or reinforcement learning, might also reduce manual tuning efforts.
Impact of Fine-tuning Epochs. BRIDGE's shot fine-tuning demonstrates strong results with limited labeled data
40

---

How Much Can Transfer? BRIDGE: Bounded Multi-Domain Graph Foundation Model with Generalization Guarantees
and epochs. However, its performance under prolonged fine-tuning remains unexplored. For example, understanding the trade-offs between overfitring and incremental gains with additional epochs larger labeled datasets would provide practical guidance for deploying BRIDGE in scenarios with varied data availability.
J. Future Directions
Building on the advancements presented in this work, several promising directions can further enhance and expand the impact of BRIDGE.
Enhancing Generalization for Extreme Few-Shot Scenarios. Although BRIDGE demonstrates strong performance in one-shot and five-shot tasks, real-world scenarios often involve domains where labeled data is either completely unavailable (zero-shot) or extremely sparse (e.g., fewer than 5 samples per class). Future research could explore unsupervised prompt initialization techniques that leverage unlabelled target domain data. Additionally, meta-lcarning approaches could be integrated to pre-train models with better generalization capabilities across unseen domains, enabling more effective transfer with minimal supervision.
Integration with Graph Agents. Future research could explore integrating BRIDGE with graph-based agents that leverage traph-structured data for decision-maken -making in interactive environments. For example, combining BRIDGE with reinforcement learning or multi-agent systems could enable applications such as autonomous traffic management, supply chain optimization, or real-time network monitoring. This would involve adapting BRIDGE's pre-training and prompt learning mechanisms to dynamically update graph representations based on agent interactions and feedback. Additionally, policy-guided graph prompting could be introduced to optimize agents' actions by focusing on fomain-relevan graph substructures, enhancing both efficiency and effectiveness in graph-driven environments.
Extending to Dynamic and Temporal Graphs. Real-world graphs, such as social networks, traffic systems, and financial transactions, often evolve over time. Future extensions of BRIDGE could incorporate temporal graph neural networks (e.g., temporal attention mechanisms) to capture dynamic relationships. Designing time-sensitive feature aligners that adapt to evolving graph structures and align temporal features across domains would further enhance the framework's 's utility for applications like anomaly detection or forecasting.
Incorporating Multi-Modal Graph Data. Many practical applications involve graphs enriched with multi-modal data, such as images, textual descriptions, or sensor readings. Future research could focus on joint feature extraction mechanisms that integrate these modalities into the BRIDGE framework. For example, leveraging pre-trained language vision models to embed auxiliary information and align it with graph features could improve performance on tasks requiring holistic understanding of entities and their relationships, such as biomedical knowledge graphs or e-commerce recommendation systems.
Exploration of Domain-Specifte Variations. Although BRIDGE aligns features across domains, certain specialized graphs, such as biological pathways, chemical reaction networks, or transportation systems, have unique properties that may benefit from tailored techniques. Future research could investigate Nomain-speeciff pre-training objectives, such link prediction for protein-proter interaction graphs path optimization for logistics networks. Additionally, customizable aligners that leverage domain knowledge, such as hierarchical clustering or molecular graph fingerprints, could improve transferability in these specialized contexts.
Expanding the Range of Downstream Tasks, While this study focuses on node and graph classification tasks, BRIDGE has the potential to address a broader range of graph-related problems. Future work could adapt the framework for link prediction (e.g., for knowledge graph completion or social network analysis), graph generation (e.g., molecule generation in drug discovery), or graph clustering (e.g., community detection in social graphs). This expansion would require task-specific prompts and evaluation protocols to demonstrate adaptability across diverse use cases.
41


Reviews:
[{"review_id": "4UHS4Tgaed", "reviewer": "Reviewer_vj96", "summary": "This paper introduces BRIDGE, a graph foundation model framework designed for multi-domain knowledge transfer using domain-invariant feature aligners, a MoE prompt initialization, and a theoretical generalization bound. It addresses feature heterogeneity across domains, efficient adaptation via prompt tuning, and theoretical quantification of transferability. The framework is validated on 6 diverse graph datasets with convincing empirical results.", "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": "Thank you for your insightful and constructive review, and for recognizing the contributions of BRIDGE. We address your questions briefly below.\n\n**Q1: The empirical validation of the generalization bound.**\n\n**A1:** To empirically validate the effectiveness of the bound, we conducted specific ablation studies by removing the spectral regularizer (Section 5.3). Removing the spectral regularizer resulted in **a clear performance drop**: specifically, accuracy decreased from **42.18% to 41.33% (−0.85%)** for 1-shot node classification, and more significantly from **60.20% to 54.51% (−5.69%)** for 5-shot node classification on the CiteSeer dataset. This substantial performance degradation empirically demonstrates the effectiveness of our generalization bound in improving transferability. These findings explicitly confirm the bound’s role in guiding practical model design.\n\n---\n\n**Q2: Empirical verification of Lipschitz behavior reduction should be included.**\n\n**A2:** Thank you for raising this important point. While our experiments clearly show that including the spectral regularizer improves transfer performance (Section 5.3), the original submission did not explicitly measure the Lipschitz constant's empirical evolution. Theoretically, the spectral regularizer is specifically designed to constrain the Lipschitz constant by penalizing large variations in learned representations. Given your valuable suggestion, we plan to explicitly measure and report the evolution of the empirical Lipschitz constant in future version, to more clearly demonstrate how our method reduces the representation sensitivity across domains.\n\n---\n\n**Q3: Still requires at least one labeled example for good performance.**\n\n**A3:** Indeed, BRIDGE is specifically designed for **text-free graphs**, where zero-shot knowledge transfer based solely on structural information is inherently limited, which is a fact well-supported by prior works. Pure structural information lacks sufficient semantic cues, leading to poor zero-shot performance, as extensively confirmed in the literature.\n\nTo demonstrate that BRIDGE can also effectively leverage textual semantics when available, we conducted additional experiments integrating textual information generated by an LLM as an enhancement strategy (Section 5.6, Table 3). Results clearly showed notable improvements: e.g., node classification accuracy on Reddit improved from **55.72% (structure-only)** to **57.16% (structure + LLM)** under zero-shot conditions. These experiments indicate that BRIDGE, though originally targeted for text-free scenarios, can effectively incorporate textual semantics, significantly improving zero-shot transfer performance when such information is available.\n\nWe will explicitly clarify BRIDGE’s intended scope (text-free graphs) and discuss the potential benefits of textual augmentation thoroughly in the revision.\n\n---\n\n**Q4: Spectral regularization may be expensive. Runtime analysis.**\n\n**A4:** Thanks for bringing up this important issue. We reply the same concerns for [**Reviewer Efax (Q2)**](https://openreview.net/forum?id=bjDKZ3Roax&noteId=e2XOFb3spO). To explicitly address this, we adopted efficient approximation techniques (lines 969–989) based on truncated eigenvalue decomposition and eigenvalue approximations.\n\n---\n\n**Q5: Tuning challenging.**\n\n**A5:** Your point regarding the complexity of hyperparameter tuning due to multiple moving parts is valid. Recognizing this issue, we provided a detailed sensitivity analysis of key hyperparameters (Figure 6, Section 5.5), covering: $K$, $\\alpha$, $\\beta$, and $\\gamma$. These clearly presented sensitivity analyses significantly simplify practical hyperparameter tuning by offering explicit recommendations and robust ranges. We also have provided the suggested hyperparameter settings in Appendix F.1 and the configuration files.\n\n---\n\n**Q6: Difference between training all graphs together versus one by one.**\n\n**A6:** Thank you for this insightful question. In BRIDGE, all graphs from different domains were trained simultaneously in the pre-training phase, explicitly enabling cross-domain alignment and knowledge transfer. This approach encourages the model to learn shared domain-invariant features across multiple domains simultaneously, resulting in enhanced transfer performance. However, we recognize that training all graphs jointly can pose substantial memory overhead.\n\nTraining graphs separately (one domain at a time) reduces memory consumption significantly but **sacrifices explicit cross-domain feature alignment**, potentially weakening transferability. While this sequential training strategy was not explicitly investigated in the original submission, we acknowledge your point and will explicitly clarify our current training strategy and discuss potential trade-offs in the revision. Detailed empirical comparisons of joint vs. sequential training strategies will be included in future work to thoroughly address this important point.", "claims_and_evidence": "Most of the claims are clearly supported with some minor problems.\n\n- Multi-domain feature alignment: the authors propose a two-stage aligner that unifies domain features. Though no direct alignment metric is provided, performance gains over baselines support this claim.\n- Prompt-based fine-tuning with MoE: ablations confirm MoE improves low-shot learning, and BRIDGE outperforms 15 baselines in few-shot settings.\n- Generalization bound informs model design: the bound integrates Wasserstein distance and Lipschitz constraints, operationalized via a spectral regularizer that improves transfer performance. The claim is theoretically sound, though empirical validation of the bound’s direct impact would strengthen it.", "methods_and_evaluation_criteria": "Technical contributions: BRIDGE is proposed by combining prompt learning, domain adaptation, and spectral regularization.\n- Baselines & benchmarks: evaluated against 15 baselines across 6 datasets in three domains (academic, e-commerce, social), covering node and graph classification.\n- Ablation & sensitivity studies: component-wise analysis, showing MoE and spectral regularizer are critical.", "theoretical_claims": "- Domain adaptation bound for graph transfer: extends Wasserstein distance-based domain adaptation theory to graph learning, where the bound depends on the Lipschitz constant of the GNN encoder.\n- Spectral analysis for transferability: derives a lower bound on the Lipschitz constant using graph spectral properties. Uses this to construct $R_{spec}$, penalizing over-sensitive representations.\n- Mathematical correctness: the proofs follow graph domain adaptation theory, adapting results from Mohri (2018) and Gama (2020). However, empirical verification of Lipschitz behavior reduction should be included.", "experimental_designs_or_analyses": "-BRIDGE achieves SOTA performance on all tasks: outperforms all 15 baselines in one-shot and five-shot classification. Cross-domain generalization (e.g., social graphs) is improved over existing pre-training methods. MoE prompting performs better in extremely low-shot settings, confirming its role in knowledge transfer.\n-Ablation studies validate each component: removing the feature aligner, MoE, or spectral regularizer significantly reduces accuracy. Faster convergence than standard fine-tuning.\n-Hyperparameter sensitivity: performance varies with aligner strength ($\\alpha$), MoE routing entropy ($\\beta$), and spectral regularization ($\\gamma$). Some tuning is needed, but trends remain consistent across datasets.", "relation_to_broader_scientific_literature": "This paper contributes to multiple areas in graph representation learning, domain adaptation, few-shot learning, and graph foundation models. It integrates ideas from these domains into a novel framework that enhances transferability across graph datasets.", "essential_references_not_discussed": "There are no critical missing references in the paper. The authors provide a comprehensive review of related work, covering graph pre-training, domain adaptation, prompt learning, and theoretical generalization bounds. They cite relevant prior works. The citations potentially sufficiently contextualize BRIDGE’s contributions, and nearly no essential prior work appears to be missing.", "other_strengths_and_weaknesses": "Strength:\n- Combines multi-domain learning, prompt-based transfer, and theoretical regularization into a unified framework.\n- Outperforms all baselines across 6 datasets and multiple domains, with robust few-shot generalization.\n- First work to derive a domain adaptation bound for prompt-based graph transfer, integrating Lipschitz constraints.\n- Includes full proofs, hyperparameter details, and additional experiments, enhancing transparency.\n \n\nWeakness:\n- Many moving parts (aligners, MoE, spectral regularizer) and multiple hyperparameters, making tuning challenging.\n- Performance drops significantly when no labeled target examples are available (m=0 setting). A true zero-shot graph foundation model remains an open challenge.\n- Eigen-decomposition for $R_{spec}$ could be expensive on large graphs. A runtime analysis would be helpful.", "other_comments_or_suggestions": "See Weakness.", "questions_for_authors": "During the pre-training stage, what is the difference between training all graphs together versus training them separately one by one? How do their performances compare? (Considering that training all graphs together poses a significant memory challenge.)\n- Given limited labeled data, how were hyperparameters optimized? Are there default recommendations?\n- Did $R_{spec}$ actually reduce domain divergence or Lipschitz constants? Can you show this empirically?", "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": "Dvx33SbmNK", "reviewer": "Reviewer_QW95", "summary": "BRIDGE is a bounded multi-domain graph pre-training and prompt learning framework that enhances knowledge transfer in graph foundation models. It integrates domain-invariant alignment, a lightweight MoE for selective knowledge transfer, and a graph spectral-based generalization bound. Experiments show state-of-the-art performance in few-shot classification across datasets and domains, with further improvements when combined with LLMs. I consider this paper as a new bench for more generalizable and trustworthy GFMs, as existing works on GFMs lacks theoretical analysis of cross-domain transfer of pre-trained knowledge, to the best of my knowledge.", "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": "Thank you for your thoughtful review and positive feedback. We appreciate your recognition of BRIDGE’s contributions and address your comments and suggestions point by point below.\n\n**Q1: The scalability and computational efficiency.**\n\n**A1:** Thank you. For scalability, we address this in **Reviewer Efax (Q3)** with new results on **ogbn-arXiv** (source) and analysis on **Reddit** (target). For efficiency, see **Reviewer Efax (Q2)**, where we provide end-to-end training and fine-tuning time comparisons.\n\n---\n\n**Q2:  Theoretical validation of the generalization error bound tightness.**\n\n**A2:** The bound tightness primarily depends on: **(1)** how accurately the Wasserstein distance $d_W$ captures true domain divergence, and **(2)** how precisely the Lipschitz constants ($L_f$, $L_g$) reflect actual smoothness of the learned graph representations. Lower $d_W$, tighter estimates of $L_f$ and $L_g$ yield a tighter bound, more closely approximating empirical generalization errors. We will explicitly evaluate these factors empirically in future revisions.\n\n---\n\n**Q3: The practical impact of the generalization bound.**\n\n**A3:** The generalization bound indicates how effectively BRIDGE transfers knowledge across domains. A smaller bound means better transfer (e.g., between similar graphs like Cora and PubMed), whereas a larger bound signals difficulty (e.g., between citation networks and social graphs), guiding practical adjustments like additional fine-tuning.\n\n---\n\n**Q4: How minimizing the bound directly affects transfer performance.**\n\n**A4:** Directly minimizing the bound improves BRIDGE’s transfer by explicitly constraining domain divergence ($d_W$) and smoothness ($L_f$, $L_g$). Practically, this means representations from different domains become closer and smoother, significantly enhancing generalization to unseen graphs. Experiments (Section 5.3, Figure 4) confirm that optimizing this bound via the spectral regularizer consistently leads to higher accuracy on downstream tasks.\n\n---\n\n**Q5: Whether real-world graphs satisfy optimal transport-based alignment assumption.**\n\n**A5:** The optimal transport-based alignment assumption has been widely adopted in prior studies, and our work follows this common practice. The underlying reason is that real-world graphs often **share intrinsic structural similarities** (e.g., similar community or connectivity patterns across social or citation graphs), allowing optimal transport methods to meaningfully measure domain divergence.\n\n---\n\n**Q6: Missing comparisons to general domain adaptation baselines.**\n\n**A6:** Since BRIDGE targets **graph-structured data**, we compare with strong **graph-specific** adaptation baselines (e.g., GCOPE, MDGPT). General domain adaptation methods lack structure-aware design, making them less relevant for fair comparison.\n\n---\n\n**Q7: LLM feature alignment lacks qualitative validation.**\n\n**A7:** Thank you for this valuable comment. We clarify that the LLM-enhanced alignment is **not part of BRIDGE's core contributions**. Rather, it serves as an **auxiliary experiment** to demonstrate BRIDGE’s applicability to text-attributed graphs, highlighting BRIDGE’s broad adaptability. We agree qualitative validation would further support this applicability and will consider it in future extensions.\n\n---\n\n**Q8: Statistical significance tests for ablations.**\n\n**A8:** Thank you for raising this point. Explicit statistical significance tests (e.g., t-tests) for ablation studies are relatively uncommon for papers in AI research communities. Ablations generally aim to intuitively illustrate each component’s individual contribution rather than establish statistical differences rigorously. Consistent with common practice, our ablations provide results averaged over multiple runs (with std), clearly showing reliable performance differences. \n\n---\n\n**Q9: Compare different MoE architectures.**\n\n**A9:** Our MoE is a lightweight internal module of BRIDGE. Rather than comparing MoE variants in isolation, we focus on end-to-end comparisons with full graph adaptation methods (e.g., MDGPT, GCOPE) to reflect overall effectiveness.\n\n---\n\n**Q10: Compare against full fine-tuning methods.**\n\n**A11:** BRIDGE focuses on **efficient prompt-based tuning**, not full model updates. We compare with strong graph prompt baselines (e.g., ProNoG, GraphPrompt), which align better with our design and goals.\n\n---\n\n**Q11: Negative transfer issues for multi-domain learning.**\n\n**A11:** Negative transfer is a key challenge in multi-domain learning. BRIDGE addresses this via **expert assignment weights ($\\mathbf{S}$)**, which down-weight less relevant domains. As shown in our response to **Reviewer Efax (Q5)**, $\\mathbf{S}$ effectively favors helpful sources. We acknowledge BRIDGE doesn’t explicitly handle extreme cases where **all** source domains are harmful, which will be explored in future work.\n\n---\n\n**Q12: Typos and Grammars.**\n\n**A12:**  We will carefully proofread the manuscript.", "claims_and_evidence": "The claims made in this paper are generally supported by clear empirical results and theoretical analysis, while some aspects could be more thoroughly validated. \n\n- The well-supported claims include the effectiveness of BRIDGE in few-shot learning, theoretical justification for multi-domain knowledge transferability, effectiveness of key modules or components, etc.\n\n- The potentially problematic claim probably lies in the scalability and computational efficiency. BRIDGE claims superior efficiency compared with other baselines in down-prompting, but comparisons only focus on fine-tuning convergence speed rather than end-to-end training costs. But I believe this is not a major issue, as GFMs do not place a strong emphasis on the training cost of the pre-training phase but rather focus on whether they can achieve good performance quickly during the fine-tuning stage.", "methods_and_evaluation_criteria": "I agree that the proposed methods and evaluation criteria align well with multi-domain graph transfer learning under the GFM settings. The “pretrain-then-prompt” paradigm, domain-invariant alignment, and spectral-based generalization bound basically seems to effectively address more generalized and trustworthy knowledge transfer in multi-domain GFMs. The evaluation on few-shot node and graph classification with cross-dataset and cross-domain settings is promising, and comparisons with 15 baselines strengthen credibility. The only concern lies in the computational efficiency are not fully analyzed. A more comprehensive evaluation on larger real-world graphs and detailed cost analysis would further validate BRIDGE’s practicality.", "theoretical_claims": "I have reviewed the theoretical claims and proofs presented in the paper, particularly focusing on the generalization error bound derivation and its spectral regularization component.\n\n- generalization error upper bound: an optimizable upper bound for the generalization error built upon the graph spectral theory and Lipschitz continuity. While the bound is theoretically sound, the paper lacks in-depth theoretical validation of how tight this bound is. Some additional visualization or sensitivity analysis of the bound’s impact on performance would strengthen the authors’ claim.\n\n- Lipschitz constant lower bound: relies on the graph eigenvalue decomposition to express the Lipschitz constant in a spectral form. The bound assumes optimal transport-based alignment of graph structures on both sides, but the practical effectiveness of this assumption is somewhat unclear. More discussion on whether real-world graphs under GFM settings satisfy these assumptions would be beneficial.", "experimental_designs_or_analyses": "I reviewed the experimental design and analyses in the paper, focusing on the few-shot learning setup, baseline comparisons, ablation studies, and efficiency evaluations. \n\n**Pros:**\n- The few-shot experimental design is well-suited for evaluating multi-domain knowledge transfer under the GFM settings. \n- Baseline selection is comprehensive and covers multiple paradigm related to the paper scope.\n- Ablation studies effectively isolate key components, and independently analyze their contributions, respectively.\n- The LLM integration experiment for the initial feature enhancement is novel and promising. \n\n**Cons:**\n- Missing comparisons to general domain adaptation baselines, not only resitricted to graph learning.\n- No full computational cost analysis, including training and inference time, memory for both pre-training and fine-tuning.\n- LLM feature alignment lacks qualitative validation.", "relation_to_broader_scientific_literature": "The key contributions of this paper build upon and extend several existing areas of multi-domain graph pre-training, prompt learning for graphs, generalization bounds, and LLMs for GFM understanding.\n\n- multi-domain graph pre-training: BRIDGE introduces a domain-invariant alignment strategy that aligns both dimensions and semantics across domains, addressing the cross-domain generalization challenge more explicitly than prior work. Unlike prior works relying on textual metadata for alignment, BRIDGE is agnostic to text attributes, making it applicable to a wider range of graphs.\n\n- prompt learning for graphs: BRIDGE extends prompt learning to multi-domain graph settings, introducing a lightweight MoE network to selectively transfer domain-specific knowledge, which prior graph prompt methods lack. Unlike GraphPrompt and GraphPrompt+, which manually design prompts, BRIDGE learns self-supervised prompt initializations, making it more adaptive.\n\n- generalization bounds: BRIDGE integrates generalization bounds into its optimization, providing an explicit, optimizable spectral regularizer that controls transferability constraints. Unlike past works that assume smooth transfer functions, BRIDGE derives an explicit bound in the graph spectral space, which is novel in the filed of GFMs.\n\n- LLMs for graph understanding: BRIDGE incorporates LLM-generated node descriptions as a feature alignment mechanism, demonstrating improved multi-domain adaptation. Unlike MDGPT, BRIDGE’s graph-specific prompting enables more efficient knowledge transfer without relying on extensive LLM augmentation.", "essential_references_not_discussed": "I believe that the related work section and other parts of the paper have included the necessary references, allowing me to understand the current state of research in the relevant field and to objectively evaluate the contribution of this paper.", "other_strengths_and_weaknesses": "**Strengths:**\n- The pretrain-then-prompt approach for multi-domain graph learning is an promising extension of prior works that primarily focused on either pretraining or prompting separately.\n- The graph spectral-based generalization bound provides a new theoretical perspective on quantifying transferable knowledge, which is not commonly explored in existing works.\n- The integration of domain-invariant alignment with a lightweight MoE for selective knowledge transfer is a novel contribution that improves adaptability across domains.\n\n**Weakness:**\n- The paper does not analyze how minimizing the bound directly affects transfer performance, making it unclear how well the bound correlates with real-world generalization.\n- The MoE mechanism is a key component of BRIDGE, facilitating selective knowledge transfer, but the paper does not compare different MoE architectures. It is unclear why this specific MoE design was chosen over alternatives such as soft attention-based routing or adaptive gating mechanisms.\n- Multi-domain learning can suffer from negative transfer, where knowledge from one domain hurts rather than helps performance in another. The paper assumes all knowledge is beneficial, but does not analyze whether certain source domains degrade performance in the target domain.", "other_comments_or_suggestions": "**Typos and Grammars:**\n- The symbol “≜” (Eq. (12) and Eq. (13)) is not commonly used. \n- “The graph learner compound of the encoder …” should be “The graph learner is composed of the encoder …”\n- “We consider as benefiting from the spectral regularizer …” should be “We attribute this improvement to the spectral regularizer …”\n- “between domains introduces graph pre-training variations …” should be “between domains introduces variational inference in graph pre-training …”", "questions_for_authors": "- Can you provide an empirical analysis or more easy-to-follow understanding demonstrating whether minimizing the generalization upper bound directly improves transfer performance?\n- Did you conduct any experiments to quantify negative transfer? Are there cases where knowledge from one domain degrades performance in another?\n- How does BRIDGE compare against full fine-tuning methods (e.g., LoRA, Adapter-based fine-tuning) in terms of performance vs. computational cost?\n- Other questions, please refer to the aforementioned 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": "pc92mjC8Ma", "reviewer": "Reviewer_Efax", "summary": "The authors argue that while the \"pretrain-then-prompt\" framework has been extensively studied in other fields, its application in the graph domain remains underexplored, particularly from a theoretical perspective.\nTo address this gap, this paper introduces BRIDGE, a pretraining and prompt learning framework designed for multi-domain graph data, incorporating a theoretically driven regularization loss.\nDuring the pre-training phase, BRIDGE performs dimension-wise alignment and semantic-wise feature alignment, inspired by the Independent Causal Mechanism.\nIn the fine-tuning phase, alongside the downstream task loss, BRIDGE introduces an aligner routing network that selectively assembles the pre-trained source domain aligner and proposes a spectral regularizer to minimize the upper bound of generalization error.\nThey conduct experiments in both cross-dataset and cross-domain settings to demonstrate the effectiveness of the proposed framework.\n\n## update after rebuttal\nAs I mentioned in the 'Rebuttal Comment', my concerns have been well addressed; therefore, I have updated my score.", "questions": "", "limitations": "", "rating": 3, "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 for the positive evaluation regarding its theoretical foundation for constructing GFMs. Response:\n\n**Q1: Why work well in zero-/one-shot but not in two-/three-shot.**\n\n**A1:** Thank you for the insightful question.\n\n- **0-shot limited for text-attributed graphs:** Knowledge transfer based solely on structural information is inherently challenging in zero-shot scenarios. BRIDGE primarily targets **text-free graphs**, with experiments (RQ5) integrating an LLM as an enhancer to show effectiveness when textual semantics are available.\n- **Why excels at 1-shot but not 2-/3-shot:** BRIDGE’s MoE-based selective knowledge assembly effectively utilizes precise domain-invariant knowledge, providing a principled advantage at **extremely sparse supervision** (1-shot). At 2-/3-shot, the MoE routing faces **increased uncertainty**, while simpler alignment strategies (e.g., ProNoG, GCOPE, MDGPT) temporarily gain an advantage with limited additional supervision. When supervision further increases (≥5-shot), BRIDGE’s **spectral regularizer dominates** by constraining generalization error, restoring its superior performance.\n\n---\n\n**Q2: The training and inference time.**\n\n**A2:** Thanks for bringing up the important issue.\n\n- **Pre-training Phase:** We adopt an **early-stopping strategy** based on **loss**, so the maximum training time depends on a preset maximum of 1000 epochs. No explicit inference is conducted during pre-training. Results (s):\n\n| Source | Time / Epoch | Total Time |\n| --- | --- | --- |\n| CS, PM, Ph, Com | 1.67 | 1653.57 |\n| CR, PM, Ph, Com | 1.53 | 1624.01 |\n| CR, CS, Ph, Com | 0.92 | 938.38 |\n| CR, CS, PM, Com | 1.20 | 1147.02 |\n| CR, CS, PM, Ph | 1.31 | 1698.75 |\n- **Fine-tuning Phase:**  We report time per epoch, total fine-tuning time, and inference time separately. The spectral regularizer (highlighted as potentially time-consuming) is only computed during fine-tuning loss, particularly at small shot numbers ($m$). To reduce computational costs in eigenvalue decomposition, we implemented approximate techniques (lines 969–989). Results (5-shot Node Cls., s):\n\n| Target | Time / Epoch  | Inference Time | Total Time |\n| --- | --- | --- | --- |\n| Cora | 0.13 | 0.02 | 142.37 |\n| CiteSeer | 0.18 | 0.02 | 183.15 |\n| PubMed | 0.82 | 0.10 | 1002.62 |\n| Photo | 0.36 | 0.09 | 384.26 |\n| Computers | 0.69 | 0.05 | 755.39 |\n| Reddit | 6.62 | 2.30 | 579.16 |\n\n---\n\n**Q3: Including larger datasets (e.g., ogbn-arxiv) for both source and target domains.**\n\n**A3: Thanks for this valuable suggestion.**\n\n- **In source domain:** Following your advice, we have now included **ogbn-arXiv** as an additional dataset for pre-training. Results (Acc. %):\n\n| S → T | 1-shot Node Cls. | 1-shot Graph Cls. |\n| --- | --- | --- |\n| CS, Ph, arXiv → Cora (cross-dataset) | MDGPT: 37.21, **BRIDGE: 43.62** | MDGPT: 45.35, **BRIDGE: 48.20** |\n| CS, Ph, arXiv → Reddit (cross-domain) | MDGPT: 58.82, **BRIDGE: 64.16** | MDGPT: 68.22, **BRIDGE: 71.74** |\n- **In target domain:** We have already validated BRIDGE on **Reddit**. Notably, Reddit has **232,965** nodes and **114,615,892** edges (Table D.1), substantially **larger than ogbn-arXiv** (169,343 nodes and 1,157,799 edges). Therefore, the strong performance on Reddit sufficiently demonstrates BRIDGE’s capability for knowledge transfer on large-scale graphs.\n\n---\n\n**Q4: The ablation studies regarding the prompt.**\n\n**A4:** Thank you for the insightful suggestion. We have conducted an ablation concerning the graph prompts for fine-tuning. Results (Acc. %):\n\n| T: CiteSeer | 1-shot Node Cls. | 1-shot Graph Cls. | 5-shot Node Cls. | 5-shot Graph Cls. |\n| --- | --- | --- | --- | --- |\n| BRIDGE | 42.18 | 50.49 | 60.20 | 59.80 |\n| BRIDGE (w/o prompt) | 35.23 | 34.12 | 50.34 | 49.17 |\n\nNote that, prompts $\\boldsymbol{\\mathcal{P}}$ and routing weights $\\mathbf{S}$ are the only tunable parameters. Without prompts, only $\\mathbf{S}$ is trained, significantly restricting fine-tuning flexibility and degrading performance, highlighting prompts' necessity in BRIDGE.\n\n---\n\n**Q5: How does 'S' favors same-domain over cross-domain knowledge for selective assembly?** \n\n**A5:** We calculate the learned assignment weights for source experts from the same domain versus cross-domain sources. Results (1-shot, Node Cls.):\n\n| S | T | Weights |\n| --- | --- | --- |\n| CS, PM, Ph, Com | CR | [0.5742, 0.2950, 0.0486, 0.0822] |\n| CR, CS, PM, Ph, Com | Reddit | [0.3001, 0.2855, 0.0634, 0.1254, 0.2256] |\n\n**Analysis:** For **Cora** (CR),  **Academic** experts (CS, PM) receive significantly higher weights (0.5742, 0.2950) compared to **E-Commerce** sources (Ph, Com: 0.0486, 0.0822). For **Reddit**, the structurally similar **E-Commerce** expert (Computers, 0.2256) and **Academic** experts (CR, 0.3001; CS, 0.2855) dominate. These results validate that BRIDGE effectively favors knowledge transfer from same-domain or structurally similar-domain experts, supporting our selective assembly design.", "claims_and_evidence": "I agree with their claim that prompt learning is under-studied in the graph domain from both practical and theoretical perspectives.\nMoreover, each module in BRIDGE is intuitive, and the proposed regularizer is supported by a strong theoretical foundation, which is a significant advantage of this work.", "methods_and_evaluation_criteria": "The authors' evaluation criteria for cross-dataset and cross-domain settings are valid. However, I suggest incorporating larger datasets (e.g., ogbn-arxiv) for both the source and target domains. This is important for two reasons: 1) assessing whether the model's performance improves as the amount of training data increases, and 2) evaluating the model's ability to effectively perform inference on large datasets.\n\nAdditionally, while the paper does not report training and inference times, I am concerned that calculating the spectral regularizer could be time-consuming for large datasets.", "theoretical_claims": "I did not find any issues with the theoretical claims.", "experimental_designs_or_analyses": "The experimental design is valid, but an ablation analysis comparing the model with and without the prompt is needed.", "relation_to_broader_scientific_literature": "I believe the key contribution of this paper is the proposed theoretical analysis bounding knowledge transfer generation, along with the derived regularization loss, which can serve as a foundation for future work in this field.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "A major concern with this paper is the low performance observed in the few-shot case (m=2,3) in Figures 2, 3, E.1, E.3, and E.4. Since pre-training is particularly beneficial when supervision is limited, this represents a critical scenario for the model's effectiveness.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1) It would be helpful if the authors could clarify why BRIDGE performs well in the zero and one-shot settings but not in the two or three-shot cases.\n2) I would like to know the training and inference times.\n3) Please consider including larger datasets for both the source and target domains.\n4) Please show the ablation studies regarding the prompt\n5) Please show how the learned 'S' gains significantly more knowledge from the same domain compared to cross-domain sources to validate the selective knowledge assembly.\n\nFor checking the validity of selective knowledge assembly, showing the learned 'S' that it is truly gains high knowledge from same domain than cross domains.\n\nSelectiveKnowledgeAssembly\n\nIf all of these concerns are addressed, I would be happy to reconsider my score.", "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: This paper received three effective reviews, and all of them are positive. Overall, the paper is of good quality and should be accepted.
Decision: Accept (poster)