Transfer Learning on Edge Connecting Probability Estimation Under Graphon Model
Authors: Yuyao Wang
Yu-Hung Cheng
Debarghya Mukherjee
Huimin Cheng
Venue: NeurIPS 2025 poster ()
Keywords: Graphon Estimation
Transfer Learning
Gromov-Wasserstein Transport
Neighborhood Smoothing
Adaptive Debiasing
Primary area: general_machine_learning
TL;DR: We propose GTrans, a transfer learning framework that enhances graphon estimation for small graphs via neighborhood smoothing, Gromov-Wasserstein transport, and adaptive debiasing, boosting link prediction and graph classification.

Abstract: Graphon models provide a flexible nonparametric framework for estimating latent connectivity probabilities in networks, enabling a range of downstream applications such as link prediction and data augmentation. However, accurate graphon estimation typically requires a large graph, whereas in practice, one often only observes a small-sized network. One approach to addressing this issue is to adopt a transfer learning framework, which aims to improve estimation in a small target graph by leveraging structural information from a larger, related source graph. In this paper, we propose a novel method, namely GTRANS, a transfer learning framework that integrates neighborhood smoothing and Gromov-Wasserstein optimal transport to align and transfer structural patterns between graphs. To prevent negative transfer, GTRANS includes an adaptive debiasing mechanism that identifies and corrects for target-specific deviations via residual smoothing. We provide theoretical guarantees on the stability of the estimated alignment matrix and demonstrate the effectiveness of GTRANS in improving the accuracy of target graph estimation through extensive synthetic and real data experiments. These improvements translate directly to enhanced performance in downstream applications, such as the graph classification task and the link prediction task.

Transfer Learning on Edge Connecting Probability Estimation Under Graphon Model
Yuyao Wang Boston University yuyaow@bu.c edu
Yu-Hung Cheng Boston University yhcheng@bu. edu
Debarghya Mukherjee
Huimin Cheng
Boston University mdeb@bu. edu
Boston University huimin230bu edu
Abstract
Graphon models provide a flexible nonparametric framework for estimating latent connectivity probabilities in networks, enabling range of downstream applications such as link prediction and data augmentation. However, accurate graphon estima- tion typically requires large graph, whereas in practice, one often only observes small-sized network. One approach to addressing this issue is to adopt a transfer learning framework, which aims to improve estimation in a small target graph by leveraging structural information from a larger, related source graph. In this paper, we propose novel method, namely GTRANS, transfer learning framework that integrates neighborhood smoothing and Gromov- Wassoossteserttinn optimal transport to align and transfer structural patterns between graphs. To prevent negative transfer, GTRANS includes an adaptive debiasing mechanism that identifies and corrects for target-specific deviations via residual smoothing. We provide theoretical guarantees on the stability of the estimated alignment matrix and demonstrate the effectiveness of GTRANS in improving the accuracy of target graph estimation through extensive synthetic and real data experiments. These improvements translate directly to enhanced performance in downstream applications, such the graph classification task and the link prediction task.
Introduction
Graph data increasingly common in numerous applications, from social networks [54, 14, 17] to biological systems [5] and power electronics networks [64]. A fundamental task in graph learning is estimating the probability of connections between nodes, facilitating various downstream analyses. Traditional methods for estimating connection probabilities often rely on specific parametric random graph models, including Erdōs-Rényi (ER) model [20, 18]. the stochastic block model (SBM) [24] and SBM variants [27, 31, 8], the exponential random graph model (ERGM) [35], and latent position model [23]. While useful, these can be restrictive and suffer from model misspecification.
Graphon. To address this limitation, the graphon model provides a powerful non -on-aarmettiic framework [34, 42]. A graphon is symmetric, measurable function f : [o, 1]2 - [0, 1], where nodes are assigned latent positions u, ~ Unif(0, 1], and edge probabilities are given by P.j = f(u, uz). This model includes classical models special cases-e.g., constant functions yield ER graph, and step functions yield SBMs. Graphon models have been widely used for various downstream applications, such as data augmentation [22], link prediction [67], causal inference under network interference [32], dataset distillation [61]. and assessing vulnerabilities in the smart grid [3]. In the existing literature, various methods have been developed for estimating the connection probability under graphon model [10, 1, 9, 11, 65, 45J. The estimation accuracy of these methods typically improves as the network size (i.e., number of nodes) increases.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Motivating example. Small graphs severely limit graphon estimation accuracy, posing significant challenges for graphon-based downstream analysis. For example, in the PROTEINS-Full dataset ([22)), each graph represents protein structure and consists of only 25 nodes on average. Therefore, employing standard graphon estimation strategies may lead to poor accuracy. Fortunately, similar domains often offer larger graphs with related structures, e.g., the D&D dataset averaging 284 nodes per protein graph. Both datasets encode similar biological relationships, creating an ideal transfer learning opportunity. Therefore, it would be highly beneficial to develop a statistically sound methodology for transferring knowledge from a related large graph. Such an approach could substantially enhance inference on smaller graph, while preserving the flexibility of the graphon framework.
Transfer learning. Transfer learning has emerged as powerful framework for leveraging knowledge from data-rich domains to improve learning in dcta-scarce domains [43]. In graph settings, it has been used for GNN meta-learning, pre-trainingffie-tunee
and adversarial adaptation, as well as for network regression [63, 25, 47, 7, 62, 12]. To the best of our knowledge, there is only one work [26] proposing transfer learning method for graphon estimation. However, their methodology is constrained to scenarios where the target network exists as a subset of the source network, thereby leveraging known node correspondences between networks. We aim to tackle the more practical scenario where node correspondences are unknown. To our knowledge, no previous research addresses this gap, which presents two key challenges: (1) Aligmment problem: Without known node correspondences, structural patterns may transfer to non-corresponding regions, potentially degrading estimation quality. (2) Unsupervised learning setting: Traditional transfer learning relies on labeled data, but graphon estimation provides no such signals for formulating clear transfer loss functions.
Transfer learning for graphon estimation. To address these challenges, we propose novel transfer learning method for graphon estimation via optimal transport and neighborhood smoothing. abbreviated as GTRANS. Our method consists of three key steps. Initial estimation step: Given source and target adjacency matrices A, E {0, 1) and A E {0. 1 1 ( n, > n4, we first apply neighborhood smoothing to obtain initial graphon estimators pini and Pini for both source and target. Transferring step: We then employ Gromov- Wasseersttntt (GW) optimal transport to compute the alignment matrix Å [0, 1]",,** using the aforementioned initial graphon estimators rather than the original adjacency matrices which contain Bernoulli noise. The resulting alignment matrix is then used to map the initial source graphon estimator into the target domain's latent space projection. Debiasing step: When the source and target graphons
through a structure-preserviin differ significantly (indicated by large GW distance in the previous step), we implement adaptive debiasing mechanism to mitigate potential negative transfer.
Our contributions can be summarized as follows: (1) To the best of our knowledge, GTRANS is the first method for graphon estimation that transfers knowledge across graphs without any known node correspondence. (2) We establish consistency results for the proposed method, showing that the alignment matrix computed using smoothed graphon estimates converges (under mild conditions) to the true optimal transport map. (3) Extensive experiments on synthetic datasets show method consistently achieves lower estimation error than state-of- fhate-of-the-e the-art alternatives. On real-world networks, it outperforms existing approaches in both graph classification via data augmentation and link prediction tasks. Our implementation is publicly available at https:/ //gi thub. com/ homm/oliie olivia3395/ GTRANS.
Preliminary
In this paper, we let A f0, 1 denote adjacency matrix, and P denote the corresponding probability matrix that generates A, i.e., Aij ~ (Pr A graphon model typically assumes that each node has latent position u, E [0, 1] and P.j [(u, uz), where I is a symmetric, measurable function on [0, 1 Graphon estimation refers to the estimation of the probability matrix P under the graphon model. The accuracy of the graphon estimation is typically measured using the mean squared error: MSE(P, P) = =|P PPP -PPP12 P1l.//?. MSE is widely used as a standard evaluation metric in existing literature [65, 45, 9].
2.1 Graphon Estimation
Broadly speaking, graphon estimation methods fall into three categories: (1) Global low-rank techniques, including Universal Singular Value Thresholding [11, 60): (2) Combinatorial optimization, which directly searches for node assignments [19, 29] (3) Smoothing banoothing-based methods, which pool

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nodes with similar degree [9] or similar latent neighborhoods [65, 40]. Among these methods, neighborhood smoothing (NS) [65] stands out because it can achieve nearly optimal MSE among all estimators computable in polynomial time. The key idea of NS method is to estimate pairwise connection probabílities by measuring the proportion of edges between the respective neighborhoods of node pairs, effectively leveraging local structure to infer global patterns. Further algorithmic details of NS are provided in Appendix G.2.
2.2
Soomov-Wasserstein Distance
The Waanoo-Wassersteinn (GW) distance [37, 38] provides a principled framework for comparing metric-measure spaces, making it particularly suitable for comparing graphs of different sizes without requiring node correspondences. GW distance and couplings (and their variations, e.g., fused GW distance [50, 41], sliced GW distance [51]) have proven to be very useful for comparing/aligning graphs, shapes, or distributions supported on different domains [15, 2, 58, 21, 36, 56]. In its most general formulation, given cost/distance matrices C E Rmxm and D E Raxn the GW distance between these two distance matrices is defined as: minxell(p.e) E.j.k. [(C(i,, k), D
Thi where [1( v) is the set of all couplings E ===x" such that Taj 2 0. 2ij Wij = 1 and Σ, mij- pi the loss function, for example L(x,)) y) (r - yer.e μ and v denote the uniform measures on the source and target nodes, respectively; that is, μ (1/ns....,, 1/nns) and v (1/nr,-. ..... 1/nr). The minimizer л of the above equation coupling. While the GW distance offers a powerful framework, the exact computation of the GW distance is a quadratic assignment problem, which is known to be computationally intensive [37, 52]. The GW distance has been widely used for graph alignment and transfer tasks, demonstrating its effectiveness in applications such as node embedding, cross-domain alignment, and subgraph matching [58, 56, 13, 30, 59].
and Σ; Tay = Fj, and
called an optimal GW
Entropic wasserewass distance for scaling up to large datasets. To alleviate the compu- tational difficulty, [49] proposed adding an entropic regularization to the GW objective function as follows: min Ei,,,k. [C k), 1)) Thy + cKL (I 14 & v) The product μ @ the product measure assigning a uniform weight of (ngnr))- Although the addition of an entropic regularizer does not make the optimization problem convex (unless e exceeds a certain threshold), it significantly enhances computational efficiency. As elaborated in [49], this regularization enables a Sinkhorn-like algorithm, adapted from [16] for the standard optimal transport problem, which is simple to implement and typically exhibits fast convergence [46].
3 Transfer Learning on Connecting Probability Estimation for Graph
In this section, we first formalize the problem setup for transfer learning on connecting probability estimation, then present our method GTRANS in detail.
3.1 Problem Setup
We consider the problem of estimating the graphon for target graph of relatively small size, leveraging information from a larger source graph that shares structural similarities. Formally, we have a source graph with adjacency matrix A, E {0, 1) where nodes, and target graph with adjacency matrix A, (0, 1 where n, = |V is the number of nodes. We assume n. > ne, i.e., the source graph is larger than the target graph. We assume these networks are generated by the following graphon models.
[V. is the number of
Agaj ~ Ber(P...)) where Pa.ú = fs Ms.), usss E [0, 1]. us.j E [0, 1].
1 $ 1
< j S n.
Ar. ~ Ber (Ptr(),))) (,i)); where Pr.sy = ft (u,,s, ue.s), ues E [o. 1], u.j E [0, < < S ni-
Here f. and f. represent the latent graphon function in the source network and target network. Each node i E (1....,n,, n,, of the source and target network has a latent position u, and u,,,, respectively.
Our goal is to estimate the target probability matrix P, by leveraging both the observed target graph A, and knowledge transferred from the larger source graph A,, adaptively, when the source is indeed similar to the target.
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3.2 Proposed Method
Overview. Figure shows the workflow our method GTRANS, which consists of three main steps: (1) Initial graphon estimation: GTRANS begins with observed source and target networks and computes their respective initial estimates Piné [0, 10,, Pins E [0, 1 Xms, We obtain these initial edge probability matrices using the NS method [65]. These initial estimators capture the basic structure of the respective graphs but may have limited accuracy for the target graph due to its smaller size. (2) Transferring step: We employ optimal transport to calculate the alignment matrix between between the source and target domains. Using this alignment,
transfer the source estimate the target domain obtain Ptrans which is further smoothed to ohtain Prrans If the source-targe domain shift (measured by the transport distance d) is below a threshold 6, our final estimate is P = Prrans (3) Debiasing step: When d > 6, indicating significant domain shift, direct transfer may miss surget-specifce structures. To correct this, we perform debiasing step. We compute the residual matrix R, Pimi Ptrans which captures structural patterns present in the target graph but not explained by transfer from the source. However, R, is noisy due to the small target sample. To extract meaningful signal, we apply neighborhood smoothing on R, to obtain a denoised residual estimate pre. This smoothed correction then added back to the transferred estimate, yielding the final output P Parans2 pres
Original Networks
Initial
Optimal Transport
Alignment
Transterring
Smoothing
Source Natwork
prrans
pireasz
Output
Target Network
Debissing step Debiased Smoothed Reaidual
Residdst
Output
pras
Figure 1: Workflow of GTRANS. Initial estimates from source and target networks are aligned via optimal transport. If source-target distance d < 8, the smoothed transferred estimate is returned. Otherwise, debiasing step applied to produce the final output.
3.2.1 Transferring Step
In this step, we align nodes between the source and target graphs using optimal transport and transfer the source graphon estimator the target domain. The key idea is to align nodes that exhibit similar connection patterns. We define the oracle alignment as the theoretical correspondence that would be obtained if had access the true underlying graph probability matrices. practice, we don't know true underlying graph probability, researchers have been using observed adjacency matrices for computing alignment estimate. However, each adjacency matrix represents just a single realization of random graph, which is subject to Bernoulli noise. To address this limitation, we propose using neighborhood smoothing technique to obtain the initial graphon estimates pint and Pins, which are shown to be consistent [65].
In this work, we allow flexibility in choosing between the standard GW distance and its entropic regularized variant EGW for computing the optimal transport plan between two graphs. The choice between GW and EGW depends computational considerations: For larger source networks with limited computational resources, we recommend using EGW; Otherwise, the standard GW formulation may be preferred for its sharper alignment results. Mathematically. in the GW setting, we compute the optimal transport plan à E [0, 1|s, Xn, by minimizing the following objective function: min Eijj,k. L(Pini k), pirni ()) Tyy For the EGW variant, we calculate similarly but add an entropic regularization term.
Ad Advantage of using initial graphon estimates for optimal transport. The advantage of using Pini and pini is clearly demonstrated in Figure We illustrate this with a simple example: 100-node target graph and 500-node source graph, both generated from the same base graphon function.

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We compare the alignment matrices calculated using the original noisy adjacency matrices A, and A,, the initial estimated probability matrices pink and Pi and the true underlying probability matrices P. and P.. As shown in Figure 2, the alignment based on raw adjacency matrices is noisy and poorly structured, deviating substantially from the ground- truth alignment. In contrast, the alignment obtained Figure 2: Visualization of alignment matrix. using the initial estimated probability matrices closely approximates the true alignment.
Normalization of optimal transport plan. Each entry ft. quantifies the strength of the learned correspondence between source node i and target node j. Nevertheless, as transport plan that distributes a total mass of 1 across n, X n, entries, the individual values of ftz can often be quite small. Thus, we apply column normalization to the transport plan. The resulting matrix, denoted by # E [0, 1)),xRi, is scaled such that each column sums to one.
Projection. Using the estimated alignment matrix, we transfer the source graphon estimator to the target domain: Perans # E.mps т. This matrix transformation acts as projection operator, effectively mapping the connectivity patterns encoded in the source probability matrix Pini onto the target domain's structure. It produces a weighted aggregation of source graphon values, where the weights reflect the learned node correspondences. To further refine the transfer, we apply neighborhood smoothing to the transferred graphon estimator, obtaining a refined version Perans? This additional smoothing step preserves the smoothness properties of the graphon and ensures that the transferred knowledge is not only structurally aligned but also smooth.
3.2.2 Debiasing Step
When source and target graphons differ significantly, negative transfer can occur. We quantify this domain difference using the GW between Pine and Pins. When the distance is smaller than the threshold (d < 8), we simply use the smoothed transferred estimator Ptrans2 as our final estimator. When this distance exceeds predetermined threshold (d > 8), we implement the following debiasing procedure to mitigate potential negative transfer effects.
Specifically, first compute residual matrix: R, Piii - This residual matrix captures (a) target-specific structural patterns not explained by the transferred estimator and (b) random noise due small sample size of the target graph. To keep the meaningful structure from part (a) but remove the noise from part (b), we then use the neighborhood smoothing method [65]. This method averages information from nearby nodes with similar connection patterns, which smooths out random fluctuations and keeps the stable, informative patterns. The result is a cleaner, smoothed residual estimator, Pres. Finally, we combine the transferred estimator with the smoothed residual to get our final graphon estimator for the target graph: P, Pirans2 + Pres Adding the smoothed residual back allows the model to recover unget-specific structural information that was missing from the transferred estimator, without reintroducing the random noise filtered out during smoothing.
Algorithm 1 summarizes our complete procedure, where the optimal transport can be calculated using either GW or its entropic variant EGW, and the choice depends primarily on the computational considerations.
4 Theoretical Properties
In this section, we present a stability analysis of the estimated alignment matrix obtained via minimizing the entropic Gromov- Wasserstein distance. Although EGW has been successfully applied in a variety of domains [53, 28), the question of the stability the optimizer x* with respect to perturbations in the cost or distance matrices (C, D) has received limited attention. Very recently, [66] first established duality theory for EGW, deriving optimal n 1/2 empirical convergence rates and proving stability with respect to the regularization parameter and [46] analyzed the a certain aspect of stability of the estimator, when both C' and D are euclidean distance matrices based on and observations respectively. These analyses do not address the case when (i) C and D are arbitrary cost matrices without the inner-produce structure of Euclidean distances, and (ii) the stability of the
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EGW estimator under perturbations of the cost functions, both of which are particularly relevant to our transfer learning framework. Recall that in the GTRANS algorithm, the estimated alignment matrix it is obtained using EGW with inputs C = Pini and = Pini As a result, these matrices cannot be interpreted as Euclidean distance matrices between point clouds, since they do not arise from pairwise distances in an inner-product space. is therefore natural to ask how far ft from the oracle alignment matrix n², which minimizes the same objective using the true connectivity matrices Pt. While the stability of convex optimization problems has been extensively
C' = P. and studied (see, e.g., [6] for overview of techniques and results). these tools do not directly apply to our setting due to the nonconvex nature of the EGW objective. Our subsequent theorem showcases that the distance between ât and n can be upper bounded by distance between Pini and P.. and Pini and Py:
Theorem 4.1. Let ñ and x¹ is the solution of EGW optimization problem with (C, D) (Pini P and (C, D) (Ps: P.) respectively. If the penalty parameter e satisfies: || ll C. P.
for some C₁ > 2, then,
(P 50 P.) - -(Piné
Pimé
a| T 1 SC₂ C2
[P & Pellop
as soon as ||P PPnn Ilx + APP--inij is less than certain cutoff (mentioned explicitly in the proof), for some universal constant C2 that depends on C₁.
We would like to highlight that our analysis is quite general as it does not rely on any structure of (P PI), which makes analysis different from [66, 46]. The lower bound the is needed to bring some local convexity structure of the problem near the oracle optimizer π². Such a lower bound condition for is typically adopted in literature, e.g., see [49].
The following result shows that the empirical GW distance between initial estimators Pigit and Pipit closely approximates the population GW distance between Ps and Pr, up to small estimation error. Theorem 4.2. Let Rne and ōn,T denote the estimation error of IP's and Pr respectively with respect to average squared Frobenious norm, Lee.
1
IP
P -P S S6ns)
21-6 IP
-Pr S 16n P
1 En
n
n1T
Define 5n = ong + 8wy Then, on the intersection of the above events, we have:
GW B1 Pv) < (GW2 Pinit
(P
Pipit + 5,)
GW Pimit) < (GW2 P.))
We next justify the theoretical validity of the alignment step. The following result shows that applying the optimal GW coupling preserves L2 proximity between the source and target graphons. Theorem 4.3. Let x² be the optimal coupling of (f.9) 3) for GW-distance, i.e.
J ,(v, 1))² (x, v) dn dr(
n" =rg mig
Define ğ (population analogue of # P as:
Ō(v.v)) = /
T 10.11
xf dx dx'
Then we have
||ğ g12 < GW2
As a consequence, when (W2(f.g) g) small, then ğ is close to g. which validates the alignment strategy.
Remark 4.4. The estimation error 8ng, 88g depends the estimation strategy and the underlying data- nauta-generatin process. For example, ofP, and Pr are generated from a o-Hölder graphon, then it is possible to construct P. and PT such that 8, x n. -aa/(a+1)) for 0 < a < 1 and ón x (log for Z 1 (e.g., see [19]) for {TT albeit they are NP-hard compute. Later, [65] proposed a neighhorhood smoothing based approach to obtain (Pss P.) with 8, x /log ni { nT).

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Experiments
5.1 Simulation Experiments
Simulation setup. To evaluate the effectiveness of our proposed method, GTRANS, we conduct simulation studies using ten representative graphon functions. Figure 3 illustrates the probability matrices P of 500 nodes generated for Graphons to 10, where the rows and the columns are ordered by latent position used in graphon functions used in ICE [45]. The details about the structure of these graphon functions can be found in Section G.I.
Graphons 1-5 are graphon functions used in SAS [9), and graphons 6-10 are
0.33
0.50
0.45
0.64
0.33
0.13
0.22
0.33
0.50
0.19
Figure 3: Heatmap of true probability matrix for graphon 1-10 (from left to right), where high values of probability are colored in red, and low values of probability are colored in blue. The number in each heatmap means the average connecting probability.
Baseline. To evaluate the effectiveness of our proposed method GTRANS, we compare its perfor- mance with four other existing graphon estimation methods which use target data only: neighborhood smoothing (NS) [65], sorting and smoothing (SAS) [9], universal singular value thresholding algo- rithm (USVT) [11], and iterative connecting probability estimation (ICE) [45]. We report results for our method implemented with both standard and entropic Gromov- Wassoo--Wassrteiin alignment, denoted as GTRANS--OE GW and GTRANS-EGW respectively. For all experiments, we evaluate performance using the MSE. We performed 50 independent simulation runs to calculate the averaged MSE.
In this simulation, we aim answer three key questions: (1) Impact of source sample size: How does the performance of GTRANS improve as the sample size in the source domain increases? (2) Impact of domain shift: How robust is GTRAAN against negative transferring when transferring between two different graphons? (3) Impact of density shift: How effective is GTRANS when transferring from a dense network to a sparse network or vice versa?
Asymptotic performance. To investigate the impact of source sample sizes, we vary the sample size of source data n, E {100, 200, ... 1000} while fixing other parameters: size of the target data n, 50. We introduce a small perturbation between source and target graphons: fe(u, v) = fu(u, v) + 5 where generated uniformly from (-0000 0.01). Figure 4 displays MSE results across ten different graphon types. Several key observations: (1) Both GTRANS-GW and GTRANS-EGW consistently outperforms all baseline methods across all graphon types, demonstrating the effectiveness of our transfer learning approach.
(2) We observe a clear decreasing trend in MSE as source sample size increases for both of GTRANS-GW and GTRANS-EGW: while other target only methods (including NS, USVT, ICE, SAS) show flat pattern because they can not borrow strength from source sample. (3) GTRANS-GW and GTRANS-EGW achieve comparable performance overall, though in some cases, the entropic regularization in GTRANS-EGW yields lower MSE, benefiting from entropic regularization to enhance alignment stability. Figure 5 further illustrates our method's effectiveness (using GW for demonstration) when transferring from a 500-node source to 50-node target network (both from graphon 8). Panel (a) displays the noisy target-only estimation. Panel (b) shows the accurate source estimation (left), significantly improved transferred result (middle), and final estimation (right). Since the source and target domains are identical, the optimal transport distance is small, and the final estimate equals the transferred result without requiring debiasing.
Transfer between different graphons. To assess the robustness of GTrans to domain differences. we fix n, = 500 and n. 50, and consider transfer learning between different graphon pairs. We consider transfer between similar graphon pairs (such 1 and 3, or 6 and dissimilar graphon pairs (such as 8 and 9, or 5 and 10). Table 1 presents the MSE results for these transfer scenarios compared to target-only methods. We observe several key patterns in the results: (1) For similar graphon pairs, both GTRANS- GW and GTRANS-EGW achieves substantial improvement over target-only methods. This demonstrates that when graphons share structural characteristics, knowledge transfer remains highly effective even between different graphon functions. Figure 5(c) provides visual evidence of effective knowledge transfer the (from graphon to 8) scenario. Both graphons share smooth gradient-basse based transition patterns. The middle panel in (c) shows that the
and 8), as well as
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0.44
002
2016
6015
0.00
0.000
Sumpll
Souroe Surpple Car
Starde Eumppe Buse
Stande Sampll Sice
Scurce Sample
0.00
0.020
0.05
0.000
0.04
0.11
0.00
0.015
0.010
Sourte Sample Blto
Source Bample 8.00
Sturre Earppe Bize
Source Sersçl Bioa
Scurce Sample Bioe
GTrans-EGW
GTrans-GW
NS
USVT
++-
ICE
SAS
Figure 4: MSE performance of five methods as source network size increases from 100-1000 nodes, with error bars representing #0.1 standard deviations. GTRANS-GW (red circles with solid line), GTRANS-EGW (pink circles with dashed line). NS (blue squares with solid line), USVT (yellow triangles with solid line), ICE (green diamonds with solid line), SAS (gray hollow squares with solid line). Top row: graphons 1-5; bottom row: graphons 6-10.
(a) Target only
(b) From graphon to 8
(c) From graphon 7
Target initial
Source Initial
Transferred
estimatrr
Source initial
Transferred
Final astimastt
Figure 5: Visual comparison of GTRANS performance with different source graphons when estimating target graphon (a) target-only initial estimator using only limited target data; (b-c) three panels showing source initial estimator, transferred estimator, and final estimator from different source graphons (8 > 8,7 8 respectively).
transferred estimator successfully preserves the source's smooth structural regularity while adapting to the target's diagonal gradient, evident from the emergence of grid-like texture where intensity gradually decreases diagonally. The final estimator in the right panel of (c) exhibits both the structural smoothness inherited from the source and more precise alignment with the target graphon's 's latent geometry, highlighting the advantage of debiasing step. (2) For transfers between graphon pairs with larger difference, GTRANS maintains similar performance compared its target-only counterpart NS method. This robustness is due to our adaptive debiasing mechanism, which effectively mitigates negative transfer while preserving beneficial knowledge.
Impact of density shift. To investigate the performance of GTRANS to density differences between source and target domains, we examine see- narios with varying levels of perturbation berween source and target graphons:
Grapphon density:
Graphon 0.15
0.18
MSE
0.016
MGE
fu, v) = Se(u, +
where E
UO, if
0.009
>
0, E ~ U(X,0) 0) if
0.50
0.00
0.25
0.50
Lambda
< We vary from and truncate the probability if is larger than I smaller than O. This allows us to simulate scenarios where target network is either sparser (A 0) denser
1-0.5.00.44
..., 0.5), .0.45......55)
GTrans-EGN
GTrans- GW
NS
USVT
ICE
SAS
Figure 6: MSE comparison of GTRANS-GW, GTRANS-EGW, NS. ICE, USVT, and SAS under density shifts À for Graphon (0.5) and Graphon 6 (0.13), with error bars representing +0.1 standard deviations. Bottom panels show adjacency matrix transformations -0.50, -0.55, 0.00, 0.25, 0.50.

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(A < () than the source. Figure 6 presents MSE results for two representative graphons (ID 2 and ID 6). Additional results can be found in Figure E1. For the dense Graphon 2 (density 0.5). GTRANS-Gi TTRAN--GW and GTRANS-EGW both show a symmetric U-shaped pattern centered almost around zero, indicating that performance degrades equally whether transferring from a denser or sparser source. In contrast, Graphon 6 is a sparse network (density = 0.13). Here, the performance curve of GTRANS-GW and GTRANS-EGW achieve its lowest error at a positive value of X. This indicates that when the target graph is dense, the impact of density shift on transfer learning is symmetric around zero, However, when the target graph is sparse, transfer performance becomes more sensitive to the direction of the density shift, and transferring from slightly denser source network yields better results. The underlying reason might be: (1) a denser source network typically exhibits a higher hignal-to-nois ratio, yielding a higher-quality initial graphon estimator.
(2) When the source network is sparse, Pint will also be sparse potentially to an extreme degree where it might even be (approximately) zero matrix. This severely limits the information available for transfer, regardless of how well the transport plan aligns the domains.
Table 1: MSE comparison for transfer learning between different graphons (n, = 500, n2, = 50)
Similarity Scenario
GTRANS-EGW GTRANS-GW
NS
USVT
ICE
SAS
7 + 6
0.9 + 0.2
1.0 + 0.3
2.1 0.3
3.0 0.9
1.7
+ 0.3
1.6 0.4
6 +
1.8 + 0.3
1.8 + 0.3
2.3
2.7
2.2
122+0.3 0.3
2.4
Similar
8
1.3 1 0.3
2.3
2.7
2.2 0.3
2.6
++
1.6 1
1.6
2.0=0.2
1.2
2.1
2.8
3 +
1.6 1 0.5
1.7
-.3
5.1
2.7
2.1
1 + 3
1.6 16+0. 0.3
0.3
0.3
1,2
2.2
0.3
1.6
8
1.9
0.9
2.0 0.2
2.6
1.7
1
0.2
2.1
Different
+
2.2
1.2
2.0 0.2
1.5 0.2
2.1
1
0.2
28
10 5 5
1.6
1.5
1.7 0.1
1.5
1.7
0.2
5.5 1.0
5 + 10
2.6
2.5 1 0.3
3.1 0.6 5.6
2.8
0.3
2.6 = 0.3
Selection of Hyperparameters. GTRANS-GW involves one hyperparameter: threshold ō while GTRANS-GW involves two hyperparameters: threshold 5 and regularization parameters e. In this paper, employed a network cross- validavaliotionn procedure developed by [33] for hyperparameter tuning. (1) Selection of 6: Specifically, when implementing GTRRANS with either GW or EGW, we choose from a candidate set {0.1, 0.11, 0.12 0.5), and use the cross-validation procedure to select the value that minimizes held-out prediction error. Figures E3a and E3b illustrate the optimal thresholds identified across different simulation scenarios for GW and EGW, respectively. As we can see: most GW scenarios work best when 8 = 0.15, while most EGW scenarios work best when 8 0.18. While volssavali is preferred, we recommend default values of 8 0.15 for GW and 5 = 0.18 for EGW when tuning is computationally infeasible in practice. (2) Selection of e: When implementing EGW, we select the optimal regularization parameter from candidates (0.001, 0.005, O.01..... 0.1). Figures E4 illustrates the MSE comparison across different regularization parameters e, highlighting that e 0.01 is consistently effective for most graphons.
5.2 Application to Real Data
Evaluating probability matrix estimation methods on real networks directly is difficult, since the true probability matrix is unknown. We assess the practical utility of our method by applying it to two downstream applications: graph classification and link prediction. Due to page limitations, we only show results for graph classification. Link prediction results provided in Appendix F.2.
Graph classification is fundamental to network analysis, but often suffers from limited labeled data. [22] proposed G-Mixup, which augments datasets by interpolating between graphons of different classes to generate synthetic training graphs. The quality of this augmentation critically depends on the accuracy of the underlying graphon estimation. When the network size is small, conventional graphon estimators often produce poor results. For example, IMDB-B and IMDB-M datasets used in [22] have only 19.77 and 13.00 average nodes per graph, respectively.
Datasets. To address this challenge, we implemented GTRANS to enhance G-Mixup for graph classification by transferring knowledge from larger networks. In our experiments, we consider two co-actor graph datasets as targets: two-class IMDB-BINARE and three-class IMDB-MULTI, both characterized by small graph sizes. As candidate sources, consider: (I) three-class COLLAB

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(average 74.49 nodes per graph), collaboration network derived from scientific authorship data [39]. and (2) two-class Bedditt (average 429 nodes per graph), comprising Reddit user interaction threads [39]. Additionally, we examine a bioinformatics setting by transferring from two-class D&D (average 284.32 nodes) to two-class PROTEINS-FuL (average 25.22 nodes) [39], both consisting of protein structure graphs.
Results. We adopt the same Graph Convolutional Network (GCN) architecture as used in [22], using the same hyperparameterss and training procedures for all benchmark comparisons. Full implementation details are provided in Appendix F. For each target dataset, we split the dataset into train/validdttion/ees data by 70% /10% We report the test accuracy on ten runs. Tables 2 report test accuracies for our proposed GTRANS method against baselines, including NS [65]. USVT [11], SAS [91, ICE [45], GWB [57], IGNR [55]. SIGL [4]. and graphlet [44, 48]. For GWB/IGNR/SIL we follow the original implementations and keep hyperparameters unchanged. The graphlet baseline is implemented via GraKell's Graphlet Kernel with an SVM on the precomputed kernel matrix; we use graphlet sampling with 700 subgraphs per graph for scalability. On IMDB-Binary GTRANS- GW achieves 76.30% (transfer from Reddit-Binary) and 76.25% (transfer from COLLAB), while GTRANS-EGW further improves to 76. 80% and 77.50%, respectively, outperforming all baselines by 2-5%. On IMDB-Multi, GTRANS-GW with COLLAB reaches 50.47%, and GTRANS-EGWV achieves 51.27% with Reddit-Binary -anniit--rrrryaa as the source, demonstrating clear superiority over all other methods. For PROTEINS-FUS Full. transferring from D&D yields 69.33% with GTRANS-Gr and 68.52% with GTRAAN--GG' EGW, consistently outperforming baseline methods by 3-6 points. These results confirm that both GTRANS-GW and GTRANS-EGW effectively boost classification accuracy by leveraging structural knowledge from larger networks.
Table 2: Graph classification accuracy (%) across three target datasets (mean + std) compared against graphon estimation baselines.
Source
Target
GTRANS-GW
GTRANS-EGW
NS
USVT
SAS
Reddit-B
IMDB-B B
76.30 + 2.35
76.80 + 1.52
72.90 = 2.10
73.85 = 2.40
74.25
1.93
COLLAB
IMDB-B
76,25 + 2.06
77.50 + 2.13
72.90 = 2.10
73.85 = 2.40
74.25
1.93
Reddit-B
IMDB-M
49.10 + 1.33
51.27 = 1.98
43,80 2.59
48.00 = 2.93
44.10
+ 2.05
COLLAB
IMDB-M
50.47 + 1.42
50.23 + 0.92
43,80 = 2.59
48.00 = 2.93
44.10 + 2.05
D&D
PROTEINS
69.33 + 2.55
68.52 + 1.59
63.18 = 1.94
65.11 = 2.21
65.25 + 1.85
Source
Target
ICE
GWB
IGNR
SIGL
Graphlet
Reddit-B
IMDB-B
74.30 + 2.16
75.30 + 2.19
74.35
+ 3.51
73.50 + 2.62
61.10 + 2.23
COLLAB
IMDB-B
74.30
+ 2.16
75.30
+ 2.19
74.35
+ 3.51
73,50 + 2.62
61.10 + 2.23
Reddit-B
IMDB-M
43,90
1.27
47.70
+ 2.53
47.50
3.32
49.13 + 3.04
39.37 + 1.62
COLLAB
IMDB-M
43.90
1:00-110 1.27
47.70
+ 2.53
47.50
3.32
49.13
+ 3.04
39.37 + 1.62
D&D
PROTEINS
65.38 1.84
63.45
+ 2.96
2.56 65.87
2.34 67.13
70.11 + 2.12
6 Conclusion and Future Work
In this paper, proposed GTRANS, novel transfer learning framework estimate the edge connecting probability, addressing the challenge of limited data availability in the target network. By leveraging
optimal transport, our method aligns latent node structures across source and target domains and effectively transfers graphon information to improve estimation accuracy. Extensive experiments both synthetic and real-world networks demonstrate that our approach consistently outperforms existing baselines, particularly in scenarios involving small, sparse graphs. By improving estimation accuracy, our framework facilitates various downstream tasks, including data augmentation and link prediction. Despite these promising results, several limitations remain. First, our current framework assumes single source graph. Extending to incorporate multiple source networks could further enhance robustness by leveraging diverse structural priors. Second, GTRANS is designed for static graphs; future extensions to dynamic or multi-layer networks would enable modeling of time-evolving or multi-modia dependencies. Third, our current formulation does not incorporate node-level covariates. Integrating such covariates could provide valuable auxiliary information for alignment, particularly in domains where topological structure alone may be insufficient for effective transfer. These directions can further enhance the flexibility and generalizability of transfer-based graphon estimation.
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Appendix for "Transfer Learning on Edge Connecting Probability Estimation Under Graphon Model"
Yuyao Wang Boston University yuyaow@bu.. edu
Yu-Hung Cheng Boston University yhcheng@bu. edu
Debarghya Mukherjee Boston University mdeb@bu.c edu
Huimin Cheng
Boston University huimin230bu. edu
A
Notation Table
Table Al: Notations used throughout the paper
Notation
Description
ns nr
Number of nodes the source and target graphs
As, A
Adjacency matrices of source and target graphs, A E {0, 1]mxn
P Pt
True connection probability matrices for source and target graphs
Pini Pini
Initial estimators via ncighborhood smoothing Transferred estimator before and after smoothing Smoothed residual graphon estimator
Prans Ptrans2
Pres
Pe
Final graphon estimator for the target domain Estimated connecting probability between node i and j
Pij
Ber(p)
Bernoulli distribution with parameter p
Ss. Ft
Latent graphon functions for source and target domains Latent position of node i in [0, 1]
Ua,i U4
I1 v)
Set of couplings (transport plans) between μ and The optimal transport plan estimated by Gromov- Wasserstein Normalized transport plan
A E [0,
x
# E [0, n0 xne
8
Threshold for triggering the debiasing step Entropic regularization parameter in Gromov-Wasserstee Wasserstein optimization
Density shift parameter for source-target perturbation.
KL(T p v)
Kullback-Leible divergence: KL(T 1 v) = Eis Tiy log pmv
P. & P
Kronecker product of P, and P:
Frobenius norm: X F v X
g||2 = (1o' So' (f(u, g(u, u))² du du
1/2
Le norm:
||ʃ
Infinity norm: | 30 maxi,j Xisl
30
op
Operator norm: X op Tmax ()
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

B Proof of Theorem 4.1
In this proof, we will the notation (x; to denote ball of radius around æ; with respect some appropriate distance, which will be clear from the context. Before delving into the proof, let us first recall the definition of strong convexity:
Definition B.1. A function f is said to be strongly convex in a neighborhood around 2, (namely B( if satisfies:
f(v) 2 f(z) + tv (v- Vf(z)) 1 2 fly x
Our proof relies on an application of the following lemma:
Lemma B.2. Suppose is comvex function which is minimized at 2, and furthermore it is pa- strongly convex on (xx;;T) {æ || z. x=11e 112 T}. Consider a perturbation g of f such that Ilf - glloo I 5 and ii) is Lipschitz with Lipschitz-constnnt K. If ò S. UT² we have:
2k 2 - 9 g||Lip
|1.. --
S
1
Proof. We divide the proof of Lemma B.2 into two smaller lemmas. The first lemma is follows:
Lemma B.3. Consider two fanctions and g, such that J - is k-Lipschitz Suppose I. and y. are minimizer of I and respectively. If is strongly convex on B(x,, T) and E B(xa;T), then
2x
||.. - yoll2
Proof. The proof essentially follows from Proposition 4.32 of [6]. Here, we sketch the proof for the ease of the readers. As y- the minimizer g. we have:
f(v.) f(x.) = (f - -9)(u-) (S (x)(xx) g(v.) g(x.)
$ (f - ()())) - -f - g)(x)) S <l|I= y lll. On the other, from the strong convexity of B(x,;T) and the assumption that ye B(xaa:)) have:
+ 11 | - v.112
f(y.) 2 f (x..) where we the fact that vs(x.) Combining these two equations, we conclude:
2
54 -11--
2x
1 v-112 < f(v)) - f(x.) I <x|lx, Voll2 = || -/-l2 S
This completes the proof.
One of the requirements of Lemma B.3 is that ys E B(x,, r). The following lemma gives a sufficient condition for this condition be satisfied:
Lemma B.4. Assume f is p-strongly convex on B(z,;T). If - glooo s 8, 8 < TH2 then the minimizer of g also lies in B(x,;T).
Proof. Suppose ys & B)x Then ||,, - y.ll > т. Therefore, (0,1) such that z(() = tæ, + (1 t)y, lies the boundary, i.e., ||z, xzt)ll2 By the strong convexity of f B(x+; T), we have:
f(x(1) 2 f(x.) + 2
On the other hand:
f(x(t)) s g(x(1)) + 8 tg(xx) ) + (1 11- t)g((=)) + + g(xx) +
Hence, can conclude that:
g(x.) f(x.) ) + 1u7
-
2
This immediate contradicts the fact that (g(x.)) f(x.) proof.
S 8 as 72 > 48/p. This completes the
D
23

---

Finally, the claim in Lemma B.2 is established by combining the arguments of B.3 and Lemma B.4.
We use Lemma B.2 to complete the proof of Theorem 4.1. With = (1/ng: ... 1/n,) and (1/ne,-.. 1/me) The oracla EGW optimization problem (with respect to Ps. Pt) can be
written as:
((T)
Σ
(P
Paik PIjI +
Tíj log (n Tij))
2
kjkel
ij
1
2
P
8,sk
Tiy TAl
+ p
2
Tf
Thl
Σ ijkl
P
P..skPe,h P. te Σ Tij log + +og (nona)e
E Te5
kyke
8jkl
P. + Pell. Pe +
Σ
mis log Taj + log (n,7)..
2
ijkl
immediate that the first, second, and fourth terms do not contribute to the estimation of Ignoring them, we redefine the oracle loss function as:
f(T) EE P. THI + Σ Tij log may 2tr( P.TP,) + Σ
Tij log Taj
Similary, we define g(-) as the sample version (with respect to (Pinit, ,)nitt::
g( = Pe.ak Pa Pes + E Σ
Tsj log = 22r((T T P.TP,) + Σ
Tsy log Tej
kjkl
1j
13
For notation simplicity define ||p Pollo0,o and 82 ||p -Pellooc,o
We have:
f(() .(T) tr( P. P.TP..) -r((m P. P.TP.))
Σ is
Tij P.TPP. P. P
i5
P Pam P.
P P
< max ij
Tij
max
P. PT
P.TPT P.P.
ij
ij
P.TP. P.TP. kj
P P.TP.. -P.TP.
< max ij
max i.
ij
P P..T) - -P)gkk
+ max [.j
I(P, (P. P.)x (mP. ks
i.j
< |P; - Pilloo,00 max (P. + |PP
Ps Palloo,oo max
(TT Deyl
S 6g max
(P ( + 6g max Σ [TTP] Dayl
< 82 max Σ Σ P..aTTu + di max Σ Σ TRP Pa.il
< 52 P 00 Σ
Thi + 81 |PPlloo Σ
T&i = 8z P. Pol + 81 |PPolooo 81 + 82.
k!
kl
Therefore, we conclude that
|| - ol
P. - -P
+IPP OP.-Pr Pilloc,00 := Apert
24

---

Now, from the definition of f and 9, we also have:
Vxf(m) (P. @ P. + P
P.
vec( + c((1 + log Ttz 1i.
Vxg(T) [ P P.
P vec(a) <((1 + log Ttj hhd
1
(TT(T) (P. + P
P.
ediag
Ttj 1
(T (P. P
P. P. + cdiag
Ttj }
As consequence, we have:
(9 f)(=ll s (P. &P) S
(P. PP))
||T|E
Hence, we have:
K = x @ P (P. & P.)
Now, from the Hessian, know is met as per our assumption that
(²S(T*) if max, m.j < < Prllop). This condition
|| 00
Ci Ci| @ Pellop
with C > 2. Fix C2 such that for any || π F S T,
< (1/2), and T = e & Prllop). Then,
max Myy T + max Tij -
C2 P. X Pellop
Ci @ Prllop
P & Prllop
In this neighborhood, have:
)/ -
1 |P. =
inf inf (2)(T)
P
p.lpop =
(( |el|a==
c
C3
Therefore, using Lemma B.2 we can conclude:
20 PP
P.)
(P. P.)
T2
|| -
- >
as soon
pert S
IP
X P lop
4
By our definition of and 14, we have:
T2p
P.OP
P. @ Pr C3
4
Recall that our assumption is:
|P, - -P + - 5 J2p
A pert
4
Substituting the expression for r2p:
2
Apai S 2C3Cs(IIP, & Prllop)
This guarantees that the perturbed solution म remains in the neighborhood B(T", r) of the true solution n². Thus, the stability of the optimal transport plan ensured under perturbations of the graphon estimators, completing the proof.
25

---

C Proof of Theorem 4.2
Proof. The proof simple, which depends on the estimator error of Ps and Pt. Recall that:
155
WT
GW
Pluit)
inf
Pinit S.i
P P Tiy
s
me
1,2'
j
GW Pt) = inf
Psae Pros 1 Tay
TE x
1,2f
33
where In is the set of couplings, i.e.
1
r,
Σ mej
nT
Now let ân and n. denotes the minimizer/optimal coupling of GW3(Pinit Pinit) and GW2 P.) respectively. Then we have:
GP P.)
ng
DT
inf Σ
Psw Pr
rel'n :-=='j===1
ng
R
Σ
Ps P ( Pnnns
(,1=1 ++==1
Σ
Ps
Pinit + S,8
Pinit
Pinit
T33 +
Pinit
T55 PT. P fnniittnn
f111
1='=1
<
Σ
P Pinit S
P Ps. 2 "n
1-==11==1
Ds
+
Pinit 8.
Pinit
22 Toj
Pray Pinit Pnsssttnn
1,7 3=1 =
iinit S.st
- -s 1 +
Σ
Pinit
2 Pnm
Prar Pinit T.3
S.
T33
1,1==1
1,1=1
=1
j='=1
< (Ing (Piiit pinit) + SNT) 4 (6n (Piii p (
26

---

Similarly, for the other side:
GW Pinit
P
RT
inf
Pinit
Pinit
T.ji T
Tey
S,
=elnn
n.
Pinit
Pinit 12 Tn,
S
2:55
4,4= jd'=1
Pinit
Ps + Ps -P + P P T338 1 PnnifTnn
S
1,1== j='=1
As
W
<
Σ
P
init S
P Ps P &n
j+'==
72
12.
Ps Pr Tn
Pras
Pinit Tai
1,===ju²== 1
ff'=1
n2
Pinit SaF
Ps +
Σ
PsN Pr. Proy Paany
Pror
Pipit T33
1==1
1,'=1 111'=
3='=1
(8ng + GW3 (P., P.) + Sur) (6, GWP(P.,PP.))
D Proof of Theorem 4.3
Proof. We start with the definition of the GW distance:
GW / (f(x,2')- - v1)2 du dx (r, dr (z',v/)
/ (lx,z') dr de' 9² (v. dy dy' -2 2/ ʃ f(x, 2')gy. d') v) dx dz'(2', (2'.8)
/ Pla,z') do de' + 1 ((( dy dy' - / [ (ay.s")(lu. v/) dy dyf
Furthermore, we have:
2 I ((..')) dy dy / / (1 xx(,,))(a, af)" (a', v) dx dr'
dydy
/ (ax,2') (U,,z)) (x' v) dxdx dydy'
/ Pla, da da'
Therefore, we have:
||ğ -g133 [ /O.P (0(v.v') g(u. dydy'
-ʃ (l( dydy + (w. dydy/ f Dol. af) dydy
/ P(a,d)) drde' + Plu.d') dydy' - 2 Max, ('),,))))( v/) dr (x. vdd ") (f.g). This completes the proof.
27

---

E
Additional Simulation Results
E.1
Additional results for Impact Density Shift
To evaluate the robustmess of GTRANS under density discrepancies between the source and target
domains, we introduce a density shift parameter À E |-0.5, 0.5] to simulate structured perturbations. defined as fu, [u, v) = fu( v) + Ç. where & ~ ( A) if X > 0 and & ~ U( 0) if X < 0. We investigate the performance of GTRANS--ON functions (IDs: 1 to 10) with a fixed target sample size of 50 and source sample size of 200. Each experiment is repeated 50 times to evaluate the adaptability of the models under varying levels of density shift. The results are summarized in Figure E1.
and GTRANS-EGW across 10 different graphon
0.05
0.00
0o10
2230
0.04
M86
0015
3.516
0.310
8:0
000
0/0
0.00
6,09
0.00
0.20
0.50
2.2
0.00
0.29
835
32:0
333
020
0.33
123
0.00
850
0.29
0.90
0.29
Lanooda
Lambda
Larbol
0.05
6.00
0216
0.00
0.0
0.25
0.00
025
0.0
0.30
0.20
0.00
0.25
0.50
40:00
-223
0.00
0.25
850
-353
025
0.33
3.3
0.50
-0.00
03.25
0.30
0.25
0.50
GTrans-EGW EGW
GTrans-GW
NS
USVT
ICE
SAS
Figure E1: MSE performance of five methods as the density shift parameter À varies from -0.5 to 0.5, with error bars representing 10.1 standard deviations. GTRANS-GW (red circles with solid line), GTRANS-EGW (pink circles with dashed line), NS (blue squares with solid line), USVT (yellow triangles with solid line), ICE (green diamonds with solid line), SAS (gray hollow squares with solid line). Top row: graphons 1-5; bottom row: graphons 5-10.
Figure E1 shows the average MSE of different methods as the density shift parameter varies from -0.5 to 0.5. Both GTRANS-GW (red solid line) and GTRANS-EGW (pink dashed line) demonstrate substantial improvements over the baseline methods (NS, USVT, ICE, and SAS), which remain constant across all À values. Notably, À shifts from negative to positive, GTRANS methods effectively adjust their alignment, capturing structural changes and minimizing MSE. This result underscores the capability of GTRANS-Gr -W and GTRANS-EGW to dynamically adapt to source- target discrepancies, achieving consistently lower MSE across all scenarios. Also, we notice that for dense graphons like Graphon 2 (density = 0.5) and Graphon 9 (density 0.5). GTRANS-GW and GTRANS-EGW exhibit symmetric U-shaped MSE curves around zero indicating balanced performance regardless of source density. In contrast, for sparse graphons like Graphon 6 (density= 0.13) and Graphon 10 (density = 0.19). the lowest MSE is achieved with slightly denser source, highlighting the need for stronger initial signals when the target sparse. Unique structural patterns in Graphon 5 (anti-diagonal) and Graphon 7 (oscillatory) cause non-monotonic fluctuations under density shift, reflecting sensitivity to structural misalignments.
E.2 Additional Simulation for Transfer Between Different Graphons
We conduct simulations across ten distinct graphon structures, indexed from 1 10. For each graphon, we generate a target domain with fixed sample size of = 50 and source domain with n, = 500, simulating realistic structural perturbations by adding uniform noise sampled from [-0.01, 0.01]. The evaluation includes USVT, ICE, and SAS, and our proposed transfer learning frameworks: GTRANS-GW and GTRANS-EGW (e = 0.01). For comprehensive analysis, we compute the Mean Squared Error (MSE) for each method and aggregate the results.
28

---

Table E2: Comparison of different methods for various transfer scenarios. Best-performing methods are bolded.
Scenario
GTRANS-GW
GTRANS-EGW
NS
USVT
ICE
SAS
1 + 2
1.4 0.3
1.3 + 0.3
2.1
1.5 + 0.3
2.1 + 0.2
1,4 + 0.4
+ 3
1.6 16-00 0.3
1,5 + 0.3
0.3
1.2 1 0.2
2.2 0.3
1.6 1 0.3
+
1.3 0.2
2.1
1.0 0.1
2.2
1.1 - 0.2
1 - 5
1.2
1,5 0.2
1.7
3.5 1.5
1.2
5.5
1 + 6
1.0
1.1
2.3
3.0
1.6 1.7
+ 7 1 + 8
1.4
0.9 0.4
0.3
5.7
2.
2.6
1.5
1,4 0.3
2.0
1.5
2.1
2.8
+ 9
1.8
1,7 0.3
2.0
2.6
1.7
2.1
+ 10
2.3
2.20..
3.6
5.6
2.7
2.
2 +
1.2
1.2
0.
5.1
2.7
2.1
2 3
0.3
1.6
0.3
122
2.2
1.6
2 44
0.2
0.2
2.2
1.0
2.2
1.2
2 4
1.2
1,5 0.2
1.7
1.5
1.7
1.0
4
1.0
1. 0.2
2.1
3.0
1.7
1.6
2 7
1.6
1.6
0.3
5.7
2.3
0.6
2 +
1.2
1,5 0.2
2.2
1.5
2.2
2.7
2 9
0.5
2.2
2.6
1.7
2.10..1
2 + 10
2.2
2.2
0.6
0.6
0.3
0.6
3 +
1.6 0.5
1,7 0.2
0.3
5.6
2.7
2.1
3
1.3
1,403
2.2
1.3
21
1.404
3 +
0.2
1.3
2.2
1.0
23
1.1
3 3
1.4
1.2
1.1
1.5
1.7
5.5
3 46
0.9
1.2
2.1
3.9
1.7
1.6
3 +
1.5
1.5
0.3
5.9
2.2
2.6
3 8
0.3
1.4
2.0
1.2
2.1
2.
3 +
1.9
0.3
0.2
2.6
1.7
2.1
3 + 10
0.3
0.3
0.6
0.6
2.7
2.6
4-1
1.7
1.7
0.3
5.1
2.7
2.1
4 +
1.3
0.3
0.1
0.3
2.1
1.4
4 +
0.4
0.3
0.3
1.2
0.3
1.6
1.2
132
1.7
3.5
1.7
5.5
4 + 6
1.0
1.2
2.1
0.9
1.7
1.6
4 +
1.5
1.2
2.3
5.9
2.2
2.4
4 8
1.4
1.4
2.2
1.2
2.1
2.8
4 +
1.3
1.3
2.2
2.1
1.7
2.1
+ 10
0.3
2.3
0.6
0.6
2.3
2.
5 1
0.3
0.3
2.3
4.6
2.6
2.
5 3-2
1.7
1.6
2.1
1.5
2.1
1.4044
3
1.7
1.3
2.3
2.
1.6
5
1.4
1.2
2.1
1.0
0.2
1.2
5 + 6
1.2
1.2 0.2
0.3
3.9
1.7
1.600.
5 ++7
1.3
1.3
2.3
5.9
2.2
0.6
5
1.5
1.2
2.2
1.5
2.1
2.7
5 3 9
2.3
2. 0.3
2.2
2.1
1.
2.10.11
5 10
0.3
2.5 0.3
0.6
5.3
2.3
2.
6 + 1
2.
2.40.3
5.1 0.3
0.7
2.1
6 +
1.3
1.3 1.3 0.3
2.1
1.5 0.3
2.1
1.4
6 3
0.3
1.5 1 0.3
2.3
1.2 1 0.2
0.
1.6
6 + 4
1.2
1.2 0.2
2.1
10 0.1
2.
1.
5
0.2
1.5 0.2
1.7 - 0.1
3.5 1.5
1.7 - 0.2
5.5 + 1.0

---

Scenario
GTRANS-GW
GTRANS-EGW
NS
USVT
ICE
SAS
6 ++
1.8 + 0.3
1,8 + 0.3
2.3 0.3
5.9 2.7
2.2 2.2 0.3
2.4 + 0.6
+
1.6 0.2
1,5 - 0.2
2.2
1,5 0.2
1,5
2.1 2.1
2.8 + 0.7
6 9
1,5 0.4
1.6
2.6 2.0
2.1 1.7
6 + 10
2.3
2.3 0.3
3.6
0.6
2.7
2.3
7 +
0.3
1,9 0.3
2.3
5.1
2.7
2.1
7 + 2
1.3
1
2.1
1.3
2.1
1.4
7
1.6
1.3
2.3
1.2
2.
1.6
7
1.2
0.2
2.1
1.0
2.
1.102.
7 + 5
1.6
1.2
1.7
3.5
1.2
5.5
7
0.9
0.3
2.3
0.9
1.7
1.6
7 8
0.2
0.2
2.2
1.1002
2.1
2.7
7 + 9
1.4
1.7
2.0
2.6
1.7
2.1
+
0.4
2.3
3.6
5.6
2.3
2.
8 1
0.3
1.5
2.3
4.6
2.3
2.3
8 + 2
1.4
1.3
2.1
1.3
2.1
1.4
8 +
0.3
1.3
2.3
1.2
2.2
1.6
8 + 4
0.2
0.2
2.1
2.
1.1
8 + 5
1.2
1.5
1.7
1.5
1.7
1.0
8 +
1.2
1.0
2.1
3.0
1.7
1.6
8 + 7
1.3
1,4
0.3
5.9
0.
2.6
8 + 9
1.9
1.3
2.2
2.1
1.7
2.1
8 + 10
0.3
2.3
0.6
0.6
2.7
0.3
9 + 1
0.3
0.2
0.3
4.1
2.3
2.
9 +
0.3
1.3
2.2
1.5
2.1
1.4
9 3
1.9
0.3
0.3
1.2
0.
1.6
9 + 4
1.6
1.2
2.2
1.0
2.3
1.1
9 +
1.2
1.3
0.1
3.5
1.2
5.5
9 6
1.2
1.2
2.1
0.9
1.7
1.6
9 + 7
1.2
0.3
0.3
5.9
2.3
0.6
2.2
1.2
2.2
1.5
2.1
2.7
9 - 10
2.3
0.3
0.6
0.6
0.7
0.3
10 > 1
2.1
2.0 0.3
0.3
5.1
2.7
2.1
10 2
0.3
1.200
2.1
1.3
2.1
1.4
10 + 3
0.6
0.3
0.3
122
0.3
1.6
10 + 4
1.2
1.2
0.2
1.1
2.3
1.1
10 5
1.2
1.5 0.2
1.7
1.5
1.7
5.5 + 1.0
10 + 6
1.3
1
2.1
3.0
1.7
1.6 - 0.4
10 + 7
1.9
1.8 0.3
5.9
2.2 0.3
2.4 144000 0.6
10 8
1.6
1.5 0.2
2.0
1.5 - 0.2
2.1
2.8 0.7
10 9
1.5 +.3
1.5
2.0
2.6 0.1
1.7
2.1 + 0.1
E.3 Ablation Study
Figure E2 presents the average estimation errors (MSE) four configurations: GTRANS (red dia- mond with solid line), GTRANS-NonDebias (purple circle with dotted line), GTRANS-NonSmoodd (orange triangle with dashed line), and GTRANS-Adj (yellow square with dash-dotted line). The x-axis represents the source sample size, and the y-axis shows the average MSE over 50 runs.
The four configurations are derived as follows: GTRANS: This the full version, including all three steps from Algorithm 1. GTRANS-NonDebias This variant removes the debiasing step (Step 3), directly using the transferred estimator ptrans? as the final output. GTRANS-NonSmooth This variant omits the neighborhood smoothing steps in the initial estimation (Step 1). GTRANS-Adj: This variant replaces the neighborhood smoothing (Step 1) entirely with raw adjacency matrices A, and A, In this setting, the optimal transport step (Step 2) operates directly on the unprocessed adjacency representations.
30

---

From the results, we observe that the complete GTRANS consistently outperforms its ablated variants across most graphon structures. The absence of smoothing (GTRANS-NonSmoott) introduces sharp oscillations in MSE, indicating the sensitivity to noise. G'TRANS-NonDebia struggles to correct. for transfer-induced discrepancies, especially when source-target alignment imperfect. GTTRANS- Adj shows consistently higher MSE, highlighting that raw adjacency matrices lack smoothness for effective GW-based alignment. This underscores the necessity the smoothing step in GTRANS to achieve consistent and robust estimations. Therefore, the debiasing and smoothing mechanisms in GTRANS are not just additive; they are essential for robustness.
0.00
114
0.05
0.55
2.22
3.2
0.59
0.00
0.31
0.01
2.40
A031
1200
254
15.0
1000
237
569
10.00
Stzo
2.0399
3.032
MSE
0.000
MSE
520
2.0390
2.0000
583
Sample Stoe
Sample
Sample
GTrane-Nord STranieNonDebise
GTrane-NonSmooth
GTrans-Ad]
Ablation Study with Increasing Source Sample Size.
0-155
00072
3.11
0.05
1.00
230
0.00
330
0.00
330
Iada
Lamizds
Lanbads
Larbaaa
0.00-
MGE
1.50
0.04
610
000
0:01
Lanibdo
Lanodo
Lamboa
Lanbos
Lanboa
GTrane-NonDebias
Ciranne-NoSSmol
GTrans
GTrane-Ad]
(b) Ablation Study with Varying A.
Figure F2: Ablation study of different estimation methods: (a) illustrates the impact of increasing source sample size estimation accuracy, and (b) demonstrates the effect of varying density shift (A) on the performance of GTRANS, GTRANS-NonDebias, GTRANS-NonSmoott and GTRANS-Adj.
E.4 Hyperparameter Selection
E.4.1 Debiasing Threshold 5 Selection
Threshold 6. We evaluate the stability of the selected threshold 8* under matched source and target distributions. Fixing the target size at n, = 50, we vary the source size in {100, 200, 400, 800) and generate both graphs from the same graphon (graphon_ie E {1..... 10})) Source perturbations are introduced via a structural noise parameter À E 0.5, 0.5). Thresholds are evaluated using the mean squared error (MSE) between the estimated and true target graphons. We evaluate a fixed threshold 0.15 across various source perturbations. If it aligns with the side of the optimal jump point determined by the GW distance d, it is selected; otherwise, the best
(0.01, 0.02. 0.50)
31

---

threshold on that side is chosen. Empirically, 6 = 0.15 remains robust, while under EGW with e 0.01, the optimal choice converges to 8 0.18.
012
0/6
612
611
513
015
013
6.1
$11
6.12
1228
5.2
3.15
121
1.25
0.5
5.3
520
115
515
4.15
2.15
2.5
215
3.1
10
Sample Siee (Bourca)
(a) Optimal threshold 6 across Graphon IDs for GTRANS-GW.
1.2
5/3
318
815
811
314
114
4.25
0.15
3.10
111
532
512
118
$11
1.2
128
515
127
520
0.6
015
3.25
1.10
015
3.5
311
1.10
Sample (Bource)
(b) Optimal threshold 8 across Graphon IDs for GTRANS-EGW.
Figure E3: Heatmaps of the optimal threshold ò across different Graphon IDs. GTRANS-GW (top) shows that 0.15 is consistently optimal or near-optimal, while GTRANS-EGW (e 0.01 (bottom) favors 5= = 0.18 in most scenarios, with slight increases under high perturbation.
Figure E3 shows that the optimal threshold 8 depends on the graphon's inherent density and the perturbation level X. For dense graphons, 8 remains stable, reflecting robustness to structural shifts. In contrast, sparse graphons exhibit a sharp increase in under positive X, as perturbations enhance connectivity and reduce noise sensitivity. The asymmetry arises from a biri-variance trade-off. Denser sources (A > 0) yield high-quality but mismatched transfer, favoring delayed debiasing with larger 8*: sparser sources (A < 0) require earlier correction, hence smaller 8*. Besides, GW distance increases with denser sources due to structural mismatch, and decreases with sparser ones despite degraded transfer-supp supportnin larger 8 for dense, and smaller for sparse sources.
32

---

E.4.2 Regularization Parameter e Selection
Threshold e. To determine the optimal regularization parameter E in EGW, we perform K-fold cross-validation over a candidate set étist {0.001, 0.005, 0.01, 0.05, 0.1 This process evaluates the MSE between the estimated and true target graphon. We observe that e 0.01 is consistently selected as the best- pess-performng choice across different settings. To further illustrate this, figure E4 presents boxplots of the average MSE for each under varying source sizes 100, 200, 300, .... 1000 with a fixed target size of n, = 50. Each boxplot shows the MSE distribution across ten graphon types. While some graphons like Graphon 9 benefit from larger c (e.g., 0.1) due to oscillatory patterns, = 0.01 consistently achieves the lowest error, demonstrating its robustness across diverse structures.
Grapkis Sc1
Guppant). 2
O sphot B:
Grsgnon
Gropha
306
eps lon
0.001
0.005
0.01
0505
0.1
Figure E4: Boxplot comparison of different regularization parameters for GTRANS-EGW across varying source sample sizes. The results show that = 0.01 (red) consistently maintains lower variance and median MSE in most graphons, indicating its robustness.
F
Additional Real Data Results
F1
Additional results for graph augmentation task
In Transfer Learning, selecting the optimal source dataset and identifying the best class correspon- dence are crucial for effective knowledge transfer, as the structural characteristics of source labels significantly impact performance. To determine optimal class-to-class transfer pairings, we first estimate graphons separately for each source and target class using neighborhood smoothing [65]. Subsequently, we compute the pairwise Chomov-Wasserstcnn (GW) distances as well as the Entropic Gromov- Wasserseenn (EGW) distances with a commonly chosen regularization parameter c = 0.01 for each target-source graphon pair, identifying the best- matched class from the source domain for each target class based on structural similarity. We summarize the key structural statistics of all datasets used in our experiments in Table F3. Table F4 summarizes the GW distances calculated between each pair of source and target labels across the considered datasets, while Table F5 presents the corresponding EGW distances, reflecting the regularized alignment between graphon pairs.
For the MMDB-Binary dattaset, when transferring from Reddit- SaldittBinnry, the best-matching source label for target label is label 0 (GW = 0.3276, EGW = 0,3152), and for target label 2, it is also source label 0 (GW = 0.4103, EGW 0.3987). When COLLAB is used as the source, target labels and best match source label (GW = 0.1910, EGW = 1784, and GW = 0.2012, EGW = -..889. 1895, respectively). For IMDB-Multi transferring from Reddit-Binary, all three target labels (1, 2, and 3) have their lowest GW distances with source label 0 (GW = 0.4437, 0.4260, and 0.5172, respectively), while the EGW distances are slightly improved (EGW = 0.4321, 0.4155, and 0.5053). When transferring from COLLAB, the best-matching source labels are more varied: target labels 1, 2, and 3 correspond best with source label (GW = 0.15,, 0.2084, and 0.55,, respectively; EGW 0. 1.169, 0. 0969.8 and 0. 0444).). Finally, in the bioinformatics setting of PROTEINS-FIN transferring from D&D, target
33

---

labels 1 and 2 both best align structurally with source label (GW 0.0616, EGW = 0.0592, and GW 0.0652, EGW 0.0624, respectively).
These results suggest clear structural correspondences between target and source labels, enabling effective and informed transfer learning across domains.
Table F3: Statistics of the datasets used in our experiments.
Property
REDDIT-B
IMDB-B
IMDB-M
COLLAB
PROTEINS-PULL
D&D
#Graphs
2,000
1,000
1,500
5,000
1,113
1,178
#Classes
2
2
Avg. #Nodes
429.63
19.77
13.00
74.49
25.22
284.32
Avg. #Edges
497.75
96.53
63.94
2457.78
226.41
715.66
Table F4: Gromov- Wasserstein distance between each source and target label pair across datasets. Bold values denote the best-matching source label for each target label.
Source - Target
IMDB-B from REDOMIBB
IMDB-B from COLLAB
IMDB-M from REDOIT-B
IMDB-M from COLLAB
PROTEINS FULL from D&D
Target lahel
2
2
Source label 0
0.3276
0.4103
0,4710
0.399
0.4437
0.4260
0.5172
0.3521
0.3844
0.3014
0.0631 0.0616
0.0622
Source label
0,3300
0.4143
0.1910
0,2012
0.4542
0.4604
0.5326
0.1751
0.2084
0.1552
0.0652
Table F5: Entropic- Gromov- Wasserstein distance with 0.01 between each source and target
label pair across datasets. Bold values denote the best-matching source label for each target label.
IMDB-B from REDDIT-B
IMDB-B from COLLAB
IMDB-M
IMDB-M
PROTEINS FULLL from D&D
Source Target
from REDDIT-B
from COLLAB
Target label
0
2
Source labe) 0
0.3276
0.4104
0.4711
0.3996
0.4438
0.4260
0.5172
0.3523
0.3845
0.3014
0,6631
0.0584
Source Jabel
0,3300
0.4142
0.1910
0.1306
0.4549
0.4360
0.5280
0.1750
0.2085
0.1552
0.0652
0.0497
Source label 2
0.5479
0.4773
0,4092
0.4155
0.3573
We also include two transfer baselines. (1) Pooled Estimation pooling source & target graphs and estimating a graphon using the G-Mixup procedure with neighborhood smoothing (NS), and (2) golled-then-Transees using the pooled estimator as the initial source estimate piui in our transfer framework, followed by the GTRANS procedure (denoted PGTRANS-GE From Table F6, both pooled baselines underperform our method, suggesting that naive joint estimation suboptimal when source and target are misaligned, while our alignment and debiasing steps enable more accurate transfer.
PGTRANS-EOG -GWW..
Table F6: Graph classification accuracy (%) comparison between GTRANS and pooled transfer estimation methods (mean + std).
Source
Target
GTRANS-GW GTRANS-EGW
Pooled
PGTRANS-GW
PGTRANS-EGW
Reddit-B
IMDB-B B
76,30 +: 2.35
76,80 +: 1.52
74,30 + 2.52
74.65 + 2.64
75.15 + 3.03
COLLAB
IMDB-B
76.25
2.06
77.50 2.13
74.15 4: 2.26
73,90 1 2.34
74.75 : 1.89
Reddit-B
IMDB-M
49.10
51.27
3.13
48.10 3.30
3.04
COLLAB
IMDB-M
50.47
1.42
50.23 0.92
47.70
2.86
49,60 2.85
48.97 1.79
D&D
PROTEINS
69.33 2.55
68.52 +: 1.59
66.59 + 2.19
2.73 67.80
67.53 + 2.37
F.2
Application to Link Prediction
Existing literature applied graphon estimation to the link prediction task [65, 45), where the goal is to predict missing future connections between nodes in a network.
34

---

F.2.1 Transfer from a Single Source Network
Datasets. We evaluate our method on five real-world social networks spanning diverse domains: dol- phins, karate, football, firm, and wiki-vote. These datasets cover different interaction types, including animal social behavior (dolphins), human relationships and organizational affiliations (karate, firm), and communication or membership networks (football, wiki-vote). Table F7 summarizes their key statisties. The number of nodes ranges from 33 (firm) to 2,375 (wiki-vote), and the number of edges from 78 (karate) to 16,717 (wiki-vote). The average degree varies accordingly, from 4.59 (karate) to 14.07 (wiki-vote). Network density also differs widely, with firm being relatively dense (0. (7.3) while wiki-vote is much sparser (0.0074). We select wiki-vote as the source network because it is the largest. Link prediction is then performed on each of the remaining datasets treated targets, with results averaged over 50 random seeds.
Table F7: Basic statistics of the real-world datasets used in graphon estimation.
Dataset
#Nodes
#Edges
Avg. Deg.
Density
dolphins
62
159
5.13
0.0841
karate
34
78
4.59
0.1390
football
115
613
10.66
0.0935
firm
33
91
5.52
0.1723
wiki-vote
889
2914
6.56
0.0074
Experimental Setup. We simulate realistic link prediction task using a masking- based evaluation strategy following [65]. Specifically, we randomly mask subset of edges in the upper triangular portion of the target adjacency matrix to form a test set. Let M E {0, ] be a masking matrix with Mij Bernoulli(1 p). where p is the test ratio. The observed matrix is mask M,j Ar.j. meaning each edge observed with probability 1 1 p. We set = 0.1 unless otherwise specified. Both GTRANS-GW and GTRANS-EGW are applied to A mask to estimate tt and each experiment repeated 50 times with different seeds to report averaged performance.
Evaluation. To evaluate the performance of link prediction, we computed the area under the receiver operating characteristic curve (AUC). Evaluation is based on the masked (unobserved) entries where M,j = 0, using the original adjacency A, as ground truth. For a threshold >> O, the false positive rate (FPR) and true positive rate (TPR) are defined as:
Egj (Pr.s
A 0, Mij 0)
(tr(t)
Eij (A = 0, Mij = 0)
Enj ( > t, A 1, Mij = Enj (A 1, Mij = 0)
PTP (1)
These quantities are used to construct the ROC curve by varying the threshold t, and the AUC is computed as the area under this curve.
Results. We evaluate transfer performance using Wiki- Vote as the source network (n, 889). Each of the remaining datasets (dolphins. firm, football, karate) is treated as the target graph. Results are averaged over random seeds. Table F8 reports the link prediction AUC (mean + standard deviation, multiplied by 100). Our method consistently matches or outperforms baselines across datasets. Overall, these results highlight the effectiveness and robustness of wranffr-based graphon estimation, particularly in smaller noisier networks where leveraging external structure is beneficial.
Table F8: Link prediction AUC scores (mean std, 1000). Best result per dataset bolded.
Dataset
GTRANS GW
GTRANS-EGO EGW
NS
USVT
SAS
ICE
Dolphins
75.96:18.53
76.26:1..54
70.60+9.01
12.36+
50.66:16.35
72.77-9.59
Firm
11.31
71.26:11206
66.22+12.27
65.56+12.25
54.90=7.55
12.20
Football
86.6443..66
86.7413.772
86.7513.44
85.3243.56
44.56=7.38
8..83+4.60
Karate
82.47+10.36
82.53+11.466
76.74++12.01
15.26+15.20
63.88+
77.20+
35

---

F.2.2 Intra- vs. Inter-Dataset Transfer
Datasets. We evaluate intra- and inter-dataset transfer IMDB-BINARY. For each target dataset, the smallest graph (> 20 edges) is used as the target. As sources, we consider (i) the largest graph from the same dataset, (ii) the structurally closest graph within the same dataset (smallest GW distance), and (iii the largest graphs from Reddit-BINARY or COLLAB.
Experimental Setup. Graphons are estimated using GTRANS-GW and compared against NS, ICE, SAS, and USVT.
Results. Table F9 reports AUC scores. Transfer from the most structurally similar IMDB-B graph achieves the best performance (AUC 0.96), surpassing both the largest IMDB-B source and all inter-dataset sources. This highlights that transfer is most effective when the source and target share high structural similarity (small GW distance).
Table F9: Link prediction AUC results on target graphs with various transfer sources. For brevity, standard deviations are omitted.
Source
Target
Tis
Thc
GW
GTRANS-GW
NS
ICE
SAS
USVT
IMDB-B (largest)
IMDB-B
136
12
0.39
0.94
0.91
0.79
0.59
0.75
IMDB-B B (GW smaller)
IMDB-B
0.31
0.91
0.59
0.75
Reddit-B
IMDB-B
12
0.40
0.94
0.91
0.79
0.59
0.75
COLLAB
IMDB-B
191
12
0.43
0.94
0.91
0.79
0.59
0.75
IMDB-M
IMDB-B
89
12
0.40
0.95
0.91
0.79
0.59
0.75
G Additional Details
G.1
Graphon functions
We implement 10 distinct graphon structures (see Table G10) ranging from simple bilinear forms to highly structured oscillatory and piecewise functions.
Table G10: Graphon functions implemented.
ID
Graphon Function
1
exp(-x0: 7077)
2
exp(- - max (z, y) 0.75
3
exp (-0.5 [min(sr, V vJl)
4
T+exp(
( +min
5
[
6
xy
2
7
x t"
3
COS
x- +ッ
+ 0.15
8
atw cos
+ 0.15
2
9
(in
+ 0.5
10
min
exp
sin
(1-2)-47²
exp
sin x+1(1- m/
G.2 Neighborhood Smoothing Details
For effective neighborhood smoothing, we must identify a neighborhood N, for each node i that contains only nodes with approximately homogeneous distribution. Here, the neighborhood. M, in this context refers to nodes that are close in the underlying latent spuce (i.e., have similar latent positions), not nodes that are connected in the observed graph. Mathematically, node i' belongs to node i's neighborhood if their graphon values are sufficiently close: Ilf(u,,) - flur, 1)12 S 17, where ll2 denotes the L2 norm with inner product < f. >= fo f())g(z) dzz, and is a tolerance parameter. Empirically, this can be estimated by Σ llp Pv, ll.. As proved in [65], under mild conditions,
36

---

this quantity is bounded by: 1 = m 1/2 -j==(AA - Aay) This bound enables efficient neighborhood definition: NN (?' #i: A.r S r), where is predefined threshold. Once neighborhoods. N₁ and Nr are identified, nodes within each neighborhood are treated as replicates.
Thus, Pij can be estimated by neighbourhood smoothing, i.e., Pij / -ren Ag DzeeN, Asy [Ng1 where [M]] represents the number of nodes in the neighborhood N,, This averages two empirical proportions: edges between node and nodes in Mi. and edges between node and nodes in NNj.
G.3 Algorithm
G.3.1 Algorithm: GTRANS
Algorithm GTRANS: Transfer Learning for Graphon Estimation
Require: Source adjacency matrix As, target adjacency matrix A,, threshold e.
Ensure: Target graphon estimator P.
1: Step 1: Initial Graphon Estimation
2: Apply neighborhood smoothing to obtain two initial estimators for both the source and target
graph: Pint < NS(A)) Pini + NS,)..
3: Step 2: Transferring Step
4: Compute the optimal transport plan á and the optimal transportation distance d between Pini and Pini
5: Apply column normalization to ât, obtaining मे
6: Transfer source graphon estimator to target domain: Ptrens +- 7 P.ri.
7: Refine the transferred estimator by applying neighhorhood smoothing Ptrans, ptrans?
NS (((tra))))
8: if d 8, then
9: Step 3: Debiasing Step:
Compute residual matrix: R. Pini ptrans2
11:
Apply neighborhood smoothing to residual, obtaining estimator Pres
12:
Combine transferred estimator with smoothed residual: P, + Ptranel + Pres
else
14:
P. + ptrans2
end if'
16: return P.
G.3.2 Algorithm: Network Cross- Validatio by Edge Sampling
Algorithm 2 Network Cross- Validation by Edge Sampling for Threshold Selection
Require: Adjacency matrices A,; A, true probability matrix Pt, threshold candidates (c1,..., } number of folds K, loss function C(:-)
Ensure: Selected threshold é, final loss, estimated graphon P.
1: Step I: Edge Sampling
Randomly partition the edges of A, into folds, ensuring that each fold is disjoint and collec- tively covers the entire set of edges.
2: for k 1 to K do
3: Mask edges in the k-th fold form the incomplete adjacency matrix (*).
4: Perform matrix completion suggested by [33], leveraging low-rank and smoothness assump-
tions recover the completed adjacency matrix.
5: Apply GTRANS with each threshold and evaluate the loss (()) on the held-out set (.()
6: end for
7: Step 2: Select Optimal Threshold
8: Average the loss across all folds: C(e:) R Σk (())
9: Select the threshold that minimizes the average loss: è arg mine; C(EL)
10: Return
37

---

G.4 Graph Augmentation Model Training Setup.
We follow modified version of the training configuration from [22]. Specifically, we train a GIN model for 200 epochs using the Adam optimizer with fixed learning rate of 0.01. The mini-batch size is set to 128, and the hidden dimension is 64. Validation loss is monitored throughout training. and test accuracy is reported at the epoch with the best validation performance.
38


Reviews:
[{"review_id": "xKHkC5tWbL", "reviewer": "Reviewer_aKUK", "summary": "The paper presents GTRANS, a method designed to estimate the graphon underlying a small target graph by transferring knowledge from a larger source graph with similar properties. The approach begins by estimating graphons for both the small target graph and the large source graph using neighbor smoothing. It then computes the Gromov-Wasserstein (GW) distance between these two graphons and uses the resulting alignment matrix to map the source graphon's structure onto the latent domain of the target graph. Finally, the method assesses potential domain shift by comparing the transferred graphon to the initially estimated one and smooths the residual if a significant difference is detected.", "questions": "Please refer to the questions mentioned in the weaknesses above.", "limitations": "yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The paper is well-written and easy to follow.\n2. Transferring knowledge from a large graph to aid in estimating the graphon of a smaller graph is a novel and interesting idea, particularly applicable to some real-world scenarios where observed graphs are small.\n3. In the experiments, the proposed method achieves better results compared to the baselines.\n\nWeaknesses:\nWhile the idea of transferring knowledge from a larger graph is compelling, the method assumes the large graph comes from a different but known to be similar graphon. \nThis raises some questions:\nHow do we determine which dataset's graphon is similar to the one at hand? In other words, how do we know the source is truly similar to the target? The method assumes access to an external large graph that is known to be similar to the smaller graph, but there is no formal or mathematical definition of this similarity.\n\nAs a follow-up to the above:\nWhy not use larger graphs from the same dataset? For example, in the G- Mixup dataset, although IMDB-B is a small dataset, it contains a few larger graphs that could be used. These might even be more beneficial than transferring from a different dataset. That is, what happens if fs=ft.\n\nIf we compare transferring from a large graph in the same dataset vs. from a large graph in a different dataset, what would the performance difference be? Intuitively, the former should yield better results. Some experiments in this direction would strengthen the paper.\n\nAlong the same lines, what if we pooled source and target graphs together and estimated the graphon jointly from multiple graphs?\n\nThe method does not compare to more recent, resolution-free graphon estimation techniques that estimate the graphon from a single graph or a set of graphs. Examples include GWB [1], IGNR [2], and SIGL [3], whereas this work relies only on simpler graphon estimation baselines.\n[1] Xu, Hongteng, et al. \"Learning graphons via structured gromov- wasserstein barycenters.\" Proceedings of the AAAI Conference on Artificial Intelligence.\n[2] Xia, Xinyue, Gal Mishne, and Yusu Wang. \"Implicit graphon neural representation.\" In Int. Conf. on Artif. Intell. and Stat.\n[3] Azizpour, A., Zilberstein, N., and Segarra, S. (2025). Scalable implicit graphon learning. In Int. Conf. on Artif. Intell. and Stat.\n\nWhat is the rationale behind smoothing the residual? While it’s true that the residual may capture domain shift, why should smoothing and adding it help improve the estimated graphon? Some justification or experimental evidence would be useful.\n\nIn the G-Mixup experiment, how is the source graph selected? There are many large graphs in the source dataset, and selecting one over another may significantly impact the alignment, especially given the diversity of the real-world graphs.\n\nThere are no experiments to validate the theoretical assumptions. This could potentially be addressed using synthetic datasets, where the ground truth alignment is available.\n\nMinor comment: There is a typo in the caption of Figure 5: \"iniial\" should be \"initial\".", "was_revised": true, "final_justification": "The idea of transferring knowledge from a large graph to improve the graphon estimation from a smaller graph is a nice addition to the graphon estimation literature. The rebuttal from the authors included several useful additional experiments and insights.", "rebuttal": "We thank the reviewer for these insightful comments. We will correct typos, add the following clarifications, discussions, and experiments in the revised manuscript. Due to space constraints, only mean values are shown in the following experiments.\n\n> Comment 1: Similarity between source and target graphs:\n\nResponse: We quantify source–target similarity using the Gromov-Wasserstein (GW) distance between their latent probability matrices $\\mathbf{P}_s$ and $\\mathbf{P}_t$ (Section 2.2). Since these are unobserved in practice, we use the GW distance $d$ between their initial estimators $\\hat{\\mathbf{P}}^{\\text{ini}}_s$ and $\\hat{\\mathbf{P}}^{\\text{ini}}_t$ as a surrogate (Section 3.2.2, line 194; ref. [62] of our method). The entropic regularized GW can also be used for efficiency. To avoid negative transfer, we apply debiasing when $d > \\delta$, with the cutoff $\\delta$ chosen via network cross-validation (ref. [32], Section 5.1, lines 321–327).\n\nWhen multiple candidate source graphs are available, we use the estimated GW distance between the initial graphon estimators as the criterion for selecting the most appropriate source graph for transfer (Section D.1 in Appendix, lines 982-992).  Our new **theoretical analysis**  (see detailed response for Comment 2 of Reviewer JN9a)  shows that $GW_2^2(\\hat{\\mathbf{P}}^{ini}_s, \\hat{\\mathbf{P}}^{ini}_t)$ **approximates** $GW_2^2(\\mathbf{P}_s, \\mathbf{P}_t)$, thereby justifying its use as a criterion for source selection with theoretical guarantees.\n\n> Comment 2: Larger graph from same dataset in G-Mixup framework:\n\nResponse: First, we would like to clarify that in the G-Mixup framework, the authors estimate a **single graphon per class** by aligning and aggregating all graphs within that class, not individual graphons per graph. Specifically, for each class, they: (i) sort nodes in each graph by descending normalized degree, (ii) pad smaller adjacency matrices to match the largest graph, (iii) average the aligned matrices, (iv) estimate a graphon from this averaged matrix. This class-level graphon is meant to capture the common structural pattern of the class, not any individual graph. Our use of GTRANS in this context aims to improve the estimation accuracy of this shared graphon.\n\nSecond, when the application demands per-graph graphon estimates (e.g., for individual graph prediction), intra-dataset transfer can be effective. As per your suggestion, we conducted additional experiments on link prediction for single graphs. Taking IMDB-BINARY as target dataset, we select the smallest graph (with $>20$ edges) as the target. For intra-dataset source graph, we considered: (1) the largest graph in IMDB-BINARY, and (2) the graph within IMDB-BINARY that has a small GW distance to the target. For comparison, we also included the largest graph from a different dataset as an inter-dataset source. Graphons were estimated using GTRANS-GW and baselines (NS, ICE, SAS, USVT). Link prediction was performed by the same procedure as described in **Appendix D.2**. \n\nAs we can see from **Table R1**,  the best results are achieved by choosing the IMDB-B source graph with the smallest GW distance to the target, even if it is not the largest. Specifically, transfer from the most structurally similar IMDB-B graph yields the highest link prediction accuracy (AUC 0.96), outperforming both the largest IMDB-B source and all cross-dataset sources. This demonstrates that **transfer is most effective when the source and target are structurally similar** (small GW distance). \n\nTable R1: Link prediction AUC results on target graphs with various transfer sources. \n|Source|Target|$n_s$|$n_t$|GW|GTRANS-GW|NS|ICE|SAS|USVT|\n|:-----------------:|:------:|:------:|:---------:|:-------------:|:-------------:|:---------:|:---------:|:---------:|:---------:|\n|IMDB-B (largest)|IMDB-B |136|12|0.39|0.94|0.91|0.79|0.59|0.75|\n|IMDB-B (gw smaller)| IMDB-B |84|12|0.31| **0.96**|0.91|0.79|0.59|0.75|\n|Reddit-B|IMDB-B|41|12|0.40|0.94|0.91|0.79|0.59|0.75|\n|COLLAB|IMDB-B|191|12|0.43|0.94|0.91|0.79|0.59|0.75|\n|IMDB-M|IMDB-B|89|12|0.40|0.95|0.91|0.79|0.59|0.75|\n\n> Comment 3: Joint estimation by pooling:\n\nResponse:  To address your concern, we conducted additional experiments comparing our method to two alternatives: (1) **Pooled Estimation**: pooling source and target graphs together, estimating a graphon using the G-Mixup procedure with neighborhood smoothing (NS) (see our response of your comment 2), and (2) **Pooled-then-Transfer**: using the pooled estimator as the initial source estimate $\\hat{P}^{\\text{ini}}_s$ in our transfer framework, followed by the GTRANS procedure (denoted PGTRANS-GW and PGTRANS-EGW). Following Section 5.2, we evaluated all methods within the G-Mixup framework for graph classification. \n\nAs shown in **Table R2**, GTRANS-GW/EGW consistently **outperforms pooled estimators**.  Naive pooling obscures domain-specific features and introduces negative transfer; While Pooled-then-Transfer partially recovers through post-pooling adaptation, it cannot correct initial information dilution. In contrast, GTRANS maintains domain separation until adaptation, leveraging source information more effectively. \n\nTable R2: Graph classification accuracy (\\%) across three target datasets.\n|Source|Target|GTRANS-GW|GTRANS-EGW|Pooled NS|PGTRANS-GW|PGTRANS-EGW|\n|:----:|:----:|:-------:|:--------:|:-------:|:--------:|:---------:|\n|Reddit-B|IMDB-B|76.30|**76.80**|74.30|74.65|75.15|\n|COLLAB|IMDB-B|76.25|**77.50**|74.15|73.90|74.75|\n|Reddit-B|IMDB-M|49.10|**51.27**|46.50|48.10|48.67|\n|COLLAB|IMDB-M|**50.47**|50.23|47.70|49.60|48.97|\n|D\\&D|Proteins|**69.33**|68.52|66.59|67.80|67.53|\n\n> Comment 4: Comparison with recent methods.\n\nResponse: We conducted additional experiments comparing our method with GWB, IGNR, and SIGL in the graph classification task. As shown in Table R3, our proposed transfer learning method outperforms these three non-transfer learning methods.  \n\nTable R3: Graph classification accuracy (\\%).\n|Source|Target|GTRANS-GW|GTRANS-EGW|GWB|IGNR|SIGL|\n|:----:|:----:|:--------:|:---------:|:---:|:---:|:----:|\n|Reddit-B|IMDB-B|76.30|**76.80**|75.30|74.35|73.50|\n|COLLAB|IMDB-B|76.25|**77.50**|75.30|74.35|73.50|\n|Reddit-B|IMDB-M|49.10|**51.27**|47.70|47.50|49.13|\n|COLLAB|IMDB-M|**50.47**|50.23|47.70|47.50|49.13|\n|D\\&D|PROTEINS|**69.33**|68.52|63.45|65.87|67.13|\n\n>Comment 5: Why smoothing residual?\n\nResponse: The residual matrix $\\mathbf{R}_t = \\hat{\\mathbf{P}}^{\\text{ini}}_t - \\hat{\\mathbf{P}}^{\\text{trans2}}_t$ captures (a) target-specific structural patterns not explained by the transferred estimator and (b) random noise due to small sample size of the target graph. To retain meaningful structure while suppressing noise, we apply NS to the residual, without which it simply reduces to the target-only NS estimator. Our results consistently show GTRANS outperforming NS, demonstrated by lower MSE (**Figure 4 and Table 1**) and higher classification accuracy (**Table 2**), highlighting the critical role of residual smoothing. \n\n**Rationale for adding the smoothed residual in the debiasing step**:  When GW distance is large ($d > \\delta$), transferred estimator $\\hat{\\mathbf{P}}^{\\text{trans2}}_t$ alone may cause negative transfer. However, transfer remains useful as global properties like smoothness are often shared. Debiasing adaptively combines source structure with target-specific corrections via smoothed residuals. For example, graphons 7 & 8 (Figure 3) share smooth diagonal connectivity; the transferred estimator captures this smoothness while debiasing recovers sharper local patterns (Figure 5c).\n\n>Comment 6: Source selection in G-Mixup:   \n\nResponse: In the G-Mixup framework, a single graphon is estimated per class by aggregating and aligning all graphs within that class, rather than estimating graphons for individual graphs (see our response to your Comment 2 for details). As a result, **there is no issue of selecting a specific source graph** within the source dataset. \n\nFurther, transfer is performed at the class level: for each target class, we estimate a class-level graphon. To identify a suitable source for transfer, we estimate initial class-level graphons for all source and target classes using neighborhood smoothing. We then calculate pairwise Gromov-Wasserstein distances between these initial estimators across all class pairs, to select the source class closest to the target (see Section D.1 of Appendix).\n\n>Comment 7: Validating theoretical assumptions:\n\nResponse:  Theorem 4.1 relies on two key conditions: (i) $\\epsilon > C_1|\\pi^\\star|_\\infty |\\mathbf{P}_s \\otimes \\mathbf{P}_t|{\\rm op}$ for some $C_1 > 2$, and (ii) the estimation errors for $\\mathbf{P}_s$ and $\\mathbf{P}_t$ must be sufficiently small. \n\nWhile the **first condition** involves unobservable population-level quantities and cannot be directly verified, our simulations suggest it holds for a broad range of $\\epsilon$. The following **Table E** reports computed lower bounds on feasible $\\epsilon$ values (with $C_1 = 2.01$) for Graphons 1–3 (defined in Section E.1), each scaled to have operator norm 1, with $n_s = 100$ and $n_t = 50$. Any $\\epsilon$ above the reported thresholds satisfies the condition, including $\\epsilon = 0.01$ used in our real data analysis.\n\n**The second condition is well-supported by established graphon estimation literature**, as it requires $\\hat{\\mathbf{P}}_s^{\\text{ini}}$ and $\\hat{\\mathbf{P}}_t^{\\text{ini}}$ to be consistent. For example, Ref. [62] shows that NS-based estimators achieve rates of $\\sqrt{\\log n_s / n_s}$ and $\\sqrt{\\log n_t / n_t}$ for the source and target, respectively.\n\nTable E: Lower bound of $\\epsilon$ for various choices of source and target graphons. \n\n|Source|Target 1|Target 2|Target 3|\n|:----:|:-------:|:-------:|:-------:|\n|Source 1|5.5e-4|5e-4|6e-4|\n|Source 2|4e-4|4.5e-4|4e-4|\n|Source 3|5e-4|4.5e-4|5e-4|"}, {"review_id": "5fmZ6U7DcM", "reviewer": "Reviewer_xFxn", "summary": "This paper explores the graphon estimation issue for small graph and proposes the method to transfer the more accurate estimation of graphon from a larger similar source graph to the small target graph. Specifically, their method involves three stages being the initial estimation of the graphon for both graphs, optimizing for the alignment matrix and the final debiasing step for significant distribution shift. This paper claims to be the first work targeting small graph graphon estimation without known node correspondence and exceed the traditional baselines.", "questions": "**Questions:**\n- Are there any criterions or measurements that suggest a good choice of source graph or have you encountered the case where the transferability is nearly impossible even with the designed debiasing process?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n- The paper tackles a practical issue with graphon estimation for small graph and think of the novel idea to enable the transferability of graphon from a similar larger source graph. \n- The method proposed is grounded with some theoretical guarantee and the design for each step is clear and reasonable. Also, the paper includes some evidence for method motivation, like the visualization of alignment matrix from adjacency-based and estimation based, as well as they discuss the strength and drawbacks of their design, e.g. comparing against GW and EGW.\n- The paper itself is well-written and easy to follow\n\n**Weaknesses:**\n- The success of the small target graph estimation seems to rely a bit on whether there is a large and similar graph that can serve as the source graphon estimation. Maybe, it could be better if the authors provide more real life examples to suggest that it is very common and easy to find the similar source graphs to work on. \n- It is good that we have the real dataset application being enhancing the graphon estimation used in the G-Mixup method. I'm wondering that is there any other applications and usage to showcase the benefit from more accurate small graph graphon estimation.", "was_revised": true, "final_justification": "The rebuttal that clarifies my questions, I will remain my score for the paper.", "rebuttal": "> Comment 1:  The success of the small target graph estimation seems to rely a bit on whether there is a large and similar graph that can serve as the source graphon estimation. Maybe, it could be better if the authors provide more real life examples to suggest that it is very common and easy to find the similar source graphs to work on.\n\nResponse: Thank you for raising this important point. In many real-world settings, it is actually quite common to find large graphs that are structurally similar to smaller ones, especially within the same application domain. **For example**, in our experiments (**Table 2**), we successfully transferred from large protein networks in D&D to smaller ones in PROTEINS-Full, and from large social or collaboration graphs (Reddit-B, COLLAB) to smaller IMDB-B or IMDB-M co-actor networks. **Another example** is in our link prediction task (**Section D.2 in Appendix**), we successfully transferred from large Wiki-Vote to  smaller four networks, including   Dolphins, Firm, Football and Karate. For your covenience, we put the link prediction results in Table A. \n\nBeyond our main experiments, here are some **potential examples**: (1) In systems biology, large gene regulatory or protein interaction networks (e.g., from STRING or BioGRID) can be used as source for smaller subnetworks in specific tissues or cell types. (2) In online social platforms, massive public graphs (from Facebook, Twitter, or Reddit) can be leveraged to inform smaller networks. (3) In neuroscience, large-scale brain connectomes can be sources for smaller regional networks. (4) In transportation research, traffic network of a large city (e.g., Chicago, NY) can be used to improve the traffic system of small urban areas.  Due to space limitations, further empirical studies on these additional datasets are beyond the scope of this work.  We believe our results provide a strong foundation, and we hope to systematically explore these broader applications in future work.\n\n\nWe will revise our manuscript to include discusssions about these potential application examples. \n\n\n\nTable A: Link prediction AUC (Mean ± SD, multiplied by 100) using Wiki-Vote as the source graph ($n_s = 889$).\n\n| Dataset  | GTRANS-GW | GTRANS-EGW | NS    | USVT   | SAS      | ICE     |\n| -------- | --------------- | ---------------- | ------------- | ------------- | ------------- | ------------- |\n| Dolphins ($n_t=62$)   | 75.96 ± 8.53    | **76.26 ± 8.54**     | 70.60 ± 9.01  | 72.36 ± 10.16 | 50.66 ± 6.35  | 75.36 ± 8.40  |\n| Firm ($n_t=33$)    | **71.31 ± 12.18**   | 71.26 ± 12.06    | 66.32 ± 12.27 | 65.56 ± 12.25 | 54.90 ± 7.59  | 64.26 ± 12.20 |\n| Football($n_t=115$) | 86.64 ± 3.66    | **86.74 ± 3.72**     | 86.75 ± 3.49  | 85.32 ± 3.56  | 44.56 ± 7.38  | 82.46 ± 4.59  |\n| Karate ($n_t=34$)    | 82.47 ± 10.36   | **82.53 ± 10.46**    | 76.74 ± 12.01 | 71.86 ± 15.20 | 63.88 ± 11.15 | 80.43 ± 11.22 |\n\n\n\n> Comment 2: It is good that we have the real dataset application being enhancing the graphon estimation used in the G-Mixup method. I'm wondering that is there any other applications and usage to showcase the benefit from more accurate small graph graphon estimation.\n\n\nResponse: Thank you for your encouraging comment and interest in broader applications of our method. Beyond the G-Mixup graph classification task, we have evaluated GTRANS on the **link prediction** problem, as detailed in **Appendix D.2**. Our results in **Table A** show that enhanced graphon estimation significantly improves link prediction performance compared to baselines, highlighting the method’s versatility across practical tasks. \n\n\n\nIn essence, **GTRANS can enhance any application requiring graphon estimation**. For instance, researchers have utilized estimated graphons for **community detection** [1] and **graph neural network** training [2]. Due to page constraints, we defer empirical exploration of these areas to future work.\n\n\n[1] Klimm, F., Jones, N. S., & Schaub, M. T. (2022). Modularity maximization for graphons. SIAM Journal on Applied Mathematics, 82(6), 1930-1952.\n\n[2] Hu, Z., Fang, Y., & Lin, L. (2021, December). Training graph neural networks by graphon estimation. In 2021 IEEE International Conference on Big Data (Big Data) (pp. 5153-5162). IEEE.\n\n\n\n> Comment 3: Are there any criterions or measurements that suggest a good choice of source graph or have you encountered the case where the transferability is nearly impossible even with the designed debiasing process?\n\nResponse:  Thank you for the valuable comment. \n\n\n**Criterion for Source Graph Selection**: As mentioned in Section 3.2.2 (line 194), we measure the source-target domain difference  using  the Gromov-Wasserstein (GW) distance between the initial graphon estimators $\\hat{\\mathbf{P}}^{\\text{ini}}_s$ and $\\hat{\\mathbf{P}}^{\\text{ini}}_t$, each obtained using the neighborhood smoothing method. When multiple candidate source graphs are available, we use the estimated GW distance between inital graphon estimator  as the criterion for selecting the most appropriate source graph for transfer (see Section D.1 in the appendix, lines 982-992). Our new **theoretical analysis**  (see detailed response for Comment 2 of Reviewer JN9a)  shows that $GW_2^2(\\hat{\\mathbf{P}}^{ini}_s, \\hat{\\mathbf{P}}^{ini}_t)$ **approximates** $GW_2^2(\\mathbf{P}_s, \\mathbf{P}_t)$, thereby justifying its use as a criterion for source selection with theoretical guarantees.\n\n**Cases Where Transferability is Challenging**: We have indeed encountered scenarios where transferability is nearly impossible due to high dissimilarity (large $d$). In our simulations (Table 1 and Table C2), cross-graphon transfers between highly dissimilar pairs (e.g., between graphon 8 and 9, with distinct patterns like gradient vs. striped structures; Figure 3) achieves similar performance as target-only neighborhood smoothing method. However, we would like to emphasize that the debiasing step in our method effectively mitigates the influence of the source graph when it is substantially dissimilar to the target domain.\n\nThus, while GTRANS excels with similar sources, it remains safe and non-degrading even when transfer is challenging, making it a reliable tool for practitioners."}, {"review_id": "lhdLRhZqwX", "reviewer": "Reviewer_JN9a", "summary": "This paper studies the problem of graphon matrix estimation via transfer learning, where the goal is to estimate the target probability matrix by additionally leveraging information from a larger source graph. The paper establishes an adaptive transfer learning framework under the graphon model  by combining neighborhood smoothing graphon estimation with Gromov-Wasserstein optimal transport which maps the source probability matrix estimate to the target  probability matrix estimate. When the discrepancy between the target and the transferred source estimate is larger than a pre-specified threshold, a smoothed version of the discrepancy matrix is used to improve the proposed estimator (referred to as 'de-biasing' in the paper).", "questions": "Major:\n1. Why is the debiasing step referred to as debiasing? What is the bias here and how is it being removed?\n2. What makes the debiasing step adaptive? \n3. Line 41: please explicitly define `the source-target domain shift'\n4. Why is the difference between \\hat P_t^ini  and \\hat P_t^trans2 called a `residual matrix' and not simply a discrepancy matrix? The term `residual' is commonly used in the context of linear regression where it represents the structure unexplained by the observed predictors. Is there a link/analogy that can be made?\n5. Lines 189-190: why is this additional smoothing required? Does the transfer step lead to discontinuities in the smoothed \\hat P^ini_s? Can this be theoretically justified or illustrated empirically?\n6. Line 229- where is the lower bound on \\epsilon? \n7. Theorem 4.1: Does this cover the specific case where the source graph is completely uninformative about the target graph? What would happen in that case? \n8. Experiments: The authors study the performance with n_t=50 and n_s \\in { 100,200,..1000}. It would be helpful to understand how the method compares to the baseline NS method when n_t =100, n_s=150, 200 and when n_t=150 and n_s=200, 250. This could serve as a practical guidance on when transfer learning may or may not be useful. \n9. General guidance on properties to look for when selecting the source graph would be very useful. \n\nMinor  \n\n10. Lines 54-57: Are the two problems (alignment and unsupervised learning) linked?\n11. Section 2.1: The list of references on graphon estimation could be updated to include more recent significant contributions on the topic", "limitations": "Yes. However, limitations within the framework studied in the paper are missing and could be added.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths: The proposed method is shown to work in the sense that it leads to an improved graphon estimator in some sample settings (n_s,n_t) where the target graph is small and the source graph has structure to some extent similar to the target graph. Theoretical consistency of the estimated alignment matrix to the true optimal transport map is established. The paper is clearly written.\n\nWeakness: The main motivation behind the contribution is relatively poor graphon estimates when n is small. However, it is not clear what defines a `small' n, in practice. If the underlying graphon structure is simple (e.g. an ER or a 2-blockmodel), n as small as 80 might be sufficient to estimate it. This is not discussed in the paper at all. Further, there are no theoretical results which establish sample settings (n_s,n_t) and assumptions on the underlying target and source graphons, where we may expect the proposed methodology to be useful.", "was_revised": true, "final_justification": "I thank the authors for their clear response which have improved my appreciation for the paper. I have increased my scores accordingly.", "rebuttal": "We thank the reviewer for these constructive comments. We will correct typos, add the following clarifications and discussions in the revision. Due to space constraints, only mean values are shown in the following experiments.\n\n> Comment 1: How small should the target data be? \n\nResponse: Defining universal bounds on \"small\" $n$ is challenging as it depends on graphon complexity and source-target similarity rather than fixed thresholds. Complex graphons require larger $n_t$. Existing works provide asymptotic error rates (e.g., $O(\\sqrt{\\log n / n})$ for NS method) but lack specific \"small\" cutoffs.\n\nTo provide practical guidance, we ran simulations with Graphon 1, fixing $n_s = 1000$ and varying $n_t$ from 100 to 700 (Table R4, 50 runs each).  The relative gain, calculated as $\\frac{MSE_{NS} - MSE_{GTRANS}}{MSE_{NS}}$ (where $MSE_{GTRANS}$ is the minimum of GTRANS-EGW and GTRANS-GW). The relative gain from transfer decreases as $n_t$ increases but remains positive; even at $n_t = 700$, GTRANS still outperforms target-only NS, showing that transfer learning provides substantial benefit even for moderately large target graphs.\n\nTable R4. Mean MSE  value (multiplied by 100) over 50 runs for Graphon 1 with  $n_s = 1000$.\n\n| $n_t$ | GTrans-EGW  | GTrans-GW  | NS | Relative Gain (%) |\n| ----------- | -------------- | ------------- | -------------------- | -------------------- |\n| 100         | 1.34     | 1.44    | 1.83          | 26.78               |\n| 300         | 1.11     | 1.21    | 1.40           | 20.71              |\n| 500         | 1.09     | 1.19    | 1.31          | 16.79               |\n| 700         | 1.10    | 1.20   | 1.26           | 12.70               |\n\n> Comment 2:  lack of theoretical conditions on the target and source graphons under which the proposed method is effective.\n\nResponse: We derived new theoretical results which can help us understand **how close the graphons should be** for effective transfer.\n\n(1) **Theoretical Justification for GW Distance as Similarity Measure**: We developed new theory showing that GW distance between initial estimators $\\hat{\\mathbf{P}}^{ini}_s$ and $\\hat{\\mathbf{P}}^{ini}_t$ is a theoretically justified similarity criterion. Our results establish: (i) $GW_2^2(\\mathbf{P}_s, \\mathbf{P}_t) \\lesssim GW_2^2(\\hat{\\mathbf{P}}^{ini}_s, \\hat{\\mathbf{P}}^{ini}_t) + \\delta_n$ and (ii) $GW_2^2(\\hat{\\mathbf{P}}^{ini}_s, \\hat{\\mathbf{P}}^{ini}_t) \\lesssim GW_2^2(\\mathbf{P}_s, \\mathbf{P}_t) + \\delta_n$, where $\\delta_n$ represents estimation error of order $\\sqrt{\\log(n)/n}$ for Lipschitz smooth graphons [1]. **Together**, these results establish that GW distance between initial estimators reliably proxies true graphon similarity, justifying its use for guiding effective transfer learning.\n\n(2) **Theoretical Result of Transferring Step**:  Our theoretical analysis shows that at the population level, the distance between the transferred estimator and true target probability matrix is bounded by the source-target GW distance: $|{\\mathbf{P}}_t^{\\rm trans} - \\mathbf{P}_t| \\leq \\mathrm{GW}_2^2(\\mathbf{P}_s, \\mathbf{P}_t)$. This demonstrates that the transferring step effectively transfers structural information, particularly when source and target graphons are sufficiently similar.\n\n> Comment 3: Why called  \"debiasing step\"?\n\nResponse: When source and target probability matrices ($\\mathbf{P}_s$ and $\\mathbf{P}_t$) differ, the transferred estimator may systematically misestimate target probabilities due to structural differences, creating bias that requires correction. \"Debiasing\" is standard terminology in transfer learning literature [1].\n\n**Why Debiasing helps**:  The residual matrix $\\mathbf{R}_t = \\hat{\\mathbf{P}}^{ini}_t - \\hat{\\mathbf{P}}^{trans2}_t$ captures target-specific structural patterns and noise. Neighborhood smoothing retains meaningful target patterns while suppressing random fluctuations. The final estimate $\\hat{\\mathbf{P}}_t = \\hat{\\mathbf{P}}^{trans2}_t + \\hat{\\mathbf{P}}^{res}_t$ effectively combines transferred knowledge (shared patterns) with target-specific information.\n\n**Empirical support of the effectiveness of the debiasing step**:  We conducted an ablation study comparing our method with its variant without the debiasing step. As shown in **Figure C2 in the Appendix**,  GTRANS consistently outperforms its ablated variants GTRANS-NonDebias. This highlights the importance of the debiasing step. \n\n[1] Li, S., Cai, T. T., & Li, H. (2022). Transfer learning for high-dimensional linear regression: Prediction, estimation and minimax optimality. Journal of the Royal Statistical Society Series B: Statistical Methodology, 84(1), 149-173.\n\n> Comment 4:  What makes the debiasing step adaptive?\n\nResponse: The debiasing step is adaptive because it's applied only when GW distance $d > \\delta$, where threshold $\\delta$ is chosen via cross-validation to minimize prediction error, making it data-driven based on measured source-target similarity.\n\n> Comment 5: What's `source-target domain shift'?\n\nResponse:  Following transfer learning literature [1], \"domain shift\" refers to differences in data distribution between source and target domains. In our work, we define source-target domain shift as the GW distance between true latent edge probabilities $\\mathbf{P}_s$ and $\\mathbf{P}_t$, estimated using GW distance between $\\hat{\\mathbf{P}}_s^{\\text{ini}}$ and $\\hat{\\mathbf{P}}_t^{\\text{ini}}$ (Section 3.2.2, lines 194).\n\n[1] Wilson, G., & Cook, D. J. (2020). A survey of unsupervised deep domain adaptation. ACM Transactions on Intelligent Systems and Technology (TIST), 11(5), 1-46.\n\n> Comment 6: Why called residual matrix'?\n\nResponse: We use \"residual matrix\" by analogy to regression analysis, where residuals represent the portion of observed outcomes not explained by the fitted model. In our setting, $\\hat{\\mathbf{P}}_t^{ini} - \\hat{\\mathbf{P}}_t^{trans2}$ captures the component of the target's initial estimator not explained by the transferred source estimator, isolating target-specific structural patterns and sampling noise not captured by transfer.\n\n> Comment 7: Lines 189-190: why additional smoothing? \n\nResponse: The additional neighborhood smoothing step after the projection is implemented because the transfer step, while structurally aligning the source and target, may introduce local discontinuities or irregularities, especially when the alignment matrix aggregates nodes with different local structures. As a result, the transferred estimator might inherit sharp transitions or noise not present in the original smooth source graphon.  **Appendix C.3** provides empirical justification: our ablation study shows GTRANS-NonSmooth (omitting additional smoothing) achieves higher MSE than the full GTRANS method in most cases, demonstrating that additional smoothing improves estimation accuracy.  \n\n> Comment 8: Line 229- where is the lower bound on \\epsilon? \n\nResponse:  The lower bound on $\\epsilon$ is specified in Theorem 4.1, where we require that the penalty parameter satisfies $|\\pi^*|_\\infty \\leq \\frac{\\epsilon}{C_1 | \\mathbf{P}_s \\otimes \\mathbf{P}_t |{\\mathrm{op}}}$ for some $C_1 > 2$. This condition ensures the necessary local convexity for the analysis to hold. For completeness, we will revise line 229 to explicitly reference Theorem 4.1 and clarify that the lower bound on $\\epsilon$ is given there.\n\n> Comment 9:  Theorem 4.1: Does this cover the specific case where the source graph is completely uninformative about the target graph? \n\nResponse: Theorem 4.1 establishes that the estimated alignment matrix approximates the population counterpart, i.e., minimizer of GW distance between  $\\mathbf{P}_s$ and  $\\mathbf{P}_t$, with deviation governed by the estimation error of initial estimation $\\hat{\\mathbf{P}}^{ini}_s$ and $\\hat{\\mathbf{P}}^{ini}_t$.  As long as both the source and target graphons are estimated well (e.g., via the neighborhood smoothing method, which we adopt in our work), and the parameter $\\epsilon$ exceeds a certain threshold, this theorem holds, regardless of whether the source is informative or not.  \n\n> Comment 10: Experiments when n_t =100, n_s=150, 200 and when n_t=150 and n_s=200, 250. \n\nResponse:  Following your suggestion, we have conducted additional experiments. In **Table R5**, we compared our method (GTrans-EGW and GTrans-GW) with the Target-only method NS. The results show that GTRANS consistently outperforms NS across all cases. Additionally, as the target sample size $n_t$ increases, the relative gain from transfer learning diminishes, which is expected since more target data naturally reduces the advantage of borrowing strength from a large source. \n\nTable R5. MSE (multiplied by 100) of Graphon 1. \n | $n_s$ | $n_t$ | GTrans-EGW | GTrans-GW | NS      | Relative Gain (%) |\n | ---- | ---- | ----------------- | --------------- | ------------- | -------------------- |\n | 150  | 100  | **1.40** | 1.57   | 1.83 | 23.50               |\n  | 200  | 100  | **1.39** | 1.58    | 1.83 | 24.04               |\n| 200  | 150  | **1.37** | 1.49  | 1.59 | 13.84               |\n | 250  | 150  | **1.34** | 1.43 | 1.59  | 15.72              |\n     \n> Comment 11: Lines 54-57: Are the two problems (alignment and unsupervised learning) linked?\n\nResponse: These are distinct challenges. **Alignment Challenge:** Absence of known node correspondences between graphs of different sizes can cause structural patterns to be incorrectly aligned.  **Unsupervised Learning Challenge:** Unlike conventional transfer learning with labeled data, graphon estimation relies solely on observed adjacency matrices without ground-truth edge probabilities, preventing explicit loss function definition. \n\n> Comment 12: update references on graphon estimation.\n\nResponse: We have conducted an empirical study comparing our method with GWB, IGNR, and SIGL. Please refer to **Table R3** in response to Reviewer aKUK. These references will be added."}, {"review_id": "XkLU6OSIG7", "reviewer": "Reviewer_81ne", "summary": "This paper proposes to use Gromov-Wasserstein optimal transport to improve graphon estimation in small graphs. The paper gives theoretical guarantees and illustrations using experiments.", "questions": "Please clarify whether the graph is assumed to have selfloops or not. \nWhy is the MSE divided by n^2 when the graphon is symmetric? \nIs a GW optimal coupling unique? Otherwise change `the' to `a', and specify which one you pick.]\nWhat are \\nu and \\mu in your formulation, and what is their product? \nHow do you obtain initial estimates for the GTrans method? \nin the graphon model, the edge indicators are dependent unless the U_i's are given. How is this dependence reflected in the method? \nTheorem 4.1 is phrased in terms of a cutoff (which would be interesting to mention in the main text. Could you also take randomness into account and obtain a `with high probability' statement? \nAsymptotic performance: what is the symbol combination after source data? \nThere is a vast literature on graph classification, including non-parametric methods based on graphlets. It could be interesting to see whether your method improes on these. \nThe method works only for small graphs, but that is clearly explained. It could be interesting to get a quantitative bound in terms of the graph size, or some indication of what counts as small in the paper.", "limitations": "As the  work is theoretical, the fact that  potential societal impact is not discussed is not an issue..", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The approach in the paper makes sense and is theoretically founded. Some presentation issues should be addresses, see the questions below.\n\nThe estimating method assumes that the graph is a small subgraph of a larger source graph with related structural similarities. In particular it is assumed that the initial estimates of the edge probabilities, obtained from the source graph, are informative. This assumption limits the applicability of the method. \n\nIt is a strength of the paper that it makes this assumption clear.It is a standard assumption in transfer learning. \n\nUsing a Gromov-Wasserstein distance makes theoretical sense but it is computationally intensive. The use of an entropic regulariser alleviates the computational complexity, However the theoretical results seem to be derived for the unregularised version (although it is not clearly stated in Theorem 4.1 what the EGW optimization problem is). The assumptions for the theoretical guarantees may not be easy to check as the true distribution is unknown.", "was_revised": true, "final_justification": "The authors have addressed my concerns. In my view this is a good paper which should be accepted.", "rebuttal": "We thank the reviewer for these constructive comments. We will improve figure quality, and add the following clarifications and discussions in the revision. \n\n>Comment 1: Assumption on similarity between source and target\n\nResponse: We agree that the effectiveness of GTRANS depends on the assumption that the target graph exhibits structural similarity to a larger source graph, as is standard in transfer learning. \n\nHowever, we would like to highlight the fact that our method is robust even in the absence of strong source-target similarity. When GW distance is large, our debiasing step ensures GTRANS performs on par with target-only NS, as shown in Table 1 for dissimilar pairs like 8 and 9. This robustness means that, **while structural similarity enhances transfer effectiveness, GTRANS remains a safe and broadly applicable tool even when ideal source-target matching is not available**. We will further clarify in the revised manuscript.\n\n>Comment 2: Theorem 4.1 is for Entropic GW or GW? \n\nResponse:   Theorem 4.1 is for the Entropic Gromov-Wasserstein (EGW) distance instead of GW distance. The $\\epsilon$ in the statement refers to the entropic regularization parameter. We will state it more clearly in the revised version of the manuscript.  \n\n>Comment 3: Validating assumptions of Theorem 4.1\n\nResponse:  Theorem 4.1 relies on two key conditions: (i) $\\epsilon > C_1|\\pi^\\star|_\\infty |\\mathbf{P}_s \\otimes \\mathbf{P}_t|{\\rm op}$ for some $C_1 > 2$, and (ii) the estimation errors for $\\mathbf{P}_s$ and $\\mathbf{P}_t$ must be sufficiently small. \n\nWhile the **first condition** involves unobservable population-level quantities and cannot be directly verified, our simulations suggest it holds for a broad range of $\\epsilon$. **Table E** (please find in our response to Reviewer aKUK) reports computed lower bounds on feasible $\\epsilon$ values (with $C_1 = 2.01$) for Graphons 1–3 (defined in Section E.1), each scaled to have operator norm 1, with $n_s = 100$ and $n_t = 50$. Any $\\epsilon$ above the reported thresholds satisfies the condition, including $\\epsilon = 0.01$ used in our real data analysis.\n\nThe second condition is well-supported by established graphon estimation literature, as it requires $\\hat{\\mathbf{P}}_s^{\\text{ini}}$ and $\\hat{\\mathbf{P}}_t^{\\text{ini}}$ to be consistent. For example, Ref. [62] shows that NS-based estimators achieve rates of $\\sqrt{\\log n_s / n_s}$ and $\\sqrt{\\log n_t / n_t}$ for the source and target, respectively.\n\n>Comment 4. **Whether selfloops or not, Why is the MSE divided by n^2**: \n\nResponse: **Self-loops:** Our theoretical results can be applied to both networks with and without self-loops. In our simulation experiments, for simplicity, we conducted exepriments on networks without self-loops. Our real-data also does not contain self-loop. \n\n**MSE Normalization:** The MSE is calculated as the average squared error over all $n^2$ entries in the probability matrix, following conventions established in prior graphon estimation studies [1-3]. While the graphon is symmetric, this normalization is widely adopted for consistency and comparability across the literature. \n \n[1] Zhang, Y., Levina, E., & Zhu, J. (2017). Estimating network edge probabilities by neighbourhood smoothing. Biometrika, 104(4), 771-783.\n\n[2] Chan, S., & Airoldi, E. (2014). A consistent histogram estimator for exchangeable graph models. In International Conference on Machine Learning (pp. 208-216). PMLR.\n\n[3] Qin, Y., Yu, L., & Li, Y. (2021). Iterative connecting probability estimation for networks. Advances in Neural Information Processing Systems, 34, 1155-1166.\n\n\n>Comment 5. **GW optimal coupling uniqueness**: \n\nResponse:  In general, the Gromov-Wasserstein (GW) optimization problem is **non-convex**, so multiple optimal or near-optimal coupling matrices may exist in theory. To clarify our implementation, we use the `ot.gromov_wasserstein` function from the POT library [1] in Python to compute the GW distance and corresponding coupling. While the GW problem may have multiple optima, the POT algorithm deterministically converges to a single stationary point for fixed inputs and initialization. This ensures reproducibility, though not mathematical uniqueness. We will revise the manuscript to state \"an optimal coupling\" and clarify.\n\n[1] Flamary, R., Courty, N., Gramfort, A., et al. (2021). POT: Python Optimal Transport. *Journal of Machine Learning Research*, 22(78), 1–8.\n\n>Comment 6. **\\nu and \\mu**: \n\nResponse:  In our formulation, $\\mu$ and $\\nu$ denote the uniform measures on the source and target nodes, respectively; that is, $\\mu = (\\underbrace{n\\_S^{-1}, \\dots, n\\_S^{-1}}\\_{:= n\\_S \\text{ times}})$ and $\\nu = (\\underbrace{n\\_T^{-1}, \\dots, n\\_T^{-1}}\\_{:= n\\_T \\text{ times}})$. Their product, $\\mu \\otimes \\nu$, is the product measure, which assigns a uniform weight of $(n_S n_T)^{-1}$ to each possible pair consisting of a source and a target node.\n\n>Comment 7. **How to obtain initial estimates?**:  \n\nResponse:  **Initial Estimates:** We obtain the initial edge probability matrices  using the **neighborhood smoothing (NS)** method  [1] (see lines 60–63 in our paper). \n\n**Dependence in the Graphon Model:** You are correct that, in the graphon model, edge indicators such as $A_{ij}$ and $A_{ik}$ are dependent because they share the same latent variable $u_i$. Rather than ignoring or explicitly modeling these dependencies, NS actually **leverages** them. For each node $i$, NS defines a neighborhood $\\mathcal{N}_i$ consisting of nodes $j$ whose latent positions $u_j$ are close to $u_i$, which means their edge probabilities pattern are similar. Since the latent variables $u_j$ are unobserved, in practice, NS approximates $\\mathcal{N}_i$ by identifying nodes $j$ whose observed connectivity patterns (i.e., rows of the adjacency matrix $A$) are close to that of node $i$ under the $L_2$ norm. NS averages edge values within empirically defined neighborhoods, pooling information from correlated edges to consistently estimate edge probabilities—much like kernel smoothing in nonparametric regression. \n\n[1] Zhang, Y., Levina, E., & Zhu, J. (2017). Estimating network edge probabilities by neighborhood smoothing. *Biometrika*, 104(4), 771–783.\n\n>Comment 8. **Theorem 4.1: Consider adding the cutoff and strengthening it to a high-probability statement**:\n\nResponse: Indeed as the estimation error $||\\hat{\\mathbf{P}}\\_s^{ini} - \\mathbf{P}\\_s||\\_\\infty$ and $||\\hat{\\mathbf{P}}\\_t^{ini} - \\mathbf{P}\\_t||\\_\\infty$ decays at the order $\\sqrt{\\log{n\\_s}/n\\_s}$ and $\\sqrt{\\log{n\\_t}/n\\_t}$ respectively under certain assumptions on the underlying graphon (e.g., see Theorem 2 of \\cite{}), we can leverage this type of result to argue that the conclusion of our theorem holds with probablity $\\ge 1 - C(n\\_s \\wedge n\\_t)^{-c}$ for some constant $C , c > 0$. We will update the description of the theorem in the revised version of the manuscript as per your suggestion. \n\n>Comment 9. **symbol after \"source data.\"**\n\nResponse:  We are not entirely certain what is meant by “symbol combination after source data.” We assume you are referring to the Kronecker product notation (e.g., $\\mathbf{P}_s \\otimes \\mathbf{P}_t$) that appears in Theorem 4.1. In particular, the Kronecker product between two matrices is a way of combining them to create a larger matrix. This operation expands each entry of the first matrix by multiplying it with the entire second matrix, capturing all pairwise interactions between elements—such as nodes in source and target graphs. In our context, it enables comprehensive comparison within the Gromov-Wasserstein framework. If this doesn't fully address your question, we’d appreciate further clarification.\n\n>Comment 10. **Comparing graphlet-based methods on graph classifcation**: \n\nResponse:  In response, we have included a comparison between our method and a representative graphlet-based graph classification approach. Specifically, we implemented the Graphlet Kernel method.  We use the Graphlet Kernel from the GraKeL library in Python to compute pairwise graph similarities, followed by SVM classification.  As shown in Table R6, our transfer-based graphon method significantly outperforms Graphlet on  two target data (IMDB-BINARY and IMDB-MULTI), while achieving comparable performance as Graphlet on PROTEINS target data. \n\nTable R6: Graph classification accuracy (\\%) across three target datasets \n| Source   | Target   | GTRANS-GW        | GTRANS-EGW       | Graphlet          | \n| -------- | -------- | ---------------- | ---------------- | ------------ \n| Reddit-B | IMDB-B   | 76.30     | **76.80** | 61.10  | \n| COLLAB   | IMDB-B   | 76.25  | **77.50** | 61.10 | \n| Reddit-B | IMDB-M   | 49.10     | **51.27** | 39.37  | \n| COLLAB   | IMDB-M   | **50.47** | 50.23    | 39.37  | \n| D\\&D     | PROTEINS | **69.33** | 68.52     | 70.11\n\n>Comment 11: **quantitative bound for how “small” graph means**:\n\nResponse: Defining universal bounds on \"small\" $n$ is challenging as it depends on graphon complexity and source-target similarity rather than fixed thresholds. Complex graphons require larger $n_t$. Existing works provide asymptotic error rates (e.g., $O(\\sqrt{\\log n / n})$ for NS method) but lack specific \"small\" cutoffs.\n\nTo provide practical guidance, we ran simulations with Graphon 1, fixing $n_s = 1000$ and varying $n_t$ from 100 to 700 (**Table R4**, in our response to Reviewer JN9a).  The relative gain from transfer learning decreases as $n_t$ increases but remains positive; even at $n_t = 700$, GTRANS still outperforms target-only NS, showing that transfer learning provides substantial benefit even for moderately large target graphs."}]

Author rebuttal:

Meta-review: The paper presents GTRANS, a method designed to estimate the graphon matrix underlying a small target graph by transferring knowledge from a larger source graph with similar properties. The approach begins by estimating graphons for both the small target graph and the large source graph using neighbor smoothing. It then computes the Gromov-Wasserstein (GW) distance between these two graphons and uses the resulting alignment matrix to map the source graphon's structure onto the latent domain of the target graph. Finally, the method assesses potential domain shift by comparing the transferred graphon to the initially estimated one and smooths the residual if a significant difference is detected. The paper gives theoretical guarantees and illustrations using experiments.

Strengths

S1 Three reviewers comment that the paper is well-written and easy to follow, and each step is clearly explained.

S2 The idea of estimating the graphon for a small network using a large network is novel and interesting.

S3 The proposed method is grounded in some theoretical guarantees.

S4 The results of the proposed method are better than the baselines.

S5 The paper clearly states the assumption that the graph is a small subgraph of a larger source graph with related structural similarities. Although this may limit its applicability, it is crucial to mention this in the paper.


Weaknesses

W1. The paper assumes that the small graph is similar to the large graph, but it does not provide a method to corroborate this assumption.

W2. The paper considers that the graph has a small number of nodes. Unfortunately, the definition of small is unclear throughout the paper, and it could differ among domains. There are no theoretical results that establish sample settings (n_s,n_t) and assumptions on the underlying target and source graphons.

W3. The theoretical guarantees are not empirically proven. Although this verification is challenging due to the unknown true distribution, synthetic data could be used as an alternative.

W4. The experiment and conclusions could have benefited from the use of a larger graph on the same dataset. For example, comparing the results using a large graph from the same small network, against a large one from a different network, or even combining both of them.

W5. Other baselines could have been included, such as resolution-free graphon estimation techniques that estimate the graphon from a single graph or a set of graphs. 

W6. The proposed method is computationally expensive because of the Gromov-Wasserstein distance. Even though the use of an entropic regulariser alleviates the computational complexity, the theoretical results seem to be based on the original Gromov-Wasserstein distance.

Most authors consider the paper easy to read, featuring a novel problem and solution. Also, the theoretical guarantees, despite some concerns, support the paper's acceptance. Besides the theoretical concerns, most reviewers would have expected a more helpful methodology to figure out when to apply this model, and how to choose a similar, larger graph. 

Regarding the discussion, the authors included new results comparing against other baselines, new datasets, and defined some similarity measures for comparison between graphs. All the reviews expressed gratitude for the responses, with some of them showing an increase in their final score.
Decision: Accept (poster)