Title: Transfer Learning on Edge Connecting Probability Estimation Under Graphon Model
Venue: NeurIPS 2025
Accepted: yes
Keywords: Graphon Estimation

Transfer Learning

Gromov-Wasserstein Transport

Neighborhood Smoothing

Adaptive Debiasing
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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes to use Gromov-Wasserstein optimal transport to improve graphon estimation in small graphs. The paper gives theoretical guarantees and illustrations using experiments.\n\nSTRENGTHS AND WEAKNESSES:\nThe 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.\n\nQUESTIONS:\nPlease 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.\n\nLIMITATIONS:\nAs the  work is theoretical, the fact that  potential societal impact is not discussed is not an issue..", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: 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.\n\nQUESTIONS:\nMajor:\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\n\nLIMITATIONS:\nYes. However, limitations within the framework studied in the paper are missing and could be added.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\n**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.\n\nQUESTIONS:\n**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?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\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\".\n\nQUESTIONS:\nPlease refer to the questions mentioned in the weaknesses above.\n\nLIMITATIONS:\nyes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}