Transfer Learning on Edge Connecting Probability Estimation under Graphon Model

TL;DR

This work tackles the problem of estimating latent edge probabilities in small graphs by transferring structure from a larger, related source graph within the graphon framework. It introduces GTrans, a three-step method that uses neighborhood smoothing for initial graphon estimates, Gromov-Wasserstein (GW) or entropic GW (EGW) to align source and target structures without node correspondences, and an adaptive debiasing step to prevent negative transfer. Theoretical results establish stability of the alignment under perturbations and show preservation of target structure via the GW-based projection, while extensive simulations and real-data experiments demonstrate improved estimation accuracy and enhanced downstream tasks such as graph classification and link prediction. The approach enables robust transfer-based graphon estimation, with practical implications for data-scarce networks and downstream data-augmentation applications.

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.

Figures (10)

• Figure 1: Workflow of $\textsc{GTrans}$. Initial estimates from source and target networks are aligned via optimal transport. If source-target distance $d < \delta$, the smoothed transferred estimate is returned. Otherwise, debiasing step is applied to produce the final output.
• Figure 2: Visualization of alignment matrix.
• 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.
• Figure 4: MSE performance of five methods as source network size increases from 100-1000 nodes, with error bars representing $\pm 0.1$ standard deviations. $\hbox{GTrans-GW}$ (red circles with solid line), $\hbox{GTrans-EGW}$ (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.
• Figure 5: Visual comparison of $\textsc{GTrans}$ performance with different source graphons when estimating target graphon 8. (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 ($8 \rightarrow 8, 7 \rightarrow 8$ respectively).

Theorems & Definitions (13)

• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Remark 4.4
• Definition B.1
• Lemma B.2
• proof
• Lemma B.3
• proof
• Lemma B.4