[论文解读] Graphon Neural Networks and the Transferability of Graph Neural Networks

Luana Ruiz, Luiz F. O. Chamon+1 more|arXiv (Cornell University)|Jun 5, 2020

Advanced Graph Neural Networks参考文献 31被引用 43

一句话总结

paper 将 graphon neural networks (WNNs) 定义为 GNNs 的极限对象，并证明一个界限，显示 GNNs 在同一个 graphon 生成的确定性图之间的可迁移性，误差在带限滤波下随 n^{-1/2} 下降，揭示 transferability 与 discriminability 之间的权衡。

ABSTRACT

Graph neural networks (GNNs) rely on graph convolutions to extract local features from network data. These graph convolutions combine information from adjacent nodes using coefficients that are shared across all nodes. Since these coefficients are shared and do not depend on the graph, one can envision using the same coefficients to define a GNN on another graph. This motivates analyzing the transferability of GNNs across graphs. In this paper we introduce graphon NNs as limit objects of GNNs and prove a bound on the difference between the output of a GNN and its limit graphon-NN. This bound vanishes with growing number of nodes if the graph convolutional filters are bandlimited in the graph spectral domain. This result establishes a tradeoff between discriminability and transferability of GNNs.

研究动机与目标

• Motivate transfer learning for GNNs across graphs of varying sizes but similar structure.
• Introduce graphon-based limit objects (WNNs) to study GNN transferability.
• Establish rigorous bounds on the difference between GNN outputs on different graphs from the same graphon.
• Highlight how filter bandlimiting affects the transferability-discriminability trade-off.
• Provide theoretical and numerical insights into when GNNs can generalize across graph variants.

提出的方法

• Define graph convolutions via the graph shift operator and spectral representation.
• Introduce graphons as limit objects and graphon data with the graphon shift operator (WSO).
• Formulate graphon neural networks (WNNs) using graphon convolutions and nonlinear activations.
• Show WNNs as deterministic generating models for GNNs and vice versa by mapping graphs to graphons and signals to graphon signals.
• Derive approximation bounds between WNN outputs and GNN outputs on deterministic graphs constructed from a graphon (Theorem 1).
• Characterize transferability of GNNs across graphs of different sizes drawn from the same graphon (Theorem 2).

实验结果

研究问题

• RQ1 Can GNNs trained on graphs of a given size transfer to other graphs from the same graphon family with performance guarantees?
• RQ2 How does the graphon structure and filter bandwidth affect the transferability and discriminability of GNNs?
• RQ3 What is the asymptotic behavior of the transfer error as graph size grows on deterministic graphs sampled from a graphon?

主要发现

• GNNs are transferable between deterministic graphs obtained from the same graphon, with output difference bounded by a term that decays as the graph size grows.
• The transferability bound scales with a factor LF^{L-1} sqrt(A1) (A2 + pi nc / delta_c) times (n1^{-1/2} + n2^{-1/2}).
• A fixed error term depends on the graphon signal variability A3 and also decays as n^{-1/2}.
• There is a trade-off: smaller passing-band (n_c) yields better transferability but lower discriminability, and vice versa.
• WNNs provide a generating model for GNNs, enabling viewing graph structure as a learnable architectural parameter; the results hold for deterministic graphs from graphons.
• Numerical results on MovieLens and Cora illustrate the transferability bound in practice, showing reduced output discrepancy as graph size increases.