提出连续深度图网络的收敛理论，证明模型可跨规模迁移。
On the Convergence and Size Transferability of Continuous-depth Graph Neural Networks
- 用图核微分方程建模无限节点图网络，推导其解的稳定性。
- 在两类图采样下证明解的逐轨迹收敛，给出明确收敛速率。
- 首次提供连续深度图网络跨规模迁移的理论依据，适合图学习研究者。
连续深度图神经网络（即图神经微分方程，GNDEs）结合了图神经网络的结构归纳偏置与神经微分方程的连续深度架构，为图上动态建模提供了可扩展且理论严谨的框架。本文针对带时变参数的GNDEs，在无限节点极限下进行了严格的收敛性分析，揭示其规模可迁移性。为此，我们引入图核微分方程（Graphon-NDEs）作为GNDEs的无限节点极限，并证明其适定性。借助图核理论与动力系统工具，我们证明了GNDE解到Graphon-NDE解的逐轨迹收敛性。进一步，在两种确定性图采样情形下：(1) 来自平滑图核的加权图；(2) 来自{0,1}值（不连续）图核的无权图，给出了显式的收敛速率。我们还建立了规模可迁移性边界，为将中等规模图上训练的GNDE模型直接迁移至更大、结构相似的图而无需重新训练提供了理论支持。合成与真实数据上的数值实验验证了理论结论。
原文摘要 · Abstract (English)
Continuous-depth graph neural networks, also known as Graph Neural Differential Equations (GNDEs), combine the structural inductive bias of Graph Neural Networks (GNNs) with the continuous-depth architecture of Neural ODEs, offering a scalable and principled framework for modeling dynamics on graphs. In this paper, we present a rigorous convergence analysis of GNDEs with time-varying parameters in the infinite-node limit, providing theoretical insights into their size transferability. To this end, we introduce Graphon Neural Differential Equations (Graphon-NDEs) as the infinite-node limit of GNDEs and establish their well-posedness. Leveraging tools from graphon theory and dynamical systems, we prove the trajectory-wise convergence of GNDE solutions to Graphon-NDE solutions. Moreover, we derive explicit convergence rates under two deterministic graph sampling regimes: (1) weighted graphs sampled from smooth graphons, and (2) unweighted graphs sampled from $\{0,1\}$-valued (discontinuous) graphons. We further establish size transferability bounds, providing theoretical justification for the practical strategy of transferring GNDE models trained on moderate-sized graphs to larger, structurally similar graphs without retraining. Numerical experiments using synthetic and real data support our theoretical findings.
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