On the Convergence and Size Transferability of Continuous-depth Graph Neural Networks

• URL: http://arxiv.org/abs/2510.03923v1
• Date: Sat, 04 Oct 2025 19:59:21 GMT
• Title: On the Convergence and Size Transferability of Continuous-depth Graph Neural Networks
• Authors: Mingsong Yan, Charles Kulick, Sui Tang,
• Abstract summary: Continuous-depth graph neural networks (GNNs) combine the structural inductive bias of Graph Neural Networks (GNNs) with the continuous-depth architecture of Graph Neural Differential Equations (ODEs)<n>We present a rigorous convergence analysis of NeuralEs with time parameters in the infinite-node limit, providing theoretical insights into their size transferability.
• Score: 3.3390650922528184
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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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