We study the convergence of message passing graph neural networks on random graph models to their continuous counterpart as the number of nodes tends to infinity. Until now, this convergence was only known for architectures with aggregation functions in the form of normalized means, or, equivalently, of an application of classical operators like the adjacency matrix or the graph Laplacian. We extend such results to a large class of aggregation functions, that encompasses all classically used message passing graph neural networks, such as attention-based message passing, max convolutional message passing or (degree-normalized) convolutional message passing. Under mild assumptions, we give non-asymptotic bounds with high probability to quantify this convergence. Our main result is based on the McDiarmid inequality. Interestingly, this result does not apply to the case where the aggregation is a coordinate-wise maximum. We treat this case separately and obtain a different convergence rate.

翻译：我们研究了在节点数趋于无穷时，消息传递图神经网络在随机图模型上向其连续极限的收敛性。此前，这一收敛性仅对采用归一化均值形式的聚合函数（等价于应用邻接矩阵或图拉普拉斯等经典算子）的架构成立。我们将此类结果推广至一大类聚合函数，涵盖所有经典使用的消息传递图神经网络，例如基于注意力的消息传递、最大卷积消息传递或（度归一化）卷积消息传递。在温和假设下，我们以高概率给出了非渐近界以量化该收敛性。我们的主要结果基于McDiarmid不等式。有趣的是，该结果不适用于逐坐标最大化的聚合情形。我们单独处理了这一情形，并获得了不同的收敛速率。