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 degree-normalized means. We extend such results to a very large class of aggregation functions, that encompasses all classically used message passing graph neural networks, such as attention-based mesage passing or max convolutional message passing on top of (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, we treat the case where the aggregation is a coordinate-wise maximum separately, at it necessitates a very different proof technique and yields a qualitatively different convergence rate.

翻译：我们研究了消息传递图神经网络在随机图模型上随节点数量趋于无穷时向其连续对应模型的收敛性。此前，这种收敛性仅对使用度归一化均值形式的聚合函数的架构成立。我们将此类结果扩展至极为广泛的聚合函数类，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，以及在（度归一化）卷积消息传递之上的最大卷积消息传递。在温和假设下，我们以高概率给出非渐近界限以量化这种收敛性。我们的主要结果基于McDiarmid不等式。有趣的是，我们分别处理了聚合函数为坐标极大值的情况，因为它需要截然不同的证明技巧，并产生性质上不同的收敛速率。