We revisit the classical result of Morris et al.~(AAAI'19) that message-passing graphs neural networks (MPNNs) are equal in their distinguishing power to the Weisfeiler--Leman (WL) isomorphism test. Morris et al.~show their simulation result with ReLU activation function and $O(n)$-dimensional feature vectors, where $n$ is the number of nodes of the graph. Recently, by introducing randomness into the architecture, Aamand et al.~(NeurIPS'22) were able to improve this bound to $O(\log n)$-dimensional feature vectors, although at the expense of guaranteeing perfect simulation only with high probability. In all these constructions, to guarantee equivalence to the WL test, the dimension of feature vectors in the MPNN has to increase with the size of the graphs. However, architectures used in practice have feature vectors of constant dimension. Thus, there is a gap between the guarantees provided by these results and the actual characteristics of architectures used in practice. In this paper we close this gap by showing that, for \emph{any} non-polynomial analytic (like the sigmoid) activation function, to guarantee that MPNNs are equivalent to the WL test, feature vectors of dimension $d=1$ is all we need, independently of the size of the graphs. Our main technical insight is that for simulating multi-sets in the WL-test, it is enough to use linear independence of feature vectors over rationals instead of reals. Countability of the set of rationals together with nice properties of analytic functions allow us to carry out the simulation invariant over the iterations of the WL test without increasing the dimension of the feature vectors.

翻译：我们重新审视Morris等人（AAAI'19）的经典结论：消息传递图神经网络（MPNNs）的区分能力等价于Weisfeiler-Leman（WL）同构测试。Morris等人通过ReLU激活函数和$O(n)$维特征向量（其中$n$为图的节点数）证明了该模拟结果。近期，Aamand等人（NeurIPS'22）通过向架构中引入随机性，将这一界改进至$O(\log n)$维特征向量，但代价是仅能以高概率保证完美模拟。在这些构造中，为确保与WL测试的等价性，MPNN中特征向量的维度必须随图规模增大而增加。然而实际使用的架构通常采用恒定维度的特征向量。因此，现有理论保证与实际架构特性之间存在差距。本文通过证明：对于任意非多项式解析激活函数（如Sigmoid），仅需维度$d=1$的特征向量即可保证MPNNs与WL测试等价（无需依赖图规模），从而弥合了这一差距。我们的核心技术洞察在于：模拟WL测试中的多重集时，只需利用特征向量在有理数域（而非实数域）上的线性独立性即可。有理数集的可数性结合解析函数的优良性质，使我们能够在WL测试的迭代过程中维持模拟不变性，且无需增加特征向量的维度。