In this article we present new results about the expressivity of Graph Neural Networks (GNNs). We prove that for any GNN with piecewise polynomial activations, whose architecture size does not grow with the graph input sizes, there exists a pair of non-isomorphic rooted trees of depth two such that the GNN cannot distinguish their root vertex up to an arbitrary number of iterations. The proof relies on tools from the algebra of symmetric polynomials. In contrast, it was already known that unbounded GNNs (those whose size is allowed to change with the graph sizes) with piecewise polynomial activations can distinguish these vertices in only two iterations. Our results imply a strict separation between bounded and unbounded size GNNs, answering an open question formulated by [Grohe, 2021]. We next prove that if one allows activations that are not piecewise polynomial, then in two iterations a single neuron perceptron can distinguish the root vertices of any pair of nonisomorphic trees of depth two (our results hold for activations like the sigmoid, hyperbolic tan and others). This shows how the power of graph neural networks can change drastically if one changes the activation function of the neural networks. The proof of this result utilizes the Lindemann-Weierstrauss theorem from transcendental number theory.

翻译：本文提出了图神经网络（GNN）表达能力的新结果。我们证明，对于任何使用分段多项式激活函数且架构规模不随图输入大小变化的GNN，总存在一对深度为二的非同构有根树，使得该GNN无法在任意次迭代内区分它们的根节点。该证明依赖于对称多项式代数的工具。相比之下，已有研究表明，使用分段多项式激活函数的无界GNN（即规模允许随图大小变化的GNN）仅需两次迭代即可区分这些顶点。我们的结果揭示了有界与无界规模GNN之间的严格分离，回答了[Grohe, 2021]提出的一个开放问题。接着，我们证明，若允许使用非分段多项式的激活函数，则在两次迭代内，单个神经元感知器即可区分任意一对深度为二的非同构有根树的根节点（该结果适用于sigmoid、双曲正切等激活函数）。这表明，图神经网络的表达能力可随神经网络激活函数的改变而发生根本性变化。该结果的证明利用了超越数论中的林德曼-魏尔斯特拉斯定理。