一句话：回答 GNN 中依赖源与目标状态的消息版本是否比仅依赖源状态版本表达力更强，证明非一致设置下等价、一致设置下更强。
本文研究图神经网络（GNN）中两种核心消息传递机制的表达力差异：一种消息仅依赖源顶点状态，另一种同时依赖源与目标顶点状态。GNN 的表达力可由带计数的 first-order logic 片段和 Weisfeiler-Lehman 算法刻画；这两种机制分别对应带计数一阶逻辑的 modal 片段与 guarded 片段。作者回答 Grohe（LICS'23）提出的问题：在带标签无向图上，modal 与 guarded 片段表达力相同，因此在非一致（non-uniform）设置下两种 GNN 版本表达力相同；但在一致（uniform）设置下，依赖源与目标状态的第二版严格更强。结论表明 targeted messages 的优势取决于是否允许非一致参数。
Graph neural networks (GNN) are deep learning architectures for graphs. Essentially, a GNN is a distributed message passing algorithm, which is controlled by parameters learned from data. It operates on the vertices of a graph: in each iteration, vertices receive a message on each incoming edge, aggregate these messages, and then update their state based on their current state and the aggregated messages. The expressivity of GNNs can be characterised in terms of certain fragments of first-order logic with counting and the Weisfeiler-Lehman algorithm. The core GNN architecture comes in two different versions. In the first version, a message only depends on the state of the source vertex, whereas in the second version it depends on the states of the source and target vertices. In practice, both of these versions are used, but the theory of GNNs so far mostly focused on the first one. On the logical side, the two versions correspond to two fragments of first-order logic with counting that we call modal and guarded. The question whether the two versions differ in their expressivity has been mostly overlooked in the GNN literature and has only been asked recently (Grohe, LICS'23). We answer this question here. It turns out that the answer is not as straightforward as one might expect. By proving that the modal and guarded fragment of first-order logic with counting have the same expressivity over labelled undirected graphs, we show that in a non-uniform setting the two GNN versions have the same expressivity. However, we also prove that in a uniform setting the second version is strictly more expressive.