Many Graph Neural Networks (GNNs) add self-loops to a graph to include feature information about a node itself at each layer. However, if the GNN consists of more than one layer, this information can return to its origin via cycles in the graph topology. Intuition suggests that this "backflow" of information should be larger in graphs with self-loops compared to graphs without. In this work, we counter this intuition and show that for certain GNN architectures, the information a node gains from itself can be smaller in graphs with self-loops compared to the same graphs without. We adopt an analytical approach for the study of statistical graph ensembles with a given degree sequence and show that this phenomenon, which we call the self-loop paradox, can depend both on the number of GNN layers $k$ and whether $k$ is even or odd. We experimentally validate our theoretical findings in a synthetic node classification task and investigate its practical relevance in 23 real-world graphs.

翻译：许多图神经网络（GNN）会在图中添加自环，以在每一层包含节点自身的特征信息。然而，若GNN由多层组成，该信息可能通过图拓扑中的循环返回其原点。直觉表明，与不含自环的图相比，含自环的图中这种信息的“回流”应更显著。在本工作中，我们反驳这一直觉，证明对于某些GNN架构，节点从自身获取的信息量在含自环的图中可能小于不含自环的同一图。我们采用解析方法研究具有给定度序列的统计图集合，并表明这一我们称之为自环悖论的现象，可能同时取决于GNN层数 $k$ 及 $k$ 的奇偶性。我们通过合成节点分类任务实验验证了理论发现，并在23个真实图数据集中探究其实际相关性。