Placing Degree Scales After LayerNorm

Abstract

Graph neural networks (GNNs) are widely used to learn node-selection policies on graphs, and most stack graph attention (GAT) blocks with LayerNorm. On degree-sensitive tasks, LayerNorm tends to remove the degree signal these models need to rank nodes. Much recent work addresses this by redesigning normalizers or aggregators, which changes what these components compute but does not ask where, relative to LayerNorm, a degree scale should be applied. In this paper, we show that the answer follows from a single algebraic fact about LayerNorm. When a positive per-node scale is applied before LayerNorm, LayerNorm divides it out, and it never reaches the model's output. Applied after LayerNorm, the same scale comes through and reaches the score head as magnitude. From this placement rule we derive PostDeg, a parameter-free inverse-degree scale that we add as the single change to a fixed GAT backbone. PostDeg multiplies each node's normalized representation by an inverse function of its degree, and we compare it against controls in the same position. At every evaluation size, PostDeg improves over the LayerNorm backbone on influence maximization, network dismantling, and maximum independent set, and these controls show where the improvement comes from. The same scale before LayerNorm stays at the backbone, as the absorption identity predicts, and a constant scale after LayerNorm stays there too on all but the most heavy-tailed graphs, so the effect needs both the position after LayerNorm and a degree-dependent scale. The exact form of the scale matters much less, so we recommend PostDeg, which needs no tuning.