MapPerturb: Dimension and Direction in Compact GNN Adaptation

Low-dimensional weight maps reduce the trainable parameters of a GNN, but their coordinate count does not determine which prediction changes they can realize. We study how update directions affect the benefit of adding coordinates. MapPerturb parameterizes weight updates around a pretrained anchor through a fixed affine map. We analyze first-order changes in centered logits, separating the dimension of the reachable response space from its alignment with a specified target response. On three citation graphs, gradient directions yield larger reconstruction gains from additional coordinates than random directions at matched weight-update budgets. This gradient-direction advantage persists under direct radius-constrained fitting. Additionally, weight-space retention and held-out response reconstruction also rank the tested maps differently. Low-rank gradient factors retain a reconstruction advantage over spectrum-matched random factors, use $12.97$–$15.46\times$ fewer stored basis values than dense maps, and match or exceed LoRA, VeRA, and full encoder tuning on further graphs. In transductive few-shot transfer, directions constructed from DGI or masked-feature reconstruction without task labels improve pooled accuracy over head-only adaptation by $2.65$ and $3.19$ percentage points, respectively, with eight trainable encoder coordinates and a full linear head.