Looping in Context: Node Classification with Looped Prior-Fitted Networks

Prior-fitted networks (PFNs) for graphs classify nodes on arbitrary graphs in a single forward pass, yet how their predictions form across depth is unknown. For tabular foundation models, mechanistic work shows that depth is iterative *refinement*, so one transformer block, looped, recovers a full-depth model. We ask whether this holds for graphs, where every layer also performs one hop of *propagation*. Six mechanistic experiments on the 12-layer NodePFN over 23 benchmarks show that depth plays two roles. Attention transfers the labels in the first layer and is redundant and self-repairing in later layers. Message passing stays necessary on homophilous graphs, where the loss from a removed hop persists to the last layer. The depth at which predictions form therefore grows with homophily, in NodePFN and in a second PFN for graphs, GraphPFN, and a two-class analysis explains why no fixed depth can be optimal across homophily levels. We propose **Looped NodePFN**, a single block applied repeatedly, whose depth is selected per graph *in context* from held-out labeled nodes, with no gradient steps and no extra parameters. With a tenth of the parameters it reaches 66.3% mean accuracy against 66.6% for the 12-layer model, and a two-layer block reaches 66.5%. Once its depth is chosen, the one-layer block predicts up to 5.1 times faster. A PFN for graphs thus needs only one block, but how many times it is applied, and in how many of those iterations it passes messages, should follow the graph.