Self-Tuning Graph Filters via State-Dependent Operator Composition

Most graph neural networks propagate information through a fixed-coefficient polynomial filter applied uniformly across every node and every layer. However, the appropriate filter varies along two distinct axes that such a parameterization cannot accommodate. Within a single graph, different nodes benefit from different mixtures of low-pass smoothing, high-pass contrast, and pass-through behavior. Across graphs, signal propagation dynamics themselves differ, with some graphs benefiting from long-range propagation across many hops while others require damping to prevent over-smoothing. The appropriate filter is therefore not a single fixed object at all, but a state-dependent composition whose behavior changes according to a node's current representation and its propagation history. We propose Castor, a self-tuning graph filter that realizes this composition through state-dependent decisions made at every step of propagation. A router reads each node's current state together with its initial state, and emits per-node mixing weights over three primitives, namely low-pass smoothing, high-pass contrast, and identity. A learnable per-hop coefficient then determines how much of the most recent change in the propagation is carried forward, accelerating the iteration when positive and damping it when negative. Setting the coefficient to zero recovers standard first-order propagation. Across fourteen standard node-classification benchmarks spanning the full homophily spectrum, a single Castor instance ranks top-three on every dataset and achieves an average rank of $2.07$, more than $3.5$ ranks ahead of the next-best of fourteen baselines.