Converse and Collision-Based Achievability for Node Localization with Hybrid Distance-Spectral Graph Positional Encodings

Positional encodings (PEs) enhance the power of graph neural networks (GNNs), both theoretically and empirically. Two of the most popular families of PEs - spectral (e.g., Laplacian eigenspaces, effective resistance) and walk-based (polynomials of the adjacency matrix) - are theoretically equivalent in expressive power, with expressivity between the 1-WL and 3-WL tests. However, this equivalence assumes the GNN uses the "complete" version of these PEs, which requires $O(n^3)$ time and space complexity. Instead, practitioners commonly use truncated variants of these encodings, such as the first $k$ eigenspaces or powers of the adjacency matrix. However, the theoretical properties of these truncated PEs are unknown. In this work, we initiate the study of these truncated PEs. Theoretically, we show that, under truncation, several families of PEs are fundamentally different in expressive power. As a corollary, we show that truncated spectral PEs are no longer stronger than the 1-WL test. We also study a family of spectral PEs, the $k$-harmonic distances, to highlight the differences in expressive power of even closely related truncated PEs. Finally, we experimentally show that a mix of truncated PEs is preferable to any single family on real-world datasets.

Spectral positional encodings (PEs) for \emph{directed} graphs face two obstacles: magnetic Laplacians require an $O(n^3)$ Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gauge, which prior work handles with basis-invariant architectures. We propose learnable spectral PEs of the form $h_θ(A_q)\,R$, where $A_q$ is a normalized magnetic operator, $h_θ$ a learnable scalar spectral response, and $R$ a block of random probes. Because the PE is a \emph{matrix function} of the operator, it is gauge-invariant by construction. We compute it in a Hermitian block Krylov subspace from sparse matrix--vector products only, prove that $k = O(\log(1/\varepsilon))$ block steps suffice uniformly over heat--resolvent response families, and give a covering-number argument for why low-dimensional structured families generalize where free per-eigenvalue weights overfit. On a directed SBM whose symmetrization is uninformative by construction, direction-blind PEs stay at chance while magnetic Krylov PEs converge to the exact-eigendecomposition oracle as the depth grows. The same probes yield gauge-invariant pairwise features with $1/\sqrt{s}$ Monte-Carlo error, and the undirected $q{=}0$ case improves heterophilous benchmarks over no-PE and polynomial baselines.

Message-passing graph neural networks are bounded by the 1-WL test and can miss topological structure that distinguishes non-isomorphic graphs. Positional and structural encodings (PSE) inject such topology-derived signals, and learned PSE encoders such as GPSE pretrain a single encoder to produce these signals from random node probes, which can then be frozen and reused as inputs across downstream graph models. We present CondPSE, a learned PSE encoder that applies a learnable polynomial graph filter bank to standard Gaussian node probes and refines the resulting structural-response branches through FiLM-style modulation conditioned on cross-filter, local message-passing, and graph-level signals. CondPSE is pretrained to reconstruct node-level positional/structural targets and graph-level invariants, and is then frozen for use as a downstream input encoding. On synthetic structural-discrimination benchmarks, CondPSE separates graph structures that 1-WL-bounded message passing cannot: it raises CSL accuracy from 42.9% to 97.3% and EXP accuracy from 68.3% to 99.9% relative to GPSE, and ablations show that the polynomial filter bank accounts for most of this gain. On real molecular property prediction, the picture is more limited. With a hybrid local-message-passing/global-attention backbone, CondPSE performs comparably to GPSE without surpassing it, and a ZINC backbone sweep shows no consistent ordering between the two encoders. We report these results and discuss why strong synthetic structural discrimination does not, on its own, yield a downstream advantage for frozen learned PSE encoders, including the role of downstream integration and possible mismatch between structural pretraining targets and molecular property labels.