Communicability-Inspired Positional Encoding (CIPE)

Positional encodings (PEs) are essential for Transformers. Yet designing effective PEs for non-Euclidean graphs remains challenging. Such encodings should ideally induce an Attention-Compatible Geometry for self-attention: not merely describing graph structure, but defining a geometry whose inner products reflect meaningful structural relatedness. To realize this geometry, we propose Communicability-Inspired Positional Encoding (CIPE), built from communicability, a measure between pairs of nodes that aggregates contributions from paths of all lengths. By construction, CIPE inner products recover communicability, converting global multi-path connectivity into an attention-ready similarity geometry. For practical Transformer training, we introduce dimensionality alignment, mapping graph-size-dependent CIPE representations to prescribed dimensions while faithfully preserving the induced geometry. Empirically, CIPE improves structure-agnostic Transformers by 35.5% on average across seven benchmarks, outperforming representative PEs; it also consistently improves structure-biased graph Transformers, where competing PEs often yield only marginal benefits. These results position CIPE as a principled framework for attention-compatible graph positional encodings.

Collapsed Effective Operators for Higher-order Structures

Higher-order structures are powerful relational modeling tools, yet existing spectral operators decompose the topology into separate ranks, leaving practitioners to fuse the information back to vertices through ad hoc choices. We introduce Collapsed Effective Operators, which condense higher-order degrees of freedom into a single vertex-level operator via Schur complementation of a graded Laplacian. This yields a (generally dense) operator that encodes long-range interactions mediated by topology and is applicable to arbitrary higher-order constructs. We show it preserves positive semi-definiteness with a spectral upper bound relative to the rank-0 Hodge Laplacian, effectively lowering system energy under higher-order connectivity. Empirically, our operator improves spectral clustering, signal smoothing, and enables the inclusion of topological features in neural network architectures via positional encoding. The project page can be found http://circle-group.github.io/research/CollapsedEffectiveOperators