# HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations

arXiv: https://arxiv.org/abs/2505.15405 · HTML: https://mlsanity.com/paper/2505.15405/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 72 references that are in the map.
- First posted on arXiv 21 May 2025.

## Builds on (most cited first, 30 of 51)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Recipe for a General, Powerful, Scalable Graph Transformer](https://mlsanity.com/paper/2205.12454/) (2022)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [A Generalization of Transformer Networks to Graphs](https://mlsanity.com/paper/2012.09699/) (2020)
- [Hypergraph Neural Networks](https://mlsanity.com/paper/1809.09401/) (2018)
- [Principal Neighbourhood Aggregation for Graph Nets](https://mlsanity.com/paper/2004.05718/) (2020)
- [Rethinking Graph Transformers with Spectral Attention](https://mlsanity.com/paper/2106.03893/) (2021)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Graph Neural Networks with Learnable Structural and Positional Representations](https://mlsanity.com/paper/2110.07875/) (2021)
- [Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting](https://mlsanity.com/paper/2006.09252/) (2020)
- [SIGN: Scalable Inception Graph Neural Networks](https://mlsanity.com/paper/2004.11198/) (2020)
- [Long Range Graph Benchmark](https://mlsanity.com/paper/2206.08164/) (2022)
- [Weisfeiler and Lehman Go Cellular: CW Networks](https://mlsanity.com/paper/2106.12575/) (2021)
- [Vision GNN: An Image is Worth Graph of Nodes](https://mlsanity.com/paper/2206.00272/) (2022)
- [Training Graph Neural Networks with 1000 Layers](https://mlsanity.com/paper/2106.07476/) (2021)
- [Graph Inductive Biases in Transformers without Message Passing](https://mlsanity.com/paper/2305.17589/) (2023)
- [Relational Pooling for Graph Representations](https://mlsanity.com/paper/1903.02541/) (2019)
- [Topological Deep Learning: Going Beyond Graph Data](https://mlsanity.com/paper/2206.00606/) (2022)
- [Decoupling the Depth and Scope of Graph Neural Networks](https://mlsanity.com/paper/2201.07858/) (2022)
- [Equivariant and Stable Positional Encoding for More Powerful Graph Neural Networks](https://mlsanity.com/paper/2203.00199/) (2022)
- [How Powerful are K-hop Message Passing Graph Neural Networks](https://mlsanity.com/paper/2205.13328/) (2022)
- [Cell Complex Neural Networks](https://mlsanity.com/paper/2010.00743/) (2020)
- [What are higher-order networks?](https://mlsanity.com/paper/2104.11329/) (2021)

## Built on by (most cited first, 5 of 5)

- [Have Graph -- Will Lift? The Case for Higher-Order Benchmarks](https://mlsanity.com/paper/2605.07397/) (2026)
- [No Triangulation Without Representation: Generalization in Topological Deep Learning](https://mlsanity.com/paper/2605.06467/) (2026)
- [Look Before You Lift: Visual and Quantitative Diagnostics for Topological Deep Learning](https://mlsanity.com/paper/2608.15388/) (2026)
- [Higher-Order Positional Encodings for Graph Representation Learning](https://mlsanity.com/paper/2610.01903/) (2026)
- [Batch Before You Lift: Scalable Topological Deep Learning on Large Graphs](https://mlsanity.com/paper/2610.12247/) (2026)

## Abstract (arXiv)

While Graph Neural Networks (GNNs) have proven highly effective at modeling relational data, pairwise connections cannot fully capture multi-way relationships naturally present in complex real-world systems. In response to this, Topological Deep Learning (TDL) leverages more general combinatorial representations--such as simplicial or cellular complexes--to accommodate higher-order interactions. Existing TDL methods often extend GNNs through Higher-Order Message Passing (HOMP), but face critical scalability challenges due to the steep complexity overhead of propagating messages through combinatorial structures. To overcome this limitation, we propose HOPSE (Higher-Order Positional and Structural Encoder), a framework free of message passing layers that uses Hasse graph decompositions to derive efficient and expressive encodings over arbitrary higher-order domains. Notably, HOPSE scales linearly with the size of combinatorial representations while preserving the expressive power and permutation equivariance of the HOMP approaches. Experiments on molecular and topological benchmarks show that it matches or surpasses state-of-the-art performance while consistently achieving speedups over HOMP-based models, opening a new path for scalable TDL. The code is available at https://github.com/geometric-intelligence/topobench.git.