Higher-Order Message Passing (HOMP)

• Higher-Order Message Passing (HOMP) is a framework that generalizes standard GNNs by modeling multi-way interactions in complex structures like hypergraphs and simplicial complexes.
• It employs modular, permutation-invariant aggregations, attention mechanisms, and tensor or category-theoretic methods to update node or cell embeddings.
• Empirical studies show HOMP achieves significant accuracy gains in graph classification, molecular property prediction, and complex biological data modeling.

Higher-Order Message-Passing (HOMP) formalizes the propagation of information in complex relational domains—such as hypergraphs, simplicial complexes, and combinatorial cell complexes—by generalizing the message-passing paradigm beyond pairwise (edge-based) interactions. HOMP architecturally and mathematically extends standard graph neural networks (GNNs) to accurately model higher-order interactions, permit richer expressivity, and suit the topology and data dependencies found in natural, scientific, and engineered systems. Multiple formulations exist, including modular neural architectures, particle-system-inspired dynamics, permutation-equivariant tensor algebra, and categorical/functional semantics.

1. Formal Definitions and Architectural Schemes

HOMP is instantiated differently across relational structures. In hypergraphs, the core structure is $H = (V, E)$ with node set $V$, hyperedge set $E$, and incidence matrix $H \in \{0,1\}^{|V| \times |E|}$ with $H_{i,e} = 1$ iff $v_i \in e$ (Arya et al., 2021). Each node $v_i$ carries feature vector $x_i$, yielding node feature matrix $X_v$.

In general combinatorial complexes, a HOMP layer updates cell-level embeddings $h_\sigma$ for each cell $V$0, using permutation-invariant aggregations over neighborhoods of cells defined by face/coface incidence, adjacencies, or motif-based constructs. The canonical HOMP update for a cell $V$1 is: $V$2 where $V$3 ranges over incident, adjacent, or combinatorially-defined neighbors, $V$4 is a message function (MLP or spectral filter), $V$5 a set aggregator, and $V$6 a node update MLP (Carrasco et al., 21 May 2025, Joeres et al., 2024).

In particle-system- and motif-based variants, message passing is formulated as dynamics on continuous states, e.g., velocity-coupled ODEs on local tree motifs, with updates governed by localized Laplacians and kernelized interactions (Ma et al., 24 May 2025, Han, 2024).

Specialized formulations include the MultiSet formalism (assigning node–edge-specific hidden states), tensor methods on paths and cycles, and message-passing layers with categorical or sheaf-theoretic semantics (Telyatnikov et al., 2023, Sun et al., 2023, Hume et al., 27 Sep 2025).

2. Core Principles and Key Mechanisms

The essential design of HOMP is modular, multi-stage, and leverages explicit higher-order topology. For example, in HyperMSG (Arya et al., 2021):

• Intra-hyperedge aggregation (node→edge): Aggregate features from all nodes in a hyperedge using attention:

$V$7

with attention $V$8 computed from node features, degree, and optional edge features.

• Inter-hyperedge update (edge→node): Aggregate messages from all incident hyperedges and update node states:

$V$9

with permutation-invariant set functions (mean, attention) as aggregators.

• Attention mechanism: Use degree centrality and current node/hyperedge features as input to a small MLP to quantify node importance within hyperedges.

In higher-order path- or motif-based schemes, message passing occurs over enumerated paths of length $E$0, tree-like motifs, or P-tensors representing subgraphs, with permutation-equivariant linear or MLP-based operations (Flam-Shepherd et al., 2020, Sun et al., 2023, Han, 2024).

Category-theoretic and sheaf-theoretic approaches encode updates as diffusion by higher-grade Laplacians, with the inductive bias tightly connected to data-aware cohomological structures (Hume et al., 27 Sep 2025).

3. Theoretical Guarantees and Expressivity

HOMP architectures attain strictly higher expressivity than pairwise MPNNs:

• Universal Approximation: MultiSetMixer and P-tensor-based models are universal approximators for permutation-invariant functions on finite cell multisets, subsuming all set-based HNNs (Telyatnikov et al., 2023, Sun et al., 2023).
• Inductive Bias and Homophily: Message-passing homophily, parameterized by Δ-homophily (stepwise persistence of class signals), correlates strongly with the empirical gain from higher-order connectivity, surpassing clique-based expansions. Sheaf Laplacians encode a more general, data-driven bias by aligning features with data-informed cohomological classes, overcoming degeneracy of combinatorial Hodge Laplacians in pure topological spaces (Telyatnikov et al., 2023, Hume et al., 27 Sep 2025).
• Oversquashing and Stability: Relational and physical ODE/SDE-based frameworks (e.g., HAMP, motif-based MPNN) establish sensitivity bounds for information propagation, showing that higher-order interactions can mitigate oversquashing: Dirichlet energy exhibits a provable positive lower bound even as layers deepen, supported by attraction–repulsion kernels and damping forces that avert exponential feature collapse (Ma et al., 24 May 2025, Han, 2024, Taha et al., 6 Jun 2025).
• Permutation and Equivariance: All core HOMP updates are designed to be equivariant to permutations of input objects and, in molecules, to O(3) rotations via spherical harmonics and symmetrized tensor contractions (Batatia et al., 2022, Han, 2024).

4. Empirical Performance and Topological Insights

HOMP outperforms or matches state-of-the-art baselines in domains exploiting higher-order relational information:

• Node and graph classification: HyperMSG delivers 5–10 points accuracy gains over HGNN, HyperGCN, UniGNN on citation, co-authorship, neuroimaging, and multimodal (Flickr) datasets (Arya et al., 2021, Telyatnikov et al., 2023).
• Molecular property prediction: Path-MPNN and P-tensor approaches show 15–20% error reductions on benchmarks such as QM8, ZINC-12K, OGBG-MOLHIV, with clear improvements on structure-sensitive targets (e.g., solubility, reactivity, quantum energies) (Flam-Shepherd et al., 2020, Sun et al., 2023). MACE achieves state-of-the-art force-field accuracy with only two layers by leveraging explicit four-body messages (Batatia et al., 2022).
• Complex biological data: On glycan representation tasks, GIFFLAR’s full combinatorial complex message-passing achieves substantial boosts in AUROC, MCC, and OOD performance over atom-only and tree-based GNNs, highlighting the necessity of higher-order modeling for class- and taxonomy-level prediction (Joeres et al., 2024).
• Training efficiency: Training-free HOMP modules (SHNN, TF-MP-Module) match or improve on the accuracy of learned HNNs with as little as 1%–20% of the training time by analytically folding the entire structural propagation into a fixed operator (Tang et al., 2024).

5. Limitations, Complexity, and Scalability

HOMP methods face several challenges:

• Combinatorial Explosion: The number of possible message-passing routes across $E$1 ranks in a complex grows as $E$2, rapidly making architecture search and implementation intractable for $E$3 (Carrasco et al., 21 May 2025).
• Per-layer Computational Cost: For general complexes, per-layer cost scales with the total number of neighborhood incidences and is significantly higher than edge-based GNNs—e.g., $E$4 per layer in HyperMSG; $E$5 for motif- and incidence-based HOMP (Arya et al., 2021, Carrasco et al., 21 May 2025).
• Memory Overhead: Storing augmented Hasse graphs and all motifs or subgraph tensors can easily exceed practical memory budgets, especially for high-rank complexes or large hyperedges.
• Expressivity–Cost Tradeoff: Depth, hidden dimension, and aggregation order improve expressiveness but increase risk of overfitting, training time, and memory usage. Some approaches (e.g., HOPSE) propose message-passing–free encodings to overcome these bottlenecks, providing linear scaling and comparable expressivity (Carrasco et al., 21 May 2025).
• Domain Limitations: Pure Laplacian/Hodge-based HOMP can fail in topologically trivial domains (ker Δ_k = 0), and directed or typed hyperedges or nontrivial channel polarities complicate standard construction, sometimes requiring categorical enrichment (Hume et al., 27 Sep 2025, Cockett et al., 25 Mar 2025).

6. Methodological Variants and Extensions

Diverse HOMP techniques exploit the underlying mathematical structure and target domain:

• Attention and sampling: Attention mechanisms based on degree centrality and local features, along with stochastic node/hyperedge sampling, regularize aggregation and enable scalability for large or noisy hyperedges (Arya et al., 2021, Telyatnikov et al., 2023).
• Particle system integration: Attraction–repulsion and Allen–Cahn–like forces, in continuous ODE or SDE dynamics, deliver class-dependent equilibrium and suppress over-smoothing, with deep stacking enabled by velocity–momentum stabilization and stochastic exploration (Ma et al., 24 May 2025).
• Sheaf cohomology and enrichment: Sheaf-theoretic message passing accommodates data-driven cohomological bias, enabling fine-grained control over the alignment of features and topological classes, especially in high-grade simplicial or cell complexes (Hume et al., 27 Sep 2025).
• P-tensor algebra: All linear permutation-equivariant maps between subgraphs are enumerated and parameterized by set partitions (“Bell basis”), enforcing maximal symmetry and expressivity (Sun et al., 2023).
• Category-theoretic interpretation: In concurrent systems, HOMP semantics are formalized via actegories with copowers and hom-objects, enabling higher-order process passing, recursion, and negation in message-passing programming (Cockett et al., 25 Mar 2025).

7. Applications and Outlook

HOMP is foundational in topological deep learning and underlies major advances in:

• Hypergraph learning for citation/classification, multimodal recommendation, and neuroimaging.
• Chemical and materials modeling, where higher-order interactions are critical for force fields, quantum targets, and functional group recognition.
• Computational biology and bioinformatics, notably for glycan structure–function mapping and taxonomy inference.
• Epidemic modeling, e.g., higher-order dynamic message passing for source detection in group-infection hypergraphs (Ke et al., 3 Jul 2025).
• Theoretical advancements in over-squashing analysis, universality, inductive bias, and category-theoretic programming principles.

Limitations in scalability, memory, and the combinatorial design space persist. Message-passing–free positional encodings, stochastic sampling, and categorical/homological formalization represent active directions for making HOMP both more tractable and expressive.

References:

• (Arya et al., 2021) Adaptive Neural Message Passing for Inductive Learning on Hypergraphs
• (Telyatnikov et al., 2023) Hypergraph Neural Networks through the Lens of Message Passing: A Common Perspective to Homophily and Architecture Design
• (Ma et al., 24 May 2025) How Particle System Theory Enhances Hypergraph Message Passing
• (Tang et al., 2024) Training-Free Message Passing for Learning on Hypergraphs
• (Taha et al., 6 Jun 2025) Demystifying Topological Message-Passing with Relational Structures: A Case Study on Oversquashing in Simplicial Message-Passing
• (Joeres et al., 2024) Higher-Order Message Passing for Glycan Representation Learning
• (Carrasco et al., 21 May 2025) HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations
• (Ruggeri et al., 2023) Message-Passing on Hypergraphs: Detectability, Phase Transitions and Higher-Order Information
• (Batatia et al., 2022) MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields
• (Hume et al., 27 Sep 2025) On the Sheafification of Higher-Order Message Passing
• (Sun et al., 2023) P-tensors: a General Formalism for Constructing Higher Order Message Passing Networks
• (Cockett et al., 25 Mar 2025) Categorical Semantics of Higher-Order Message Passing