Higher-Order Message Passing Models

• Higher-Order Message Passing models are techniques that generalize traditional graph message passing to capture multi-way interactions in structures like hypergraphs and simplicial complexes.
• They leverage permutation-invariant aggregation from both lower-dimensional faces and higher-dimensional cofaces, integrating algebraic topology with neural network updates.
• HOMP models have demonstrated improved performance in tasks such as molecular property prediction, community detection, and topological analysis while addressing scalability challenges.

Higher-Order Message Passing (HOMP) Models

Higher-Order Message Passing (HOMP) models generalize standard graph message passing paradigms to encompass multi-way interactions present in hypergraphs, simplicial complexes, and other combinatorial structures. Unlike traditional message-passing neural networks (MPNNs), which propagate information along edges (i.e., pairwise relations), HOMP frameworks enable information transfer over arbitrary higher-dimensional substructures, permitting the direct modeling of complex relational patterns such as group interactions, higher-order motifs, and topological features.

1. Mathematical and Conceptual Foundations

HOMP formalizes representation learning over domains characterized by higher-order relations. In hypergraphs, a hyperedge links an arbitrary subset of nodes, while in a $k$-simplicial complex, $k$-simplices encode $(k+1)$-way affiliations. HOMP schemes aggregate information not only from lower dimensional "faces" but also from higher cofaces, generalizing the pullback/pushforward procedures in algebraic topology and the sum-product paradigm of belief-propagation on graphical models.

Core HOMP equations, as instantiated in various combinatorial settings, proceed as follows. For a cell (or substructure) $C$ at layer $t$, its embedding $h_C^{(t+1)}$ updates by aggregating information from both its lower-dimensional faces $F$ and higher-dimensional cofaces $G$:

$$h_C^{(t+1)} = \mathcal{U}\big(h_C^{(t)},\ [\phi(h_F^{(t)}, h_C^{(t)})_{F \subset C}],\ [\psi(h_G^{(t)}, h_C^{(t)})_{G \supset C}]\big)$$

Here, $\phi$ and $k$0 are learnable permutation-invariant message functions and $k$1 is an update operator such as an MLP or a GRU. In hypergraph neural networks, analogous two-level updates propagate messages within the hyperedge and across the network of hyperedges (Telyatnikov et al., 2023, Arya et al., 2021).

Permutation equivariance underlies all HOMP models, so that embeddings remain consistent under relabeling of nodes or higher-order substructures (Sun et al., 2023). The P-tensor formalism rigorously characterizes the space of all equivariant linear maps between higher-order tensorized features, yielding a canonical basis for structure-preserving information flow in these models.

2. Generalized HOMP Architectures

HOMP implementations vary in their specific instantiations, but share key architectural principles:

• Hypergraph Neural Networks and Multiset MP: General schemes maintain separate representations for each node-hyperedge pair ($k$2), alternating updates between node and hyperedge states via permutation-invariant multiset functions (SUM, MEAN, attention, or learnable MLP-MIXER blocks). The MultiSet framework unifies existing hypergraph message-passing models (AllSet, UniGCNII, EDHNN) under this abstraction (Telyatnikov et al., 2023).
• HOMP with P-tensors: Features are encoded as $k$3-order permutation-equivariant tensors over the set of atoms or nodes, with message passing mediated by equivariant linear maps constructed from set partitions. A general HOMP layer aggregates information from all $k$4-order substructures to produce a $k$5-order update via basis contraction and channel mixing (Sun et al., 2023).
• Motif-Based and Many-Body HOMP: Models for graphs with rich local substructures implement higher-order messages over motifs: for each node, messages are aggregated over paths, cycles, or star-shaped ("tree") motifs of arbitrary arity (Flam-Shepherd et al., 2020, Han, 2024). Local spectral filters (e.g., Chebyshev expansions on motif Laplacians with Ricci curvature weighting) generalize from pairwise adjacency to motif-level structure (Han, 2024).
• Topological and Sheaf HOMP: In topological deep learning, HOMP layers act on the cochain spaces of cellular or simplicial complexes, propagating over boundary/co-boundary operators via Hodge Laplacians or, more generally, data-adapted sheaf Laplacians. Learning restriction maps endows the model with learnable cohomological biases (Hume et al., 27 Sep 2025, Taha et al., 6 Jun 2025).

Classical message passing is recaptured in the $k$6 setting; HOMP strictly generalizes graph MPNNs (message passing neural networks) and can recover, subsume, or outperform previous GNN variants under appropriate configurations (Sun et al., 2023, Telyatnikov et al., 2023).

3. Theoretical Properties and Expressivity

Higher-order message passing enhances the representational capacity of neural models by directly exploiting complex connectivity and group structure:

• Equivariance and universality: HOMP layers constructed from all linear equivariant maps (as in the P-tensor formalism) are universal for multiset functions over substructures of arbitrary order, with the partition basis providing optimal expressivity up to $k$7-order correlations (Sun et al., 2023).
• Homophily and Heterophily: Higher-order homophily can be quantified via dynamic message-passing metrics, reflecting the stability or variation of class agreement within $k$8-hop MP neighborhoods. This dynamic homophily predicts normalized accuracy gains over simple baselines and outperforms clique-expansion homophily metrics (Telyatnikov et al., 2023).
• Energy Bounds and Over-smoothing: Many-body HOMP provides closed-form sensitivity bounds and energy (Dirichlet) upper bounds, showing that increasing $k$9 enables higher “frequency” in learned representations, improving separability under heterophily or when resisting over-smoothing (Han, 2024). Particle system-based HOMP with attraction/repulsion and Allen-Cahn damping provably maintains a positive Dirichlet energy lower bound, resisting collapse even under deep propagation (Ma et al., 24 May 2025).
• Limitations on scalability: Full $(k+1)$0-order message passing incurs combinatorial blowup in both memory and computation ($(k+1)$1 for $(k+1)$2 nodes, $(k+1)$3-order), often restricting models to small $(k+1)$4 (typically $(k+1)$5) or necessitating sampling, pruning, or sparse kernels (Sun et al., 2023, Carrasco et al., 21 May 2025).
• Cohomological bias: Sheafified HOMP architectures tune the global bias of the model from singular to sheaf cohomology, generalizing topological diffusion and overcoming the degeneracies of pure Hodge Laplacians for $(k+1)$6 (Hume et al., 27 Sep 2025).

4. Algorithm Design and Scalability

Several key design patterns emerge across scalable HOMP instantiations:

• Multiset and MLP-Mixer-based pooling: Residual and normalization layers combined with learnable multiset aggregations (SUM/MEAN/MLP) support rich message mixing across hyperedges and nodes, and can be efficiently implemented via mini-batch sampling (Telyatnikov et al., 2023).
• Sparse and motif-based contraction: For high-order graph motifs (triangles, cycles), sparse contractions or motif enumerations are used to avoid the full exponential cost in $(k+1)$7 (Sun et al., 2023, Flam-Shepherd et al., 2020, Han, 2024).
• Training-free message passing: Closed-form, training-free message passing propagators (e.g., $(k+1)$8 for sym-normalized hypergraph adjacency) encode the same L-hop structure as full HNNs, can be precomputed, and offer dramatic training speedups with identical theoretical information content (Tang et al., 2024).
• Inductive, adaptive, and two-level aggregation: Inductive HOMP designs such as HyperMSG utilize adaptive attention weights over intra- and inter-hyperedge aggregations, enable generalization to unseen nodes, and avoid the limitations of graph conversion schemes (Arya et al., 2021).
• Hybrid designs with domain priors and knowledge adaptation: Hybrid HOMP models for molecules integrate chemistry-derived functional group hyperedges with learned pruning and adaptive propagation, balancing interpretability with expressive optimization (Chen et al., 2021).
• Avoidance of combinatorial explosion: HOPSE demonstrates that efficient positional and structural encodings over decomposed Hasse graphs can match or surpass HOMP performance without explicit message-passing, achieving linear scaling (Carrasco et al., 21 May 2025).

5. Applications, Empirical Results, and Benchmarks

HOMP models have demonstrated substantial empirical and application-driven impact:

• Node and graph classification: Higher-order MP models exhibit consistent and sometimes significant improvements over classical GNNs on node classification tasks (Cora, CiteSeer, PubMed, DBLP, molecular benchmarks), especially when higher-order local structure is predictive (Sun et al., 2023, Telyatnikov et al., 2023, Tang et al., 2024, Arya et al., 2021).
• Molecular property prediction: Capturing motifs such as rings and functional groups via $(k+1)$9-order HOMP has yielded state-of-the-art accuracy on ZINC-12K, QM9, Alchemy, and other chemistry datasets, with careful ablation revealing the key performance gain comes from incorporating higher $C$0 interactions (Sun et al., 2023, Flam-Shepherd et al., 2020, Dupty et al., 2020, Chen et al., 2021).
• Synthetic and topological benchmarks: On tasks like Dirichlet energy regression, trajectory prediction, and distinguishing complexes with nontrivial cohomology, HOMP models with motif-aware spectral filters and sheafification demonstrate robustness and outperform Hodge-only or GCN-based baselines (Han, 2024, Hume et al., 27 Sep 2025, Joeres et al., 2024).
• Hypergraph community detection: Cavity-based HOMP belief propagation enables detectability analysis tied to hyperedge-size moments and overlap entropy, producing explicit phase transitions sharpened by higher-order structure (Ruggeri et al., 2023).
• Epidemic source inference: Dynamic HOMP algorithms for SI dynamics on hypergraphs propagate susceptible/infectious probability messages across higher-order groupings, improving accuracy and ranking over standard pairwise, centrality, and simulation-based baselines (Ke et al., 3 Jul 2025).
• Scalability: Training-free and positional-structural methods have reduced wallclock times by $C$1–$C$2 on large real and synthetic hypergraphs, with competitive or improved accuracy (Tang et al., 2024, Carrasco et al., 21 May 2025).

Notable empirical patterns include the preservation of performance by discarding small hyperedges, strong performance correlation with dynamic homophily, and improved resistance to over-smoothing and oversquashing with proper HOMP structure and rewiring (Telyatnikov et al., 2023, Taha et al., 6 Jun 2025).

6. Future Directions and Open Challenges

Research on HOMP models continues to advance along multiple axes:

• Expressivity versus scalability: Managing the combinatorial complexity of higher-order architectures without compromising expressive capacity is a primary concern, motivating message-passing-free schemes, motif-based sparsification, and constrained substructure enumeration (Carrasco et al., 21 May 2025, Sun et al., 2023).
• Task-adaptive structure and optimization: Leveraging domain priors (functional groups, chemical motifs), local geometry (Ricci curvature, motif weights), and learned restriction maps (sheaf theory) supports improved alignment with downstream tasks (Chen et al., 2021, Han, 2024, Hume et al., 27 Sep 2025).
• Dynamic, stochastic, or uncertain settings: Generalizations to stochastic dynamics, temporal complexes, multi-source diffusion, and partially observed settings drive development of robust, uncertainty-aware HOMP algorithms (Ke et al., 3 Jul 2025).
• Oversquashing and connectivity design: Systematic analysis and mitigation of oversquashing—via relational rewiring, curvature augmentation, and width-depth tradeoffs—enable deep and reliable HOMP networks even in bottlenecked or sparse regimes (Taha et al., 6 Jun 2025, Ma et al., 24 May 2025).
• Foundational connections: Further formalization of HOMP principles via category theory, cohomology, and algebraic-topological machinery will deepen theoretical understanding and guide the design of new expressivity- and bias-controlled models (Cockett et al., 25 Mar 2025, Hume et al., 27 Sep 2025).

A plausible implication is that scalable, topology-aware, and dynamically adaptive HOMP models will be critical for future learning systems operating on complex relational and multi-modal domains.