Topological Message-Passing Beyond Graphs

• Topological message-passing extends GNNs by leveraging higher-order structures like simplicial and cell complexes to capture multi-way interactions.
• It employs algebraic-topological operators such as boundary, coboundary, and the Hodge Laplacian for localized, data-driven message aggregation.
• Empirical studies demonstrate enhanced expressivity and performance in molecular modeling, many-body physics, and relational tasks over traditional GNNs.

Topological message-passing beyond graphs generalizes the classical message-passing paradigm of graph neural networks (GNNs) to higher-order topological domains, including simplicial complexes, cell complexes, and CW-complexes. These frameworks systematically extend from modeling pairwise relationships (edges) to encoding genuine multi-way (polyadic) interactions (e.g., triangles, rings, higher-dimensional cells) and topological invariants, leveraging algebraic-topological operators—such as boundary, coboundary, and Hodge Laplacian—as the substrate for localized or data-driven message computation. By unifying combinatorial topology with neural architectures, topological message-passing networks have established state-of-the-art expressivity and empirical performance across molecular modeling, many-body physics, and relational domains.

1. Mathematical Foundations: Complexes, Neighborhoods, and Operators

Higher-order message passing takes place over discrete topological domains such as simplicial or cell complexes. An abstract simplicial complex $K$ is a set of nonempty subsets of a finite vertex set $V$ closed under inclusion, so if $\sigma \in K$ and $\tau \subset \sigma$ then $\tau \in K$. Each $\sigma$ is a $k$-simplex ($|\sigma| = k+1$), representing vertices ($k=0$), edges ($k=1$), triangles ($V$0), etc. More generally, cell and CW-complexes allow $V$1-cells with flexible attaching maps, modeling structures (e.g., rings, higher-genus surfaces) outside the expressivity of simplicial complexes (Papillon et al., 2023, Kovač et al., 2024).

Neighborhoods of a cell or simplex $V$2 are defined in several canonical ways:

• Boundary $V$3: lower-dimensional faces, e.g., the set of $V$4-simplices contained in $V$5.
• Coboundary $V$6: higher cofaces, i.e., $V$7-simplices in which $V$8 is a face.
• Lower adjacency: same-dimension neighbors sharing a common face.
• Upper adjacency: same-dimension neighbors sharing a common coface.

Boundary and coboundary operators, written $V$9 and $\sigma \in K$0, are fundamental in the algebraic-topological formalism. Hodge theory provides graded Laplacians $\sigma \in K$1, generalizing the graph Laplacian to all dimensions. These operators underpin message-passing, enforcing local-to-global consistency aligned with topological invariants such as Betti numbers or (co)homological features (Papillon et al., 2023, Hume et al., 27 Sep 2025).

2. Generalized Message-Passing Architectures

Topological message-passing frameworks prescribe feature updates over cells/simplices using local (multi)relational aggregations:

• For each $\sigma \in K$2-cell $\sigma \in K$3, a learnable update combines its own features with aggregated messages from boundary (downward), coboundary (upward), and lateral (adjacent) cells of appropriate dimension (Papillon et al., 2023, Lan et al., 2023, Giusti et al., 2023).
• In Simplicial Message Passing Networks (SMPNN/MPSN) (Lan et al., 2023, Bodnar et al., 2021), a canonical $\sigma \in K$4-th layer update takes the form:

$\sigma \in K$5

$\sigma \in K$6

with $\sigma \in K$7 aggregating faces ($\sigma \in K$8), same-rank, and cofaces ($\sigma \in K$9); $\tau \subset \sigma$0 is a permutation-invariant aggregation.

• Cellular Isomorphism Networks (CIN, CIN++) (Giusti et al., 2023) on cell complexes include lateral "lower" messages, enabling direct interaction among same-dimension cells via a common boundary, further increasing representational capacity and convergence speed.

Specialized mechanisms, such as attention (Simplicial Attention Networks), orientation-equivariant or E(n)-equivariant updates (Liu et al., 2024, Kovač et al., 2024), and persistent homology vectorization (Verma et al., 2024), can be layered onto this general template for domain-specific inductive biases.

3. Expressivity and Algorithmic Implications

Topological message-passing architectures strictly generalize GNNs in expressivity. Classical MPNNs are bounded by the 1-Weisfeiler–Lehman (WL) test, unable to distinguish certain non-isomorphic graphs or higher-order motifs (e.g., triangle faces vs. cycles). Simplicial Message Passing and the associated Simplicial Weisfeiler–Lehman (SWL) test provably extend this discriminative power:

• SWL distinguishes graph pairs that 1-WL and even 3-WL cannot, especially when lifting graphs to their clique complexes (Bodnar et al., 2021).
• MPSN-like networks with injective aggregators match SWL's power and can compute low-dimensional Betti numbers, capturing subtle topological distinctions such as the presence or absence of filled cliques or higher-order cycles (Lan et al., 2023, Bodnar et al., 2021).
• The inclusion of lateral (lower and upper) adjacency relations (as in CIN++) further increases expressivity and alleviates structural oversquashing in dense topologies (Giusti et al., 2023, Taha et al., 6 Jun 2025).

Integrating topological invariants (e.g., persistent homology descriptors) as network features further extends the classical expressivity hierarchy, enabling discrimination of cases indistinguishable even by the SWL procedure (Verma et al., 2024).

4. Theoretical Limits: Oversquashing, Cohomology, and Sheafification

Higher-order message-passing frameworks inherit and extend core phenomena from graph GNNs, such as oversquashing (exponential decay of distant node influence) and oversmoothing (collapse of local features). The relational framework in (Taha et al., 6 Jun 2025) treats message passing on complexes as neural computation over general relational structures, introducing influence matrices and curvature-based characterizations:

• The sensitivity of a cell's output to distant inputs decays exponentially with topological "distance" in the extended influence graph, analogous to the spectral gap in graphs.
• Relational rewiring—adding artificial higher-arity relations—can increase long-range influence and mitigate oversquashing.

Many high-dimensional message-passing schemes admit a Laplacian (diffusional) interpretation. For $\tau \subset \sigma$1, the classical combinatorial Hodge Laplacian $\tau \subset \sigma$2 implements diffusion that biases toward harmonic $\tau \subset \sigma$3-cochains (i.e., features supported on $\tau \subset \sigma$4-dimensional "holes"), but this can be opaque or degenerate, for example when cohomology vanishes (Hume et al., 27 Sep 2025). Sheaf-theoretic generalizations replace singular cohomology with (data-driven) sheaf cohomology, using a learned sheaf Laplacian $\tau \subset \sigma$5, thus enabling data-aware and flexible inductive biases adaptable to local constraints or physical symmetries (Hume et al., 27 Sep 2025).

5. Equivariance, Geometry, and Implementation Variants

In geometric domains, it is often crucial to encode equivariance under Euclidean symmetry groups $\tau \subset \sigma$6. E(n)-equivariant message-passing cellular networks (EMPCNs) (Kovač et al., 2024) and Clifford group equivariant simplicial message passing (CSMPN) (Liu et al., 2024) use cell-wise or simplex-wise features initialized with geometric invariants (distances, volumes, areas, higher-grade Clifford multivectors) and update these through equivariant message and aggregation functions. Parameter sharing across dimensions and conditioning on simplex/simplicial type can improve scalability and preserve symmetry.

Implementation strategies for scalability include decoupling dense node-to-node communication from communication over higher-order cells (rings, faces) and using sparse adjacency or incidence data structures (Kovač et al., 2024, Lan et al., 2023). Purely feed-forward approaches (Topo-MLP) inject topological priors at training time via contrastive losses over the incidence structure, enabling robust, inference-time connectivity independence (Ramamurthy et al., 2023).

6. Empirical Performance and Domain-Specific Insights

Topological message-passing networks have demonstrated empirical gains in diverse benchmarks:

• SMPNN significantly outperforms GCN/GAT/MPNN on molecular property regression (QM9, MD17, solubility), particularly when higher-order features (triangles, rings) are prevalent (Lan et al., 2023).
• CIN++ achieves state-of-the-art or near-best accuracy on molecular regression (ZINC), peptide structure/function, and real-world graph classification (TUDatasets), with lower messages facilitating non-local communication and denser ring-dominated regimes (Giusti et al., 2023).
• E(n)-equivariant generalizations (EMPCN, CSMPN) yield substantial improvements in many-body prediction, motion forecasting, and atomic trajectory modeling over non-topological or non-equivariant baselines (Liu et al., 2024, Kovač et al., 2024).
• Integration of persistent homology (TopNets) provably boosts expressivity and yields further performance increases in molecular, protein, and antibody design benchmarks (Verma et al., 2024).

These successes are attributed to the ability of higher-order message-passing to capture both fine-grained combinatorial structure and global homological properties inaccessible to edge-based GNNs.

7. Open Questions and Future Directions

Key directions for ongoing research include:

• Developing equivariant and sheafified topological layers with learnable restriction maps, enabling end-to-end adaptation to data and physical symmetries (Hume et al., 27 Sep 2025).
• Extending frameworks from simplicial to generic cellular/CW complexes and arbitrary relational structures, enabling modeling of arbitrary group interactions and mixed set-based/part-whole relations (Giusti et al., 2023, Kovač et al., 2024, Taha et al., 6 Jun 2025).
• Formulating multi-scale and continuous-depth TNNs, leveraging persistence and dynamic complexes for temporal and multi-scale relational modeling (Verma et al., 2024, Hume et al., 27 Sep 2025).
• Quantifying and mitigating oversquashing via curvature- or spectral-based rewiring and layer-wise regularization (Taha et al., 6 Jun 2025).
• Designing faster persistent homology computations and efficient batching for higher-dimensional complexes (Verma et al., 2024).
• Unifying frameworks and benchmarks to promote systematic comparison and adoption across applied domains (Papillon et al., 2023).

Topological message passing beyond graphs thus constitutes a rapidly evolving synthesis of combinatorial topology, spectral theory, and deep learning, with demonstrable advantages in expressivity, robustness, and modeling of higher-order and topologically structured data (Papillon et al., 2023, Bodnar et al., 2021, Lan et al., 2023, Giusti et al., 2023, Liu et al., 2024, Kovač et al., 2024, Verma et al., 2024, Taha et al., 6 Jun 2025, Hume et al., 27 Sep 2025).