Directed Simplicial Neural Networks

• Dir-SNNs are topological deep learning models that extend message-passing to higher-order directed simplicial complexes with asymmetric relations.
• They leverage specialized adjacency mechanisms, including boundary, coboundary, and spectral methods, to capture non-pairwise interactions in diverse networks.
• Empirical studies show Dir-SNNs excel in tasks like synthetic source localization and brain activity decoding, outperforming traditional Dir-GNNs and undirected SNNs.

Directed Simplicial Neural Networks (Dir-SNNs) are a class of topological deep learning models that generalize message-passing neural architectures to higher-order combinatorial structures with asymmetric, directed relations. Dir-SNNs operate on directed simplicial complexes or semi-simplicial sets, capturing and leveraging the complex, multi-way, and directed motifs present in diverse real-world data—such as neural, traffic, and supply networks—where interactions extend beyond simple pairwise or undirected relations. By equipping each simplex (vertices, edges, triangles, etc.) with directed structure and developing tailored adjacency and propagation mechanisms, Dir-SNNs offer strictly greater expressive power than both directed graph neural networks (Dir-GNNs) and undirected simplicial neural networks, both in theory and in empirical performance on tasks that require distinguishing higher-order directionality (Lecha et al., 2024, Lecha et al., 23 May 2025).

1. Mathematical Foundations of Directed Simplicial Complexes

A directed simplicial complex $\mathcal{K} = (V, \Sigma)$ is specified by a set of vertices $V$ and a collection of ordered $k$-simplices $\Sigma_k$, each represented as a tuple $(v_{i_0}, \ldots, v_{i_k})$ with $k+1$ vertices. The inclusion axioms ensure that all ordered faces of any simplex are themselves contained in the complex, thereby capturing higher-order directionality via tuple ordering (Lecha et al., 2024, Lecha et al., 23 May 2025).

Key combinatorial operators include:

• Face Maps ($d_i$): For a $k$-simplex $(v_0, ..., v_k)$, the $i$th face map deletes the $V$0th vertex, yielding a $V$1-simplex.
• Boundary and Coboundary Maps: In Dir-SNNs, boundary operators collect the (unordered) set of faces without alternating signs; dually, the coboundary aggregates all cofacets for which a simplex is a given face.
• Higher-Order Adjacency: Dir-SNNs define up- and down-adjacency between $V$2-simplices as pairs that are incident to shared $V$3- or $V$4-simplices via specified face indices, producing asymmetric and directed neighborhoods.

Directed flag complexes, constructed from ordered cliques in a digraph, provide a canonical example of directed simplicial complexes amenable to Dir-SNN processing (Lecha et al., 2024).

2. Dir-SNN Layer Architectures and Computational Schemes

Dir-SNN layers propagate and aggregate features along higher-order, directed relations between simplices. Dir-SNNs support both message-passing and spectral-convolutional formulations:

• Message-Passing (Spatial) Variant: Each Dir-SNN layer aggregates messages over higher-order directed adjacencies (down-adjacency, up-adjacency), boundary and coboundary relations. If $V$5 are features of simplex $V$6 at layer $V$7, then:
  • For each relation (e.g., $V$8: $V$9-simplices with matching $k$0th and $k$1th faces), features from neighboring simplices are aggregated via learned functions (MLPs or similar).
  • The outputs from all relevant relations are merged and passed through pointwise nonlinearities and learned update blocks.
• Spectral (Laplacian-based) Variant: Dir-SNNs replace the standard (real, undirected) Hodge Laplacians with Connection Laplacians sensitive to directionality and phase frustration. For the edge space $k$2,

$k$3

where $k$4 and $k$5 deploy $k$6 unitary matrix-valued weights (Pauli rotations) depending on the directed configuration. Spectral Dir-SNN layers apply polynomial filters (e.g., Chebyshev) to this Laplacian, with aggregation in the complex or real-embedded domain (Gong et al., 2024).

Stacking multiple Dir-SNN layers yields a deep architecture interleaving nonlinear, permutation-equivariant transformations at each simplex-order. Output can be read out via global pooling (for unsupervised tasks) or fully-connected heads (for supervised learning) (Gong et al., 2024, Lecha et al., 2024, Lecha et al., 23 May 2025).

3. Directionality, Expressive Power, and Theoretical Properties

Dir-SNNs are strictly more expressive than both Dir-GNNs and symmetric SNNs. The essential reasons are:

• Higher-order Directed Adjacency: By allowing features to propagate along directed, multi-way relationships, Dir-SNNs can distinguish combinatorial structures that are indistinguishable using only pairwise (edge) information.
• D-SWL (Directed Simplicial Weisfeiler-Leman) Test: Dir-SNNs simulate a higher-order, injective color refinement that aggregates colors via all directed boundary, coboundary, up- and down-adjacency relations. Non-isomorphic digraphs with identical 1-skeletons can be separated based on counts of higher-order motifs (e.g., directed triangles) (Lecha et al., 2024, Lecha et al., 23 May 2025).
• Spectral Feature Distinction: The Connection Laplacians introduce nontrivial “magnetic” gaps and mixed modes when frustration due to conflicting directionalities occurs, leading to spectra that encode nontrivial higher-order flows absent in undirected Laplacians (Gong et al., 2024).

Dir-SNNs are also capable of recovering a suite of combinatorial and topological invariants—such as higher-order degrees and directional flows—that are provably out of reach for GNNs and undirected SNNs (Lecha et al., 23 May 2025).

4. Implementation, Scalability, and Regularization

Dir-SNN implementations require:

• Complex or Real Embeddings: Complex-valued feature vectors ($k$7 per $k$8-simplex for $k$9) and block-unitary adjacency matrices; or, real $\Sigma_k$0 embeddings via realification of the Pauli blocks (Gong et al., 2024).
• Sparsity Management: The adjacency/inter-relation matrices have $\Sigma_k$1 nonzeros (for 1-simplices), with storage scales manageable via block sharing. Routing-SSN mechanisms for learnable sparsification select a subset of relations at each layer, balancing accuracy and computational cost (Lecha et al., 23 May 2025).
• Regularization: $\Sigma_k$2 penalties on layer weights and spectral filter coefficients prevent high-frequency feature explosion. Magnetic Cheeger-type losses for spectral Dir-SNNs align learned representations with global coherence by penalizing frustration (Gong et al., 2024).
• Frameworks: Complex-number support (e.g., PyTorch-Complex) is recommended; all message-passing and aggregation operations are permutation-equivariant.

Routing-SSN layers further optimize scalability by dynamically gating and sparsifying relation sets per layer, significantly reducing parameter count and runtime, with only minor loss in accuracy (Lecha et al., 23 May 2025).

5. Empirical Results and Benchmark Tasks

Dir-SNNs achieve state-of-the-art or competitive performance on multiple domains involving directed higher-order structure:

• Synthetic Source Localization: Dir-SNNs outperform Dir-GNNs and undirected SNNs by substantial margins (e.g., $\Sigma_k$3 vs. $\Sigma_k$4 accuracy at 10 dB SNR) when identifying source edge communities in synthetic directed graphs diffused by higher-order flows (Lecha et al., 2024).
• Brain Activity Decoding: On large-scale simulated rat brain data, Dir-SNNs recover stimulus identity with $\Sigma_k$5 accuracy—$\Sigma_k$6 above the second-best model and up to $\Sigma_k$7 above Dir-GNNs. Even under extreme subsampling, Dir-SNNs retain $\Sigma_k$8 improvement over all alternatives (Lecha et al., 23 May 2025).
• Standard Graph Benchmarks: Dir-SNNs remain competitive for both heterophilic (Roman-Empire dataset, $\Sigma_k$9) and homophilic (Cora, Citeseer) node classification tasks and for large-scale edge regression (Lecha et al., 23 May 2025).

A plausible implication is that Dir-SNNs are uniquely suited for domains where data exhibits directed, non-pairwise, and multi-relational motifs.

6. Limitations, Extensions, and Open Directions

Dir-SNNs reduce to Dir-GNNs or undirected SNNs in the absence of genuine higher-order directionality. The enumeration of all possible directed relations has high memory requirements on very large complexes, although mechanisms such as Routing-SSN mitigate this to a significant degree (Lecha et al., 23 May 2025). Additionally, realization of spectral Dir-SNNs using fully directed Hodge Laplacians remains an open research direction (Lecha et al., 2024).

Variations and future research themes include:

• Development of fully directed Hodge theory and spectral Dir-SNN layers for all simplex orders (Gong et al., 2024, Lecha et al., 2024).
• Attention-based propagation schemes across directed relations.
• Extensions to directed cell complexes and non-simplicial higher-order motifs.
• Integrating temporal message passing and motif discovery.
• Broadening applications to domains such as information cascades, object interaction graphs, and biological pathway inference (Lecha et al., 23 May 2025).

7. Summary Table: Key Dir-SNN Features Across Principal Works

Paper (arXiv ID)	Mathematical Core	Main Architectural Mechanism	Principal Empirical Domain
(Gong et al., 2024)	Hermitian Connection Laplacians (Pauli-twisted adjacency)	Spectral and message-passing Dir-SNN layers	Synthetic complexes, theory
(Lecha et al., 2024)	Higher-order combinatorial adjacency	General message-passing Dir-SNNs	Synthetic source localization
(Lecha et al., 23 May 2025)	Semi-simplicial sets, relation algebra	Full general SSN layers with routing SSN	Brain activity, graph benchmarks

This encapsulates the distinctive innovations and focus domains across foundational Dir-SNN literature. Dir-SNNs mark a significant advance in topological deep learning by enabling expressive, relation-aware modeling of directed higher-order phenomena inaccessible to prior GNN and SNN methodologies.