Hierarchical Equivariant Graph Encoder

Updated 30 July 2025

The paper introduces a neural architecture that encodes signals on graphs and manifolds with symmetry equivariance and preserved directional information.
It leverages directional convolutions and N-body message passing to process orientation-dependent features across multiple hierarchical scales.
The approach enhances performance in applications like shape matching, molecular modeling, and physical systems by using symmetry-preserving pooling and attention mechanisms.

A hierarchical equivariant directional graph encoder is a neural architecture designed to encode signals or features on graphs or manifolds while promoting symmetry equivariance—typically to rotations, translations, and/or permutations—while retaining and propagating directionality and hierarchical organization across model layers. Such encoders are central to geometric deep learning, particularly in non-Euclidean domains where spatial relationships, directed connections, and multiscale structure are critical. Their design incorporates mathematical operators and representations that preserve, propagate, or hierarchically pool directional information, ensuring the model's output transforms consistently with respect to underlying symmetry groups.

1. Directional Functions and Directional Convolution

The foundational element of a hierarchical equivariant directional graph encoder is the generalization of signal representations from scalar functions to directional functions—functions defined not only on points or nodes but also on a unit direction attached at each point. In the context of surfaces, a directional function $\varphi(x, v)$ is defined on point $x$ and tangent direction $v \in T_x\mathcal{X}$ . This "lifting" of scalar functions to directional functions expands the representational capacity of the encoder: at each node (or surface point), the function can encode orientation-dependent information unavailable in conventional real-valued signal processing (Poulenard et al., 2018).

Directional convolution operators enable the propagation and transformation of these directional signals. Unlike classical convolutions that require fixing a canonical orientation (which is ill-defined on curved spaces), a directional convolution $(\varphi ★ k)(x, v)$ preserves directionality by convoluting directional signals over all rotations, parameterized by the completed exponential map and parallel transport along geodesics. Formally, the convolution is

$(\varphi ★ k)(x, v) = \langle (\overline{\exp}_x)^*\varphi, \tau_{(x,v)}^*k \rangle_{L^2},$

where all angular detail is preserved. This method ensures not only position- but also orientation-dependent features are processed and propagated.

2. Equivariance Principles

Equivariance ensures that the output of the encoder transforms predictably under group actions applied to the input, such as rotations, translations, or edge-direction reversals. In graph contexts, permutation equivariance is essential; in geometric contexts, SE(3) or E(n) equivariance guarantees physical or structural consistency (Han et al., 2022, Liao et al., 2022, Du et al., 2023, Xu et al., 10 Jun 2025).

For directional convolutions, rotation equivariance is formalized as:

$(\varphi_{R_e} ★ k)(x, \theta) = (\varphi ★ k)_{R_e}(x, \theta) = (\varphi_e ★ k)(x, \theta + \theta_x),$

meaning a change in the local reference frame by rotation $R_x$ simply induces a fixed-phase shift in the output (Poulenard et al., 2018).

In more general hierarchical structures, equivariant maps for hierarchies are constructed using group-theoretic principles, notably the wreath product of symmetry groups acting at each hierarchical level. If $K$ and $H$ act on the inner and outer structure, respectively, then any linear map equivariant to $K \wr H$ symmetry can be written as:

$W_{K \wr H} = W_H \otimes (1_Q 1_Q^T) + I_P \otimes W_K,$

where $W_H$ and $W_K$ are equivariant maps at each level (Wang et al., 2020).

3. Hierarchical Graph and Surface Encoders

A defining property of hierarchical equivariant directional encoders is the multi-scale, multi-level processing of signals. Hierarchy is constructed either explicitly—via graph pooling, coarsening, and unpooling (e.g., to model atoms $\to$ residues $\to$ domains in biomolecules (Xu et al., 10 Jun 2025), functional groups in molecules (Liu et al., 17 May 2024), or clusters in collective motion (Linot et al., 31 Dec 2024))—or implicitly—via information propagation through multiple directional or matrix-valued message passing layers where intermediate outputs encode progressively global features (Han et al., 2022, Du et al., 2023).

In manifold contexts, such as MDGCNNs, information is lifted and propagated through stacks of directional convolutional layers, each producing new directional functions. Hierarchical composition allows the network to integrate local orientation information over large geodesic distances and only resolves rotational ambiguity in the final pooling operation (Poulenard et al., 2018).

In graph-structured and geometric data, hierarchical pooling mechanisms (e.g., E-Pool in (Han et al., 2022), Cross-Space Interaction Pooling in (Xu et al., 10 Jun 2025), or graph-level coarsening in (Liu et al., 17 May 2024, Luo et al., 2023)) aggregate neighborhood features into higher-order representations while preserving equivariance, typically through attention or learnable pooling functions that are themselves symmetry compatible.

4. Directionality and Higher-Order Message Passing

Capturing directionality—either in the graph (via directed edges or asymmetric roles for source and target), in geometric relations (via bond vectors, angles, and dihedrals), or in positional encodings—is critical for modelling phenomena in both discrete and continuous domains.

Techniques include:

Dual or complex embeddings (encoding amplitude and phase), where conjugate pairs of node features preserve directed roles throughout the hierarchy (e.g., DUPLEX (Ke et al., 8 Jun 2024)).
Asymmetric message passing: e.g., outgoing and incoming neighborhood embeddings (DIAGRAM (Kefato et al., 2020)) or source/target vectors in directional GCN/autoencoders (Kollias et al., 2022).
Many-body equivariant interactions: Two-body directional messages often cancel (e.g., in force fields), so extension to explicit N-body or triplet interactions is necessary for non-degenerate encoding of directional signals (Mao et al., 19 Jun 2024).
Direction-aware positional encodings for transformers, such as eigenvectors of the magnetic Laplacian or random-walk encodings that capture the asymmetry of directed graphs (Geisler et al., 2023).

These strategies are unified by the use of message passing or update functions that are sensitive to, and preserve, the direction and orientation of features across hierarchical levels.

5. Implementation Mechanics and Representational Strategies

Efficient implementation of hierarchical equivariant directional graph encoders depends critically on discretization strategies (for manifolds, meshes, or molecular graphs), tensor and matrix representations, and attention or pooling schemes:

For mesh-based manifolds: geodesic polar coordinates, barycentric interpolation tensors, and discretized parallel transport encode spatial and directional relationships (Poulenard et al., 2018).
For general graphs: matrix and tensor-valued features (for example, directional matrices in EMMP (Han et al., 2022)), or explicit pairing of incoming/outgoing or source/target channels.
For biomolecular systems: dual-space representation (Euclidean for locality, spherical harmonics for global directionality) with bidirectional cross-space message passing and cross-space pooling to combine geometric and directional context (Xu et al., 10 Jun 2025).
For deep graph transformers: hierarchical, distance-aware structural encodings are directly introduced as bias terms into the attention computation, and can be designed at multiple coarsening scales to capture hierarchy (Luo et al., 2023).

Pooling operations must themselves be equivariant under the relevant group action. Typical schemes use attention weights or softmax-normalized assignments derived from equivariant node features, with normalization and aggregation preserving global symmetry (Han et al., 2022, Xu et al., 10 Jun 2025).

6. Applications and Empirical Performance

Hierarchical equivariant directional graph encoders have demonstrated improvements across a range of tasks:

Surface-based classification, segmentation, and non-rigid shape matching with higher accuracy and smoother feature propagation than geodesic CNNs (Poulenard et al., 2018).
Link prediction, node classification, and network reconstruction in directed graphs and citation networks, with superiority to both symmetric and undirected baselines (Kefato et al., 2020, Kollias et al., 2022).
Collective dynamics prediction in physical multi-agent systems, accurately tracking local and global behaviors, and conserving motion invariants such as Hamiltonians (Linot et al., 31 Dec 2024).
Molecular and biomolecular modeling, with state-of-the-art results in property prediction, force field modeling, and recovery of hierarchical physical or functional structure (Mao et al., 19 Jun 2024, Xu et al., 10 Jun 2025).
Graph-level tasks in large-scale graphs, achieving enhanced expressivity and generalization through hierarchical biasing of attention mechanisms (Luo et al., 2023).
Antibody-antigen docking, benefiting from multiscale (atom/residue) graphs and equivariant message passing for sample efficiency and chemical plausibility (Jin et al., 2022).

Consistently, these models exhibit improved robustness, faster convergence, and superior generalization metrics compared to non-hierarchical or non-equivariant baselines.

7. Theoretical Insights and Future Directions

Central theoretical insights include:

Directional convolution and N-body message passing produce equivariant encodings that avoid information loss associated with pooled or maximized directional responses (Poulenard et al., 2018, Mao et al., 19 Jun 2024).
Hierarchical equivariant maps via the wreath product formalism provide a framework for modular, additive-complexity handling of composite graph symmetries (Wang et al., 2020).
Hierarchical representations supporting multi-scale attention provide stricter expressivity than single-resolution models (proven via generalized tests such as GD-WL) (Luo et al., 2023).
Empirical and theoretical support shows that many-body, line-graph-based constructions are necessary to prevent degeneracies in directional information transmission.

Active research areas include scaling such architectures to billion-node graphs, designing symmetry-sensitive pooling operators, incorporating directional or group-theoretical positional encoding into deep transformers, and extending frameworks to complex or irregular manifolds with composite symmetries (Du et al., 2023, Liu et al., 17 May 2024, Xu et al., 10 Jun 2025).

Hierarchical equivariant directional graph encoders represent an overview of symmetry-aware signal processing, multi-scale architectural design, and advanced representation theory, enabling robust, physically consistent, and expressive modeling of complex geometric and graph-structured domains across scales.