Hyperspherical Graph Representation Learning via Adaptive Neighbor-Mean Alignment and Uniformity

TL;DR

HyperGRL addresses the limitations of traditional GRL by operating on a unit hypersphere and eliminating negative sampling through two adversarially coupled objectives: a $k$-order neighbor-mean alignment loss $\\ ext{L}_{\\mathrm{align}}^k$ and a sampling-free uniformity term $\\text{L}_{\\mathrm{unif}}$. An entropy-guided adaptive balancing mechanism dynamically tunes the trade-off between alignment and uniformity based on a proxy entropy $H_{\\mathrm{proxy}}$, stabilizing training without manual hyperparameter tuning. The framework employs a GNN encoder (e.g., TransformerConv) to produce embeddings $\\mathbf{H}$ that are normalized to $\\mathbf{Z}$ on $\\mathbb{S}^{d-1}$, optimizing the joint objective $\\mathcal{L} = \\mathcal{L}_{\\mathrm{align}}^k + \\alpha \\\mathcal{L}_{\\mathrm{unif}}$. Across eight benchmark datasets and three downstream tasks (node classification, clustering, and link prediction), HyperGRL achieves state-of-the-art or competitive results, demonstrating strong generalization, robustness to hyperparameters, and the practical value of geometric, negative-free GRL for heterogeneous graphs.

Abstract

Graph representation learning (GRL) aims to encode structural and semantic dependencies of graph-structured data into low-dimensional embeddings. However, existing GRL methods often rely on surrogate contrastive objectives or mutual information maximization, which typically demand complex architectures, negative sampling strategies, and sensitive hyperparameter tuning. These design choices may induce over-smoothing, over-squashing, and training instability. In this work, we propose HyperGRL, a unified framework for hyperspherical graph representation learning via adaptive neighbor-mean alignment and sampling-free uniformity. HyperGRL embeds nodes on a unit hypersphere through two adversarially coupled objectives: neighbor-mean alignment and sampling-free uniformity. The alignment objective uses the mean representation of each node's local neighborhood to construct semantically grounded, stable targets that capture shared structural and feature patterns. The uniformity objective formulates dispersion via an L2-based hyperspherical regularization, encouraging globally uniform embedding distributions while preserving discriminative information. To further stabilize training, we introduce an entropy-guided adaptive balancing mechanism that dynamically regulates the interplay between alignment and uniformity without requiring manual tuning. Extensive experiments on node classification, node clustering, and link prediction demonstrate that HyperGRL delivers superior representation quality and generalization across diverse graph structures, achieving average improvements of 1.49%, 0.86%, and 0.74% over the strongest existing methods, respectively. These findings highlight the effectiveness of geometrically grounded, sampling-free contrastive objectives for graph representation learning.

Figures (7)

• Figure 1: Overview of HyperGRL. Given a graph $\mathcal{G}(\mathbf{A}, \mathbf{X})$, a graph augmentation $\mathcal{T}$ produces an augmented graph $\mathcal{G}(\mathbf{A}^\prime, \mathbf{X}^\prime)$. This augmented graph is then encoded by a GNN $f_{\boldsymbol{\theta}}$ to generate node representations $\mathbf{H}$. These representations are subsequently normalized onto a hyperspherical space to yield $\mathbf{Z}$, where training is driven by two complementary objectives: the Neighbor-Mean Alignment loss $\mathcal{L}_{\text{align}}$, which pulls each node toward the mean representation of its neighbors, and the Uniformity loss $\mathcal{L}_{\text{unif}}$, which encourages the node representations to be uniformly distributed across the unit hypersphere.
• Figure 2: Effect of adaptive $\alpha$.
• Figure 3: Impact of the target entropy $H_{\textrm{target}}$.
• Figure 4: Impact of the neighbor-mean order $k$.
• Figure 5: Performance on node classification (Accuracy %) with different hidden dimensions.