How Does Topology Bias Distort Message Passing? A Dirichlet Energy Perspective

TL;DR

This work examines topology bias in graph-based recommender systems through a Dirichlet-energy lens, revealing that standard message passing amplifies biases toward highly connected items. It introduces Test-time Simplicial Propagation (TSP), a plug-in inference approach that propagates information over higher-order simplices via a Hodge-Laplacian framework and a semantic-graph construction to rebalance embeddings. The method comprises semantic graph construction, intra-simplex smoothing, inter-simplex propagation, and multi-order fusion, leading to improved tail-item recommendations while preserving or enhancing overall quality across five real-world datasets. Empirical results show consistent tail gains, balanced embedding distributions, and favorable scalability, suggesting practical applicability for debiasing in production RS pipelines.

Abstract

Graph-based recommender systems have achieved remarkable effectiveness by modeling high-order interactions between users and items. However, such approaches are significantly undermined by popularity bias, which distorts the interaction graph's structure, referred to as topology bias. This leads to overrepresentation of popular items, thereby reinforcing biases and fairness issues through the user-system feedback loop. Despite attempts to study this effect, most prior work focuses on the embedding or gradient level bias, overlooking how topology bias fundamentally distorts the message passing process itself. We bridge this gap by providing an empirical and theoretical analysis from a Dirichlet energy perspective, revealing that graph message passing inherently amplifies topology bias and consistently benefits highly connected nodes. To address these limitations, we propose Test-time Simplicial Propagation (TSP), which extends message passing to higher-order simplicial complexes. By incorporating richer structures beyond pairwise connections, TSP mitigates harmful topology bias and substantially improves the representation and recommendation of long-tail items during inference. Extensive experiments across five real-world datasets demonstrate the superiority of our approach in mitigating topology bias and enhancing recommendation quality.

Figures (10)

• Figure 1: The degree and energy distribution of LightGCN embeddings on Gowalla dataset.
• Figure 2: The pipeline of our proposed TSP, consisting of four main steps: (1) Semantic Graph Construction, which builds a semantic graph based on node embeddings; (2) Intra-Simplex Smoothing, which aggregates information within each simplex; (3) Inter-Simplex Propagation, which performs message passing between simplices; and (4) Multi-Order Fusion, which combines embeddings from different orders of simplices for the final recommendation.
• Figure 2: Ablation study on Gowalla dataset. We report the overall and 20% tail item results of Top 20 performance.
• Figure 3: The overall performance of TSP and other baselines on Gowalla dataset.
• Figure 4: t-SNE visualization of item embeddings learned on Gowalla dataset. The top figures plot the Gaussian KDE of embeddings projected to $\mathbb{R}^2$. The darker the color, the more items are in this region. The figures below show the KDE of angles (i.e. $arctan2(y, x)$ for each point $(x, y)$).

Theorems & Definitions (20)

• Definition 1: Graph Dirichlet Energy lap
• Definition 2: Local Dirichlet Energy
• Definition 3: $k$-simplex
• Definition 4: Simplicial Complex
• Definition 5: Boundary Matrix
• Definition 6: Hodge Laplacian hodgebook
• Lemma 3.1
• Corollary 3.1
• Lemma 3.2
• Theorem 1: Hodge Decomposition hodgebook