Curvature-Guided Sheaf Diffusion for Unsupervised Community Detection on Heterophilic Graphs

Community detection in graphs is traditionally dominated by the homophily assumption—the idea that connected nodes share the same class. However, many real-world networks are heterophilic, meaning edges frequently connect nodes of different classes. Existing unsupervised methods, such as modularity maximization or standard spectral clustering, often fail on these graphs because they are either feature-agnostic or rely on assumptions that do not hold in heterophilic settings. This paper asks: can we use the local topological structure of a graph to improve unsupervised community detection without relying on node labels or homophily?

To address this, the author introduces Curvature-Guided Sheaf Diffusion (CGSD). The core innovation is the use of discrete Forman-Ricci curvature, a measure of local bottleneck structure, as a topological signal.

CGSD operates in three stages:

1. Curvature-Gated Encoder: A two-layer sheaf-diffusion encoder uses edge curvature to gate message passing. Edges with positive curvature (typically intra-community) are amplified, while those with negative curvature (inter-community bridges) are suppressed.
2. Label-Free Training: The encoder is trained using three structural losses: modularity, an anti-collapse loss to ensure distinct clusters, and a curvature-weighted reconstruction loss.
3. Curvature-Aware Spectral Clustering (CSpec): The final embedding is clustered using a modified spectral approach that re-weights the k-NN affinity matrix based on the curvature of the original edges before performing standard spectral decomposition.

CGSD demonstrates superior performance on heterophilic benchmarks compared to nine unsupervised baselines. Notably, the CSpec clusterer provides a significant performance boost over standard K-Means on the same embeddings, improving mean Normalized Mutual Information (NMI) by 15%. The mechanism is highly interpretable: on heterophilic graphs, intra-community and inter-community edges exhibit visibly distinct curvature distributions, allowing the algorithm to effectively separate communities that are otherwise difficult to distinguish.