Archetypal Graph Generative Models: Explainable and Identifiable Communities via Anchor-Dominant Convex Hulls

Representation learning is central to graph machine learning, powering tasks such as link prediction and node classification. However, most graph embeddings are hard to interpret, offering limited insight into how learned features relate to graph structure. Many networks naturally admit a role-mixture view, where nodes are best described as mixtures over latent archetypal factors. Motivated by this structure, we propose a compositional graph embedding framework grounded in Aitchison geometry, the canonical geometry for comparing mixtures. Nodes are represented as simplex-valued compositions and embedded via isometric log-ratio (ILR) coordinates, which preserve Aitchison distances while enabling unconstrained optimization in Euclidean space. This yields intrinsically interpretable embeddings whose geometry reflects relative trade-offs among archetypes and supports coherent behavior under component restriction; we consider both fixed and learnable ILR bases. Across node classification and link prediction, our method achieves competitive performance with strong baselines while providing explainability by construction rather than post-hoc. Finally, subcompositional coherence enables principled component restriction: removing and renormalizing subsets preserves a well-defined geometry, which we exploit via subcompositional dimensionality removal to probe how archetype groups influence representations and predictions.

Detecting communities in heterophilic graphs -- where connected nodes often belong to different classes -- is hard for unsupervised methods: classical modularity and spectral methods are feature agnostic, while deep graph-clustering methods rely on contrastive or generative machinery that is opaque. We propose Curvature-Guided Sheaf Diffusion (CGSD), a fully unsupervised community-detection algorithm that uses the discrete Forman--Ricci curvature of each edge as its single topological signal, propagated through every stage of an end-to-end pipeline. CGSD makes three concrete contributions: (i)~a curvature-gated sheaf-diffusion encoder that gates edge messages by $σ(κ_e)$ and is trained from three label-free structural losses (modularity, anti-collapse, curvature-weighted reconstruction); (ii)~a curvature-aware spectral clusterer (CSpec) that re-weights the $k$-NN affinity of the embedding by $σ(ακ_{e^*})$ before Ng--Jordan--Weiss; and (iii)~a unified label-free evaluation against nine truly-unsupervised baselines. On five heterophilic benchmarks (Cora, Cornell, Texas, Wisconsin, Chameleon), CGSD wins outright on Wisconsin and Chameleon and is competitive on the remaining three against nine unsupervised baselines. The gain over the strongest baseline is driven by the clusterer, not the encoder: on the same embedding, CSpec improves mean NMI from $0.091$ with $K$-Means to $0.107$ ($+15\%$, paired $t$-test $p=0.008$). The mechanism is interpretable: intra-community and inter-community curvature distributions are visibly separated. Code is open-sourced at https://github.com/woodywff/cgsd.

Community detection is a fundamental task in graph analytics that aims to identify cohesive groups of entities with similar behaviors or interests. Classic objective-driven methods struggle with complex graph structures, while deep-learning approaches improve performance at the expense of interpretability and rely on labeled data and training. Large language models (LLMs), with strong reasoning capabilities and world knowledge, are promising for interpretable, label-free community detection. To leverage these strengths, we propose LUCID, an LLM-guided, interpretable, training-free, and unsupervised community detection method. Inspired by phase-transition kinetics in natural systems, where complex structures emerge through initialization, merging, refinement, and selection, LUCID is designed as a four-stage pipeline. Within this pipeline, the LLM induces formal rules that translate implicit knowledge into explicit and interpretable logical structures. Specifically, (1) the Local-View Community Initialization stage encodes local graph structures using k-ego contexts and unsupervised node roles; (2) the Multi-factor Community Merge stage uses LLM-induced rules to iteratively merge local communities; (3) the Multi-grain Community Refinement stage applies LLM-induced coarse-to-fine rules in parallel to reduce boundary noise; and (4) the Global-view Community Selection stage identifies high-quality communities based on topological compactness and boundary clarity. Extensive experiments on real-world datasets demonstrate that LUCID, as an unsupervised approach, achieves state-of-the-art performance and consistently outperforms leading unsupervised and semi-supervised baselines.