Hypergraph Lower Ricci Curvature

• Hypergraph Lower Ricci Curvature is a discrete geometric invariant that measures the curvedness of hyperedges using local connectivity and common neighbor overlaps.
• It balances the expressivity of optimal transport methods with computational efficiency via a closed-form, bounded formulation ideal for large hypergraphs.
• HLRC supports practical tasks such as community detection, semantic label recovery, and anomaly detection by distinguishing intra- from inter-community hyperedges.

Hypergraph Lower Ricci Curvature (HLRC) is a discrete geometric invariant that quantifies the "curvedness" of hyperedges within a hypergraph, capturing both local connectivity and the higher-order organization characteristic of complex systems. HLRC is constructed to achieve a balance between the rich expressivity of optimal transport (Ollivier-Ricci) curvature and the computational efficiency of simple combinatorial metrics. This approach enables scalable and interpretable analysis of higher-order interactions in large real-world hypergraphs, supporting tasks such as community detection, semantic label discovery, anomaly detection, and generative modeling.

1. Mathematical Formulation and Definition

Hypergraph Lower Ricci Curvature (HLRC) is defined for each hyperedge in an undirected, unweighted hypergraph and reflects both the local and global structure of the surrounding topology. For a hyperedge $e$ of size (degree) $d_e > 2$, with constituent nodes $v \in e$, neighbor counts $n_v = |\mathcal{N}(v)|$ (the size of the 1-neighborhood of $v$), and $n_e$ the number of common neighbors shared by all nodes of $e$, the HLRC of $e$ is given by: $$\boxed{ \mathrm{HLRC}(e) = \sum_{v \in e} \frac{1}{n_v} + \frac{n_e + d_e/2 - 1}{\max_{v \in e} n_v} + \frac{n_e + d_e/2 - 1}{\min_{v \in e} n_v} - 1 }$$

• The first term captures local node sparsity/cohesion by aggregating the inverse neighbor sizes.
• The second and third terms account for the normalized overlap profile of the hyperedge, balancing the most and least connected nodes in $e$.
• The final subtraction ensures the curvature is strictly bounded.

HLRC is designed so that $-1 < \mathrm{HLRC}(e) \leq 1$, with maximal values corresponding to hypercliques and highly cohesive modules, and minimal values to thin, bridge-like, or boundary hyperedges.

2. Comparison with Forman-Ricci and Ollivier-Ricci Curvatures

Two prevailing hypergraph curvature methods exhibit key limitations addressed by HLRC:

• Forman-Ricci curvature (HFRC): A purely combinatorial and linear metric, $F(e) = 2d_e - \sum_{v\in e} n_v$. HFRC is fast but:
  • Only reflects local node degrees.
  • Lacks sensitivity to higher-order or mesoscopic features such as community boundaries.
  • Is not universally bounded, complicating interpretation and comparison.
• Ollivier-Ricci curvature (HORC): An optimal-transport-based generalization requiring the solution of multi-marginal transport problems among all node measures within $e$. While expressive, HORC is computationally intractable for large or dense hyperedges due to the curse of dimensionality inherent in multi-marginal transport.

HLRC combines the strengths of both:

• It is closed-form and efficiently computable, scaling linearly to mildly superlinearly with hypergraph size.
• It provides geometric sensitivity akin to HORC, distinguishing between intra-community and boundary/bridge hyperedges.
• It is bounded, interpretable, and robust across diverse datasets.

3. Structural Sensitivity and Interpretability

HLRC is specifically constructed to reveal key structural motifs in hypergraphs:

• Intra-community hyperedges: High HLRC values (approaching $1$) indicate hyperedges lying within tightly knit communities.
• Inter-community/bridge hyperedges: Negative HLRC values signify hyperedges that bridge distinct regions or function as boundaries.
• Flat/regular regions: HLRC around 0 is consistent with grid-like or "flat" geometry.

In synthetic block models, HLRC yields bimodal distributions on intra- vs. inter-community hyperedges, with statistically robust separation. In real-world datasets (such as school contact networks and co-authorship hypergraphs), HLRC identifies both known and latent boundaries, detects cohesive subgroups, and highlights functionally critical bridging hyperedges.

HLRC also supports the temporal analysis of network geometry: in time-evolving collaboration networks, mean HLRC captures the shift between fragmentation and cohesion, revealing large-scale organizational trends.

4. Practical Applications and Clustering

The closed-form and bounded nature of HLRC enables its use in several high-level tasks:

• Community Detection: HLRC cleanly distinguishes community-interior from community-bridging hyperedges, supporting accurate module identification.
• Clustering of Hypergraph Collections: Aggregated HLRC histograms serve as highly discriminative features for embedding and clustering hypergraphs by global structure, yielding higher ARI/AMI than transport-based or degree-based alternatives.
• Semantic Label Recovery: In labeled hypergraphs (e.g., co-authorship, forum threads), HLRC distributions reflect domain-specific distinctions, reliably tracking theoretical vs. applied venues or discovering latent classes.
• Anomaly Detection: Highly negative HLRC values correlate with anomalous or functionally unusual hyperedges, such as sparse cross-community links or outlying collaborations.
• Generative Modeling: HLRC provides explicit geometric constraints, guiding the design and validation of generative hypergraph models to match observed higher-order architecture.

5. Algorithmic Complexity and Scalability

HLRC is defined in closed form and leverages only local statistics (neighbor counts and their overlaps). Its evaluation requires only neighbor set operations and basic arithmetic, with total computational complexity on the order of $O(m \bar{d_e} n)$ (where $m$ is the number of hyperedges, $\bar{d_e}$ their average size, and $n$ the number of nodes). This enables application to hypergraphs with millions of hyperedges and nodes in seconds to minutes, in contrast to the combinatorial explosion faced by HORC.

Empirical benchmarks show that for large hypergraphs (e.g., co-authorship, forum, musical networks), HLRC completes in practical run times, often within a small constant factor of HFRC and vastly outpacing HORC, which becomes infeasible at scale.

6. Implications for Network Science and Data Analysis

HLRC's core properties underpin several broader uses:

• Node Classification: By exposing which nodes participate in high- vs. low-curvature hyperedges, HLRC enables the construction of geometrically principled features for node classification and semi-supervised learning.
• Robust Feature for ML Pipelines: HLRC histograms, due to their geometric sensitivity, act as rich, interpretable features for embedding, clustering, and transfer learning in higher-order network analytics.
• Generative Models and Anomaly Scoring: HLRC serves as a geometric target for generative models of hypergraphs (enforcing realistic higher-order structure) and as an outlier score for detecting structural anomalies and bottlenecks.

7. Summary Table: HLRC in the Context of Hypergraph Curvature Metrics

Property	HLRC	HORC	HFRC
Closed form definition	Yes	No	Yes
Computational scalability	High (linear)	Low (superlinear/exponential)	High (linear)
Bounded and interpretable output	Yes ($[-1,1]$)	Yes (wider)	No (unbounded)
Geometric sensitivity	High	High	Low
Bridge/community separation	Yes	Yes	No

HLRC unifies algorithmic simplicity and geometric relevance, facilitating scalable analytics and robust higher-order pattern recognition in the rapidly growing field of hypergraph-based network science.