[Paper Review] Hyperbolic triangulations and discrete random graphs

Eryk Kopczyński, Dorota Celińska-Kopczyńska|arXiv (Cornell University)|Jul 4, 2017

Complex Network Analysis Techniques20 references4 citations

TL;DR

This paper introduces the Discrete Hyperbolic Random Graph (DHRG) model, a discrete analog of the continuous hyperbolic random graph model, where nodes are mapped to vertices of hyperbolic triangulations. Using a tally counter data structure, the authors enable efficient computation of log-likelihood for maximum likelihood embedding, achieving linear-time updates and outperforming prior methods in accuracy and scalability on real-world networks like GitHub.

ABSTRACT

The hyperbolic random graph model (HRG) has proven useful in the analysis of scale-free networks, which are ubiquitous in many fields, from social network analysis to biology. However, working with this model is algorithmically and conceptually challenging because of the nature of the distances in the hyperbolic plane. In this paper we study the algorithmic properties of regularly generated triangulations in the hyperbolic plane. We propose a discrete variant of the HRG model where nodes are mapped to the vertices of such a triangulation; our algorithms allow us to work with this model in a simple yet efficient way. We present experimental results conducted on real world networks to evaluate the practical benefits of DHRG in comparison to the HRG model.

Motivation & Objective

• To address the algorithmic and conceptual challenges of working with continuous hyperbolic random graphs (HRGs), particularly in sampling and maximum likelihood embedding (MLE).
• To develop a discrete analog of the HRG model—DHRG—where nodes are mapped to vertices of regularly generated hyperbolic triangulations, enabling more efficient computation.
• To design a tally counter data structure that supports fast distance-based queries and dynamic remapping for efficient log-likelihood computation in DHRG embeddings.
• To evaluate the DHRG model on real-world networks, such as GitHub’s following graph, and compare its performance and accuracy against the state-of-the-art BFKL embedder.
• To explore the potential of using discrete hyperbolic triangulations as a foundation for more accurate and scalable hyperbolic network embedding algorithms.

Proposed method

• Propose the DHRG model, where node positions are mapped to vertices of a Gromov hyperbolic triangulation (e.g., G_{710}, G_{711}), and edge probabilities depend on graph distance in the triangulation.
• Introduce the tally counter data structure, which maintains a set of vertices and supports O(1 + deg(v)) time updates when remapping a vertex v, and O(1) time queries for the number of vertices within distance d from a given vertex.
• Use the tally counter to compute the log-likelihood of a DHRG embedding in O(n + m) time, enabling efficient MLE-based embedding.
• Implement a local search optimization procedure to iteratively improve the embedding by repositioning nodes and recomputing the log-likelihood using the tally counter.
• Apply the DHRG framework to real-world networks, including a GitHub social network with 74,946 nodes and 537,952 edges, using triangulations like G_{710} and G_{810}.
• Compare the DHRG results with the BFKL embedder, using both the original and a corrected Θ(n²) log-likelihood computation to validate accuracy.

Experimental results

Research questions

• RQ1 Can a discrete analog of the hyperbolic random graph model be constructed using regularly generated hyperbolic triangulations to improve algorithmic efficiency?
• RQ2 Can a tally counter data structure enable O(n + m) log-likelihood computation and O(1 + deg(v)) dynamic updates for node remapping in DHRG embeddings?
• RQ3 How does the DHRG model compare in accuracy and performance to the continuous HRG model and the BFKL embedder on real-world scale-free networks?
• RQ4 To what extent do coarser triangulations like G_{810} reduce runtime without significantly degrading embedding quality?
• RQ5 Is the DHRG model robust enough to improve upon existing MLE embeddings, especially when the BFKL method may suffer from approximation errors?

Key findings

• The DHRG model with the tally counter enables efficient O(n + m) computation of log-likelihood, significantly improving upon the O(n²) baseline for likelihood evaluation.
• Dynamic remapping of a node v in the DHRG model can be computed in O(1 + deg(v)) time using the tally counter, enabling fast local search optimization.
• On the GitHub network (n=74,946), the DHRG method achieved a log-likelihood of -3,527,397 after 100 iterations of local search, improving from an initial L₀ = -4,364,526.
• The DHRG approach with G_{710} achieved a log-likelihood of -170,886 after 7 iterations, outperforming the BFKL embedder’s reported -131,634 in some cases, suggesting higher accuracy when corrected for approximation errors.
• Using coarser triangulations like G_{810} reduced per-iteration runtime by ~25% with minimal loss in quality, indicating a favorable trade-off between resolution and performance.
• The authors suspect their log-likelihood values are more accurate than BFKL’s due to the latter’s reliance on an approximate likelihood function, which was validated by re-implementing the Θ(n²) version with matching results.