# Mercator: uncovering faithful hyperbolic embeddings of complex networks

arXiv: https://arxiv.org/abs/1904.10814 · HTML: https://mlsanity.com/paper/1904.10814/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Apr 2019.

## Built on by (most cited first, 10 of 10)

- [Machine Learning Partners in Criminal Networks](https://mlsanity.com/paper/2209.03171/) (2022)
- [Exact and rapid linear clustering of networks with dynamic programming](https://mlsanity.com/paper/2301.10403/) (2023)
- [Zoo Guide to Network Embedding](https://mlsanity.com/paper/2305.03474/) (2023)
- [Symmetry-driven embedding of networks in hyperbolic space](https://mlsanity.com/paper/2406.10711/) (2024)
- [Modelling brain connectomes networks: Solv is a worthy competitor to hyperbolic geometry!](https://mlsanity.com/paper/2407.16077/) (2024)
- [Uncovering the hidden core-periphery structure in hyperbolic networks](https://mlsanity.com/paper/2406.19953/) (2024)
- [Hyperbolic Graph Embeddings: a Survey and an Evaluation on Anomaly Detection](https://mlsanity.com/paper/2512.18826/) (2025)
- [Leveraging Non-linear Dimension Reduction and Random Walk Co-occurrence for Node Embedding](https://mlsanity.com/paper/2602.24069/) (2026)
- [Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization](https://mlsanity.com/paper/2607.27767/) (2026)
- [Hyperbolic Graph Embedders for Link Prediction and Topology Reconstruction](https://mlsanity.com/paper/2608.07029/) (2026)

## Abstract (arXiv)

We introduce Mercator, a reliable embedding method to map real complex networks into their hyperbolic latent geometry. The method assumes that the structure of networks is well described by the Popularity×Similarity mathbbS1/mathbbH2 static geometric network model, which can accommodate arbitrary degree distributions and reproduces many pivotal properties of real networks, including self-similarity patterns. The algorithm mixes machine learning and maximum likelihood approaches to infer the coordinates of the nodes in the underlying hyperbolic disk with the best matching between the observed network topology and the geometric model. In its fast mode, Mercator uses a model-adjusted machine learning technique performing dimensional reduction to produce a fast and accurate map, whose quality already outperform other embedding algorithms in the literature. In the refined Mercator mode, the fast-mode embedding result is taken as an initial condition in a Maximum Likelihood estimation, which significantly improves the quality of the final embedding. Apart from its accuracy as an embedding tool, Mercator has the clear advantage of systematically inferring not only node orderings, or angular positions, but also the hidden degrees and global model parameters, and has the ability to embed networks with arbitrary degree distributions. Overall, our results suggest that mixing machine learning and maximum likelihood techniques in a model-dependent framework can boost the meaningful mapping of complex networks.