# HM-LDM: A Hybrid-Membership Latent Distance Model

arXiv: https://arxiv.org/abs/2206.03463 · HTML: https://mlsanity.com/paper/2206.03463/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2022.

## Built on by (most cited first, 6 of 6)

- [Piecewise-Velocity Model for Learning Continuous-time Dynamic Node Representations](https://mlsanity.com/paper/2212.12345/) (2022)
- [Characterizing Polarization in Social Networks using the Signed Relational Latent Distance Model](https://mlsanity.com/paper/2301.09507/) (2023)
- [Aitchison Embeddings for Learning Compositional Graph Representations](https://mlsanity.com/paper/2605.00716/) (2026)
- [Rank Is Not Capacity: Spectral Occupancy for Latent Graph Models](https://mlsanity.com/paper/2605.11142/) (2026)
- [A Hybrid Membership Latent Distance Model for Unsigned and Signed Integer Weighted Networks](https://mlsanity.com/paper/2308.15293/) (2023)
- [Time to Cite: Modeling Citation Networks using the Dynamic Impact Single-Event Embedding Model](https://mlsanity.com/paper/2403.00032/) (2024)

## Abstract (arXiv)

A central aim of modeling complex networks is to accurately embed networks in order to detect structures and predict link and node properties. The latent space models (LSM) have become prominent frameworks for embedding networks and include the latent distance (LDM) and eigenmodel (LEM) as the most widely used LSM specifications. For latent community detection, the embedding space in LDMs has been endowed with a clustering model whereas LEMs have been constrained to part-based non-negative matrix factorization (NMF) inspired representations promoting community discovery. We presently reconcile LSMs with latent community detection by constraining the LDM representation to the D-simplex forming the hybrid-membership latent distance model (HM-LDM). We show that for sufficiently large simplex volumes this can be achieved without loss of expressive power whereas by extending the model to squared Euclidean distances, we recover the LEM formulation with constraints promoting part-based representations akin to NMF. Importantly, by systematically reducing the volume of the simplex, the model becomes unique and ultimately leads to hard assignments of nodes to simplex corners. We demonstrate experimentally how the proposed HM-LDM admits accurate node representations in regimes ensuring identifiability and valid community extraction. Importantly, HM-LDM naturally reconciles soft and hard community detection with network embeddings exploring a simple continuous optimization procedure on a volume constrained simplex that admits the systematic investigation of trade-offs between hard and mixed membership community detection.