HM-LDM: A Hybrid-Membership Latent Distance Model

TL;DR

The paper introduces HM-LDM, a novel latent distance model that unifies soft and hard community detection in network embeddings by constraining the latent space to a simplex, enabling accurate and interpretable community detection.

Contribution

It proposes a hybrid-membership latent distance model (HM-LDM) that combines features of existing models and allows systematic control over community detection granularity.

Findings

HM-LDM accurately embeds networks and detects communities.

The model can produce both soft and hard community assignments.

Experimental results show the model's effectiveness in various regimes.

Abstract

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.