Latent geometry explains nested patterns in complex group interactions
Latent geometry organizes higher-order interactions
Social and Information Networks
Summary
Complex systems often involve interactions among groups of different sizes, which tend to be organized in nested patterns—smaller interactions fitting inside bigger ones. The authors propose a mathematical model that uses a hidden geometric space to connect these interactions, showing how nestedness can naturally arise and persist in such systems. Their model demonstrates different behaviors depending on system size and closely matches patterns seen in real-world data. This suggests that an underlying geometric structure might be key to understanding how complex group interactions are arranged.
higher-order interactionsnestednesslatent geometryhypergraphsthermodynamic limitfinite-size effectsgeometric modelcomplex systems
Authors
Berné L. Nortier, Jasper van der Kolk, Robert Jankowski, Simon Dobson, M. Ángeles Serrano, Federico Battiston, Marián Boguñá
Abstract
Higher-order structures offer a natural representation of complex systems that involve interactions between groups of different sizes. A widespread feature of their higher-order structure is nestedness, whereby interactions involving smaller groups are contained within larger ones. Yet, why interactions of different orders organise into nested structures remains largely unexplained. Here, we introduce an analytically tractable geometric model of higher-order networks in which a single latent geometric space couples interactions across orders, leading to the spontaneous emergence of nestedness. We show analytically and numerically that nestedness undergoes a transition between a nested geometric regime, where it remains finite in the thermodynamic limit, and a regime where it vanishes with system size. In this regime, we uncover a weakly geometric range characterized by an anomalously slow finite-size decay, allowing substantial nestedness to persist in finite systems even when its asymptotic value vanishes. Finally, with a single geometric coupling parameter, the model reproduces the nestedness profiles observed in real-world hypergraphs across different domains. Our results reveal latent geometry as a simple organising principle underlying the nested organisation of higher-order interactions.