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  • Open Access

Reducibility of higher-order to pairwise interactions: Social impact models on hypergraphs

Jaume Llabrés*, Raúl Toral, and Maxi San Miguel

Federico Vázquez

  • *Contact author: jaumellabres@ifisc.uib-csic.es
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Phys. Rev. Research 8, 023250 – Published 5 June, 2026

DOI: https://doi.org/10.1103/wy1x-3px8

Abstract

We show that a general class of node-update social impact models with higher-order interactions on hypergraphs can be exactly mapped to an equivalent model with pairwise interactions on a weighted projected network. This mapping preserves the microscopic probabilities of changing the state of the nodes. As a particular case, we introduce hypergraph-voter models, for which we compute the weights of the projected network, both analytically and numerically, across several hypergraph ensembles, and we characterize their ordering dynamics through simulations of both higher-order and reduced pairwise dynamics. For a linear social impact function (hypergraph-linear voter model), the weights of the projected network are static (state-independent), allowing us to develop a pair approximation that describes with accuracy the time evolution of macroscopic observables, which turn out to be independent of those weights. The macroscopic dynamics is thus equivalent to that of the standard voter model on the unweighted projected network. For a power-law social impact function (hypergraph-nonlinear voter model), the weights of the projected network depend on the instantaneous system configuration. Nevertheless, the nonlinear voter model on the unweighted projected network still reproduces the main macroscopic trends for well-connected hypergraphs.

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Phys. Rev. Research 8, 023250– Published 5 June, 2026
  • Received 4 February 2026
  • Accepted 9 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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