# On the convergence of nonlinear averaging dynamics with three-body interactions on hypergraphs

arXiv: https://arxiv.org/abs/2304.07203 · HTML: https://mlsanity.com/paper/2304.07203/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 61 references that are in the map.
- First posted on arXiv 14 Apr 2023.

## Builds on (most cited first, 12 of 12)

- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [Nonlinear Higher-Order Label Spreading](https://mlsanity.com/paper/2006.04762/) (2020)
- [Random walks on hypergraphs](https://mlsanity.com/paper/1911.06523/) (2019)
- [A Tutorial on Modeling and Analysis of Dynamic Social Networks. Part I](https://mlsanity.com/paper/1701.06307/) (2017)
- [Fundamental structures of dynamic social networks](https://mlsanity.com/paper/1506.04704/) (2015)
- [Graph partitions and cluster synchronization in networks of oscillators](https://mlsanity.com/paper/1608.04283/) (2016)
- [Modelling Non-Linear Consensus Dynamics on Hypergraphs](https://mlsanity.com/paper/2007.09391/) (2020)
- [A Bounded-Confidence Model of Opinion Dynamics on Hypergraphs](https://mlsanity.com/paper/2102.06825/) (2021)
- [A network dynamics approach to chemical reaction networks](https://mlsanity.com/paper/1502.02247/) (2015)
- [A Tutorial on Modeling and Analysis of Dynamic Social Networks. Part II](https://mlsanity.com/paper/1801.06719/) (2018)
- [Consensus dynamics on temporal hypergraphs](https://mlsanity.com/paper/2109.04985/) (2021)
- [Comparing Graph Spectra of Adjacency and Laplacian Matrices](https://mlsanity.com/paper/1712.03769/) (2017)

## Abstract (arXiv)

Complex networked systems in fields such as physics, biology, and social sciences often involve interactions that extend beyond simple pairwise ones. Hypergraphs serve as powerful modeling tools for describing and analyzing the intricate behaviors of systems with multi-body interactions. Herein, we investigate a discrete-time nonlinear averaging dynamics with three-body interactions: an underlying hypergraph, comprising triples as hyperedges, delineates the structure of these interactions, while the vertices update their states through a weighted, state-dependent average of neighboring pairs' states. This dynamics captures reinforcing group effects, such as peer pressure, and exhibits higher-order dynamical effects resulting from a complex interplay between initial states, hypergraph topology, and nonlinearity of the update. Differently from linear averaging dynamics on graphs with two-body interactions, this model does not converge to the average of the initial states but rather induces a shift. By assuming random initial states and by making some regularity and density assumptions on the hypergraph, we prove that the dynamics converges to a multiplicatively-shifted average of the initial states, with high probability. We further characterize the shift as a function of two parameters describing the initial state and interaction strength, as well as the convergence time as a function of the hypergraph structure.