# Explosive neural networks via higher-order interactions in curved statistical manifolds

arXiv: https://arxiv.org/abs/2408.02326 · HTML: https://mlsanity.com/paper/2408.02326/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 71 references that are in the map.
- First posted on arXiv 5 Aug 2024.

## Builds on (most cited first, 8 of 8)

- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [The physics of higher-order interactions in complex systems](https://mlsanity.com/paper/2110.06023/) (2021)
- [In search of dispersed memories: Generative diffusion models are associative memory networks](https://mlsanity.com/paper/2309.17290/) (2023)
- [The statistical thermodynamics of generative diffusion models: Phase transitions, symmetry breaking and critical instability](https://mlsanity.com/paper/2310.17467/) (2023)
- [Efficient Optimization with Higher-Order Ising Machines](https://mlsanity.com/paper/2212.03426/) (2022)
- [Dense Hebbian neural networks: a replica symmetric picture of unsupervised learning](https://mlsanity.com/paper/2211.14067/) (2022)
- [Searching for collective behavior in a network of real neurons](https://mlsanity.com/paper/1306.3061/) (2013)
- [Explosive higher-order Kuramoto dynamics on simplicial complexes](https://mlsanity.com/paper/1912.04405/) (2019)

## Built on by (most cited first, 3 of 3)

- [AI in a vat: Fundamental limits of efficient world modelling for agent sandboxing and interpretability](https://mlsanity.com/paper/2504.04608/) (2025)
- [Self-Regulating Annealing in Heavy-Tailed Diffusion Models](https://mlsanity.com/paper/2606.01645/) (2026)
- [An Information-Geometric Approach to Artificial Curiosity](https://mlsanity.com/paper/2504.06355/) (2025)

## Abstract (arXiv)

Higher-order interactions underlie complex phenomena in systems such as biological and artificial neural networks, but their study is challenging due to the scarcity of tractable models. By leveraging a generalisation of the maximum entropy principle, we introduce curved neural networks as a class of models with a limited number of parameters that are particularly well-suited for studying higher-order phenomena. Through exact mean-field descriptions, we show that these curved neural networks implement a self-regulating annealing process that can accelerate memory retrieval, leading to explosive order-disorder phase transitions with multi-stability and hysteresis effects. Moreover, by analytically exploring their memory-retrieval capacity using the replica trick, we demonstrate that these networks can enhance memory capacity and robustness of retrieval over classical associative-memory networks. Overall, the proposed framework provides parsimonious models amenable to analytical study, revealing higher-order phenomena in complex networks.