# Exploration of the generative capabilities of Boltzmann machines applied to social systems under the majority rule

arXiv: https://arxiv.org/abs/2607.23349 · HTML: https://mlsanity.com/paper/2607.23349/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 27 references that are in the map.
- First posted on arXiv 25 Jul 2026.

## Builds on (most cited first, 9 of 9)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Discovering Phase Transitions with Unsupervised Learning](https://mlsanity.com/paper/1606.00318/) (2016)
- [Equilibrium and non-Equilibrium regimes in the learning of Restricted Boltzmann Machines](https://mlsanity.com/paper/2105.13889/) (2021)
- [Gaussian-Bernoulli RBMs Without Tears](https://mlsanity.com/paper/2210.10318/) (2022)
- [Thermodynamics and Feature Extraction by Machine Learning](https://mlsanity.com/paper/1810.08179/) (2018)
- [Opinion dynamics in social networks: From models to data](https://mlsanity.com/paper/2201.01322/) (2022)
- [Feature extraction of machine learning and phase transition point of Ising model](https://mlsanity.com/paper/2111.11166/) (2021)

## Built on by (most cited first, 1 of 1)

- [Classifying Directional Trajectories Near Criticality in the Three-State Majority-Vote Model with Deep Belief Networks and Bidirectional GRUs](https://mlsanity.com/paper/2608.18235/) (2026)

## Abstract (arXiv)

We study the generative capabilities of Boltzmann machines to recover systems governed by the majority rule under critical conditions. To this end, we train deep belief networks (DBNs) with different configurations, where the first layer can use Gaussian visible units with more than two states (i.e., non-binary units). We then allow the DBN to "dream" samples conditioned on visible units that we keep fixed, and we measure the deviation of this dreamed system from the real one. We also corroborate, using a discrete thermometer based on a convolutional network, that the reconstructions remain in a critical state. Across several training sessions with different architectures, we show that, despite the complexity of the problem, the DBN can recover samples that remain critical even under input noise, with a gradual degradation of physical observables relative to the original sample.