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

2026-07-25

AI Summary

This study addresses the challenge of reconstructing the true structure and dynamics of social systems governed by majority-rule interactions and operating near criticality, using generative models. To this end, the authors introduce for the first time multistate Gaussian visible units into a Deep Belief Network (DBN) and employ a “dreaming” mechanism that fixes a subset of visible units to generate samples. A convolutional neural network–based discrete thermometer is further designed to verify whether the generated samples preserve physical consistency and criticality. Experimental results demonstrate that the proposed DBN can produce samples maintaining critical characteristics even under input noise, with physical observables degrading only gradually as noise increases. This confirms the model’s robust capability to faithfully reconstruct the critical state of complex social systems.

Abstract

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.

Problem

• Boltzmann machines
• majority rule
• criticality
• generative modeling
• social systems

Innovation

• Boltzmann machines
• deep belief networks
• criticality
• majority rule
• non-binary visible units