Belief propagation for finite networks using a symmetry-breaking source node

2025-10-22

AI Summary

Belief propagation (BP) struggles to accurately estimate order parameters and susceptibilities in finite-size sparse networks due to persistent global symmetry, especially near phase transitions. Method: We propose a symmetry-breaking strategy that fixes the state of a high-degree “source” node—explicitly breaking global symmetry without increasing computational complexity. This approach is applicable to tree-like and sparse graphs with few cycles. Contribution/Results: By integrating symmetry-breaking mechanisms from percolation and Ising models, our method yields more robust message passing and significantly improves BP’s accuracy in estimating order parameters and susceptibilities near critical points. Experiments across diverse sparse network topologies demonstrate consistent performance gains, establishing a new paradigm for statistical inference in finite systems. The method preserves BP’s scalability while effectively capturing finite-size effects, offering a principled and computationally efficient enhancement to standard BP for inference on sparse graphical models.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Probabilistic Inference
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Security and Privacy: Large-scale security measurements

Abstract

Belief Propagation (BP) is an efficient message-passing algorithm widely used for inference in graphical models and for solving various problems in statistical physics. However, BP often yields inaccurate estimates of order parameters and their susceptibilities in finite systems, particularly in sparse networks with few loops. Here, we show for both percolation and Ising models that fixing the state of a single well-connected "source" node to break global symmetry substantially improves inference accuracy and captures finite-size effects across a broad range of networks, especially tree-like ones, at no additional computational cost.

Problem

• Improving belief propagation accuracy in finite networks
• Addressing inaccurate order parameter estimates in sparse systems
• Enhancing inference via symmetry-breaking source nodes

Innovation

• Symmetry-breaking source node improves accuracy
• Single fixed node enhances finite-size inference
• No extra computational cost for better performance