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

Seongmin Kim1 and Alec Kirkley1,2,3,*

  • *Contact author: alec.w.kirkley@gmail.com
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Phys. Rev. Research 8, 013202 – Published 23 February, 2026

DOI: https://doi.org/10.1103/pgxg-mhjr

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 treelike ones, at no additional computational cost.

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Phys. Rev. Research 8, 013202– Published 23 February, 2026
  • Received 31 October 2025
  • Accepted 21 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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