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

TL;DR

The paper addresses the mismatch between Belief Propagation (BP) and finite-size symmetry in networks by introducing a symmetry-breaking mechanism: fixing a single source node, yielding Source-Node Belief Propagation (SNBP). SNBP modifies BP for bond percolation and the Ising model, deriving new boundary conditions and susceptibility propagation that enable accurate estimation of practical order parameters such as the percolation strength $S_1$, magnetization $m$, and their susceptibilities in finite, often tree-like networks, at no extra computational cost. Across synthetic trees, locally tree-like graphs, and 139 real networks, SNBP significantly outperforms conventional BP and naive mean-field approximations, closely matching ground-truth Monte Carlo results, especially where cycles are sparse. The approach preserves BP’s computational efficiency while extending its applicability to finite systems with global symmetries, with potential extensions to other symmetric models and inference tasks on networks.

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.

Figures (8)

• Figure 1: (a),(c) Percolation strength $P_{\infty}$ and (b),(d) susceptibility $\chi$ as functions of occupation probability $p$ for (a),(b) a 3-regular Cayley tree ($N=94$, $M=93$) and (c),(d) the same tree with two additional edges. Symbols indicate results from MC and SNMC. Curves show message-passing results inferred by MFA, SNMFA, BP, and SNBP. The curve 'MC ($P_{\infty}=S_1$)' uses $S_1$ as the order parameter and $\chi_\mathrm{practical}$ as the susceptibility, while 'MC ($P_{\infty}=0$)' computes $\chi_\mathrm{true}$. The red node in each network marks the source node, chosen to be of the highest degree.
• Figure 2: Inference on locally tree-like networks. Order parameters for the percolation model are shown in (a), (c), and (d), and for the Ising model in (b). Simulations were run for the Norwegian board directors network ($N=179$, $M=184$), an Erdos-Renyi random graph ($N=94$, $M=139$), and a scale-free network ($N=80$, $M=156$) generated using the Barabasi-Albert model with $k=2$ out-edges per arriving node. The red node in each network marks the source node.
• Figure 3: Inference for the percolation strength on spatial networks: (a) $8\times8$ square lattice, (b) random geometric graph ($N=100$, $\langle{k}\rangle = 5.9$). Both BP and SNBP fail to reproduce MC results, showing significant deviations over a wide range of $p$. Red node in networks marks source node.
• Figure 4: Comparison of SNBP errors with BP and MFA errors for (a),(c) percolation model and (b),(d) the Ising model on 139 real networks. (a),(b) SNBP outperforms BP: $\Delta_{\mathrm{SNBP-MC}}$ is generally smaller than $\Delta_{\mathrm{BP-MC}}$, particularly for small cyclomatic numbers. (c),(d) SNBP outperforms MFA: $\Delta_{\mathrm{SNBP-MC}}$ is almost always smaller than $\Delta_{\mathrm{MFA-MC}}$. The gap increases as $\langle{k}\rangle$ decreases.
• Figure S1: Supplemental figure for Fig. 1. (a),(c) Percolation strength $P_{\infty}$ and (b),(d) susceptibility $\chi$ as functions of occupation probability $p$ for (a),(b) a 3-regular Cayley tree ($N=94$) with one additional edge and (c),(d) the same tree with five additional edges. 'MC ($P_{\infty}=S_1$)' uses $S_1$ as the order parameter and $\chi_\mathrm{practical}$ as the susceptibility. 'MC ($P_{\infty}=0$)' computes $\chi_\mathrm{true}$. Red node in networks marks source node.