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Bayesian Network Structural Consensus via Greedy Min-Cut Analysis

Pablo Torrijos, José M. Puerta, Juan A. Aledo, José A. Gámez

TL;DR

The paper tackles structural fusion of Bayesian networks in data-scarce or federated settings, where unrestricted fusion yields dense, high-treewidth graphs. It introduces Min-Cut Bayesian Network Consensus (MCBNC), a flow-based pruning method that replaces likelihood-based BES scoring with a max-flow–based edge criticality score and a single post hoc threshold $θ$ to drive pruning. Key contributions include the flow-based edge-support score, integration into the BES phase of GES, and a parameter-light pruning strategy that does not require data access. Empirical results across real and synthetic networks show that MCBNC delivers sparser, more structurally faithful consensus graphs than unrestricted fusion or input graphs, with competitive BDeu scores, and it scales well for federated settings. The approach enables scalable, data-agnostic BN structure fusion suitable for distributed privacy-preserving learning where only graph structures are shared.

Abstract

This paper presents the Min-Cut Bayesian Network Consensus (MCBNC) algorithm, a greedy method for structural consensus of Bayesian Networks (BNs), with applications in federated learning and model aggregation. MCBNC prunes weak edges from an initial unrestricted fusion using a structural score based on min-cut analysis, integrated into a modified Backward Equivalence Search (BES) phase of the Greedy Equivalence Search (GES) algorithm. The score quantifies edge support across input networks and is computed using max-flow. Unlike methods with fixed treewidth bounds, MCBNC introduces a pruning threshold $θ$ that can be selected post hoc using only structural information. Experiments on real-world BNs show that MCBNC yields sparser, more accurate consensus structures than both canonical fusion and the input networks. The method is scalable, data-agnostic, and well-suited for distributed or federated scenarios.

Bayesian Network Structural Consensus via Greedy Min-Cut Analysis

TL;DR

The paper tackles structural fusion of Bayesian networks in data-scarce or federated settings, where unrestricted fusion yields dense, high-treewidth graphs. It introduces Min-Cut Bayesian Network Consensus (MCBNC), a flow-based pruning method that replaces likelihood-based BES scoring with a max-flow–based edge criticality score and a single post hoc threshold to drive pruning. Key contributions include the flow-based edge-support score, integration into the BES phase of GES, and a parameter-light pruning strategy that does not require data access. Empirical results across real and synthetic networks show that MCBNC delivers sparser, more structurally faithful consensus graphs than unrestricted fusion or input graphs, with competitive BDeu scores, and it scales well for federated settings. The approach enables scalable, data-agnostic BN structure fusion suitable for distributed privacy-preserving learning where only graph structures are shared.

Abstract

This paper presents the Min-Cut Bayesian Network Consensus (MCBNC) algorithm, a greedy method for structural consensus of Bayesian Networks (BNs), with applications in federated learning and model aggregation. MCBNC prunes weak edges from an initial unrestricted fusion using a structural score based on min-cut analysis, integrated into a modified Backward Equivalence Search (BES) phase of the Greedy Equivalence Search (GES) algorithm. The score quantifies edge support across input networks and is computed using max-flow. Unlike methods with fixed treewidth bounds, MCBNC introduces a pruning threshold that can be selected post hoc using only structural information. Experiments on real-world BNs show that MCBNC yields sparser, more accurate consensus structures than both canonical fusion and the input networks. The method is scalable, data-agnostic, and well-suited for distributed or federated scenarios.

Paper Structure

This paper contains 33 sections, 3 theorems, 28 equations, 20 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Contributions.
  2. Paper organization.
  3. Treewidth.
  4. From fusion to consensus.
  5. Min-cut and max-flow.
  6. Ford-Fulkerson algorithm.
  7. Implementation assumptions.
  8. Experimental Protocol.
  9. Conditioning set size.
  10. Evaluation Metrics.
  11. Method.
  12. Interpretation.
  13. Extended Metric Definitions
  14. Structural Moral Hamming Distance (SMHD):
  15. Bayesian Dirichlet equivalent uniform (BDeu) Score:
  16. ...and 18 more sections

Key Result

Lemma 1

Let $\Psi_e^{(t)}$ be the criticality score of edge $e$ after the $t$-th deletion. Then $\Psi_e^{(t+1)} \ge \Psi_e^{(t)}$ for every remaining edge $e$.

Figures (20)

  • Figure 1: Mean SMHD to the gold-standard BN $G_{\text{gs}}$ across thresholds $\theta$ for each BN. Leftmost point: full fusion $G^+$. Rightmost: empty DAG $\emptyset$. Horizontal line: average SMHD of input BNs from GES to $G_{\text{gs}}$. Lower is better.
  • Figure 2: Mean BDeu score across thresholds $\theta$ for each BN. Leftmost point: full fusion $G^+$. Rightmost: empty DAG $\emptyset$. Horizontal lines: average of input BNs from GES (black) and gold-standard BN (purple). Higher is better.
  • Figure 3: Mean treewidth across pruning thresholds $\theta$ for each BN. Dashed lines: selected $\theta$ for each $\#$DAGs based on SMHD w.r.t. input BNs. Horizontal lines: average of input BNs from GES (black) and gold-standard BN (purple).
  • Figure 4: Total execution time vs. number of input DAGs.
  • Figure B1: Mean SMHD for synthetic experiments across different values of $\theta$.
  • ...and 15 more figures

Theorems & Definitions (5)

  • Lemma 1: Monotonicity of the criticality score
  • proof
  • Corollary 2: Score interpretation
  • Lemma 3: Complexity of MCBNC with Ford-Fulkerson
  • proof