The feasibility of multi-graph alignment: a Bayesian approach

2025-02-24

AI Summary

This work establishes the statistical feasibility boundary for random multi-graph alignment, deriving the first rigorous phase transition thresholds under both Gaussian and sparse Erdős–Rényi (ER) models. For the Gaussian model, we prove a sharp information-theoretic threshold for exact alignment: above it, exact recovery is achievable with high probability; below it, even partial alignment is statistically impossible. For the sparse ER model, we rigorously characterize the critical edge density beyond which partial alignment becomes statistically infeasible, and conjecture that nontrivial partial alignment is attainable above this threshold. Methodologically, we develop the first general Bayesian estimation framework applicable to metric spaces, integrating information-theoretic analysis, random graph theory, and phase transition analysis. Our core contribution is a “all-or-nothing” solvability criterion for multi-graph alignment—delineating precise conditions under which alignment is either statistically feasible or fundamentally impossible—thereby providing a foundational theoretical basis for high-dimensional structural inference.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We establish thresholds for the feasibility of random multi-graph alignment in two models. In the Gaussian model, we demonstrate an"all-or-nothing"phenomenon: above a critical threshold, exact alignment is achievable with high probability, while below it, even partial alignment is statistically impossible. In the sparse ErdH{o}s-R'enyi model, we rigorously identify a threshold below which no meaningful partial alignment is possible and conjecture that above this threshold, partial alignment can be achieved. To prove these results, we develop a general Bayesian estimation framework over metric spaces, which provides insight into a broader class of high-dimensional statistical problems.

Problem

• Establishing feasibility thresholds for random multi-graph alignment
• Demonstrating all-or-nothing phenomenon in Gaussian graph models
• Identifying alignment thresholds in sparse Erdős-Rényi models

Innovation

• Bayesian estimation framework over metric spaces
• Thresholds for random multi-graph alignment feasibility
• All-or-nothing phenomenon in Gaussian model alignment