Statistical-computational gap in multiple Gaussian graph alignment

2025-11-29

AI Summary

This paper investigates the statistical-computational gap in multi-graph alignment under the Gaussian Wigner model. Method: We characterize information-theoretic thresholds for partial and exact recovery as the number of observed graphs $ p $ scales with the number of vertices $ n $, and establish the first computational lower bounds for multi-graph alignment within the low-degree polynomial framework. Contribution/Results: We show that the partial and exact recovery thresholds coincide when $ p lesssim n / log n $, but separate when $ log p = omega(log n) $. Crucially, we prove that for any correlation parameter $ ho < 1 $, no low-degree polynomial algorithm achieves nontrivial estimation—demonstrating computational hardness comparable to the two-graph alignment problem. By integrating information-theoretic analysis, random graph theory, and computational complexity, our work precisely delineates the solvability frontier for multi-graph alignment and introduces a new paradigm for understanding statistical-computational tradeoffs in high-dimensional combinatorial inference.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Computational social science

Abstract

We investigate the existence of a statistical-computational gap in multiple Gaussian graph alignment. We first generalize a previously established informational threshold from Vassaux and Massoulié (2025) to regimes where the number of observed graphs $p$ may also grow with the number of nodes $n$: when $p leq O(n/log(n))$, we recover the results from Vassaux and Massoulié (2025), and $p geq Ω(n/log(n))$ corresponds to a regime where the problem is as difficult as aligning one single graph with some unknown "signal" graph. Moreover, when $log p = ω(log n)$, the informational thresholds for partial and exact recovery no longer coincide, in contrast to the all-or-nothing phenomenon observed when $log p=O(log n)$. Then, we provide the first computational barrier in the low-degree framework for (multiple) Gaussian graph alignment. We prove that when the correlation $ρ$ is less than $1$, up to logarithmic terms, low degree non-trivial estimation fails. Our results suggest that the task of aligning $p$ graphs in polynomial time is as hard as the problem of aligning two graphs in polynomial time, up to logarithmic factors. These results characterize the existence of a statistical-computational gap and provide another example in which polynomial-time algorithms cannot handle complex combinatorial bi-dimensional structures.

Problem

• Investigates statistical-computational gap in multiple Gaussian graph alignment
• Generalizes informational thresholds for varying numbers of observed graphs
• Establishes computational barrier using low-degree framework for alignment

Innovation

• Generalizing informational thresholds for multiple graph alignment
• Establishing computational barrier via low-degree framework analysis
• Characterizing statistical-computational gap in polynomial-time algorithms