Exact reconstruction thresholds on hypertrees over a symmetric binary alphabet

2026-06-19

AI Summary

This work investigates the fundamental limits of information propagation in symmetric binary broadcasting models on hypertrees, aiming to precisely characterize phase transition thresholds in random NAE-SAT, hypergraph bicoloring, and the stochastic block model (SBM) on hypergraphs. By integrating refined robust non-reconstruction analysis, information-theoretic tools—including the theory of binary memoryless symmetric channels and channel comparison techniques—and computer-assisted rigorous population dynamics algorithms, the study establishes for the first time that when hyperedge cardinality is at most four, the condensation threshold of random NAE-SAT and hypergraph bicoloring coincides with the Kesten–Stigum bound at all temperatures. It also exactly determines the weak recovery threshold for hypergraph SBM under specific parameter regimes, thereby confirming the theoretical predictions of Ricci-Tersenghi et al. (2019) and delineating sharp phase transitions in a broad class of random constraint satisfaction problems and community detection models.

Abstract

We establish the exact reconstruction thresholds for a class of broadcasting models on hypertrees over a symmetric binary alphabet. As a consequence, we show that the condensation threshold coincides with the Kesten-Stigum threshold for random NAE-SAT and random hypergraph bicoloring with arity at most four at any temperature, confirming a prediction of Ricci-Tersenghi et al. '19. We also determine the exact weak recovery threshold for the two-community hypergraph stochastic block model in certain parameter regimes. The proof combines an improved robust non-reconstruction analysis with a computer-assisted rigorous population dynamics algorithm. It relies heavily on information-theoretic tools, in particular the theory of binary memoryless symmetric channels and channel comparison methods.

Problem

• reconstruction threshold
• hypertree
• binary alphabet
• condensation threshold
• hypergraph stochastic block model

Innovation

• reconstruction threshold
• hypertree
• Kesten-Stigum threshold
• hypergraph stochastic block model
• channel comparison