# The non-tightness of the reconstruction threshold of a 4 states symmetric model with different in-block and out-block mutations

## TL;DR

This paper investigates the reconstruction problem in a 4-state symmetric stochastic block model with varying transition probabilities, establishing conditions under which the reconstruction threshold is not tight, revealing a complex phase where information is theoretically recoverable but computationally hard.

## Contribution

It provides the first rigorous analysis of the non-tightness of the reconstruction threshold in a 4-state stochastic block model with asymmetric transition probabilities.

## Key findings

- Identifies conditions for non-tightness of the reconstruction threshold.
- Extends understanding of phase transitions in multi-state stochastic block models.
- Highlights the complexity of the hybrid-hard phase in 4-state models.

## Abstract

The tree reconstruction problem is to collect and analyze massive data at the $n$th level of the tree, to identify whether there is non-vanishing information of the root, as $n$ goes to infinity. Its connection to the clustering problem in the setting of the stochastic block model, which has wide applications in machine learning and data mining, has been well established. For the stochastic block model, an "information-theoretically-solvable-but-computationally-hard" region, or say "hybrid-hard phase", appears whenever the reconstruction bound is not tight of the corresponding reconstruction on the tree problem. Although it has been studied in numerous contexts, the existing literature with rigorous reconstruction thresholds established are very limited, and it becomes extremely challenging when the model under investigation has $4$ states (the stochastic block model with $4$ communities). In this paper, inspired by the newly proposed $q_1+q_2$ stochastic block model, we study a $4$ states symmetric model with different in-block and out-block transition probabilities, and rigorously give the conditions for the non-tightness of the reconstruction threshold.