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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 informationof the root, as $n$ goes to infinity. Its connection to the clustering problemin the setting of the stochastic block model, which has wide applications inmachine learning and data mining, has been well established. For the stochasticblock model, an "information-theoretically-solvable-but-computationally-hard"region, or say "hybrid-hard phase", appears whenever the reconstruction boundis not tight of the corresponding reconstruction on the tree problem. Althoughit has been studied in numerous contexts, the existing literature with rigorousreconstruction thresholds established are very limited, and it becomesextremely challenging when the model under investigation has $4$ states (thestochastic block model with $4$ communities). In this paper, inspired by thenewly proposed $q_1+q_2$ stochastic block model, we study a $4$ statessymmetric model with different in-block and out-block transition probabilities,and rigorously give the conditions for the non-tightness of the reconstructionthreshold.

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