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The information reconstruction problem on an infinite tree, is to collect andanalyze massive data samples at the $n$th level of the tree to identify whetherthere is non-vanishing information of the root, as $n$ goes to infinity. Thisproblem has wide applications in various fields such as biology, informationtheory and statistical physics, and its close connections to cluster learning,data mining and deep learning have been well established in recent years.Although it has been studied in numerous contexts, the existing literatureswith rigorous reconstruction thresholds established are very limited. In thispaper, motivated by a classical deoxyribonucleic acid (DNA) evolution model,the F$81$ model, and taking into consideration of the Chargaff's parity rule byallowing the existence of a guanine-cytosine content bias, we study the noisechannel in terms of a $4\times 4$-state asymmetric probability transitionmatrix with community effects, for four nucleobases of DNA. The correspondinginformation reconstruction problem in molecular phylogenetics is explored, bymeans of refined analyses of moment recursion, in-depth concentrationestimates, and thorough investigations on an asymptotic $4$-dimensionalnonlinear second order dynamical system. We rigorously show that thereconstruction bound is not tight when the sum of the base frequencies ofadenine and thymine falls in the interval $\left(0,1/2-\sqrt{3}/6\right)\bigcup\left(1/2+\sqrt{3}/6,1\right)$, which is the first rigorous result onasymmetric noisy channels with community effects.

Big Data Information Reconstruction on an Infinite Tree for a $4\times 4$-state Asymmetric Model with Community Effects