三态对称信道的精确重构阈值
The exact reconstruction threshold for the three-state symmetric channel
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具体内容
稿件研究三态对称广播过程在正则b叉树（b≥2）与均值d>1的观测Poisson Galton–Watson树上的重构问题，作者声称重构恰在dλ²>1时发生，正则模型中d=b，且等号处无论信道参数符号如何均不重构；Poisson情形对树与自旋取平均而不以存活为条件。作者称这解决了全度数三态正则树预测，并结合已知树到图与算法结果，给出对称三社区稀疏随机块模型的精确弱恢复阈值。
- 作者声称三态对称广播在正则树与Poisson Galton–Watson树上重构阈值均为dλ²>1
- 等号处对信道参数两种符号均不重构，Poisson优势不以存活为条件
- 结合已知结果给出三社区稀疏随机块模型的精确弱恢复阈值
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英文原始摘要
We determine the exact reconstruction threshold for the symmetric three-state broadcast process on every regular b-ary tree, b ≥ 2, and every observed Poisson Galton–Watson tree of mean d > 1. Reconstruction occurs exactly when $`d\lambda^2\gt 1`$, with d = b in the regular model; there is non-reconstruction at equality for either sign of the channel parameter. The Poisson advantage is averaged over trees and spins without conditioning on survival. This resolves the all-degree three-state regular-tree prediction. Combining the Poisson theorem with known tree-to-graph and algorithmic results gives the exact weak-recovery threshold for the symmetric three-community sparse stochastic block model with independent uniform labels, fixed within- and between-community rates $`a,b\gt 0`$, and mean degree $`(a+2b)/3\gt 1`$: recovery is possible exactly when $`(a-b)^2\gt 3(a+2b)`$. Above this threshold it is achievable in $`O(n\log n)`$ time; at or below it, weak recovery is information-theoretically impossible.
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