The Annals of Applied Probability

Information flow on trees

Elchanan Mossel and Yuval Peres

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Abstract

Consider a tree network $T$, where each edge acts as an independent copy of a given channel $M$, and information is propagated from the root. For which $T$ and $M$ does the configuration obtained at level $n$ of $T$ typically contain significant information on the root variable? This problem arose independently in biology, information theory and statistical physics.

For all $b$, we construct a channel for which the variable at the root of the break $b$-ary tree is independent of the configuration at the second level of that tree, yet for sufficiently large $B>b$, the mutual information between the configuration at level $n$ of the $B$-ary tree and the root variable is bounded away from zero for all $n$. This construction is related to Reed--Solomon codes.

We improve the upper bounds on information flow for asymmetric binary channels (which correspond to the Ising model with an external field) and for symmetric $q$-ary channels (which correspond to Potts models).

Let $\lam_2(M)$ denote the second largest eigenvalue of $M$, in absolute value. A CLT of Kesten and Stigum implies that if $b |\lam_2(M)|^2 >1$, then the census of the variables at any level of the $b$-ary tree, contains significant information on the root variable. We establish a converse: If $b |\lam_2(M)|^2 < 1$, then the census of the variables at level $n$ of the $b$-ary tree is asymptotically independent of the root variable. This contrasts with examples where $b |\lam_2(M)|^2 <1$, yet the configuration at level $n$ is not asymptotically independent of the root variable.

Article information

Source
Ann. Appl. Probab., Volume 13, Number 3 (2003), 817-844.

Dates
First available in Project Euclid: 6 August 2003

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1060202828

Digital Object Identifier
doi:10.1214/aoap/1060202828

Mathematical Reviews number (MathSciNet)
MR1994038

Zentralblatt MATH identifier
1050.60082

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F05: Central limit and other weak theorems 94B99: None of the above, but in this section

Keywords
Information flow reconstruction problem tree index Markov chain Markov random field census second Eigenvalue Thompson's principle Reed-Solomon codes secret sharing

Citation

Mossel, Elchanan; Peres, Yuval. Information flow on trees. Ann. Appl. Probab. 13 (2003), no. 3, 817--844. doi:10.1214/aoap/1060202828. https://projecteuclid.org/euclid.aoap/1060202828


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