Geodesic
Ergodic behavior of graph entropy
Kieffer, John1 ; Yang, En-hui2
1 Department of Electrical Engineering, University of Minnesota, 200 Union Street SE, Minneapolis, MN 55455
2 Department of Mathematics, Nankai University, Tianjin 300071, P. R. China
Electronic research announcements of the American Mathematical Society, Tome 03 (1997), pp. 11-16 Cet article a éte moissonné depuis la source American Mathematical Society

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For a positive integer $n$, let $X^n$ be the vector formed by the first $n$ samples of a stationary ergodic finite alphabet process. The vector $X^n$ is hierarchically represented via a finite rooted acyclic directed graph $G_n$. Each terminal vertex of $G_n$ carries a label from the process alphabet, and $X^n$ can be reconstituted as the sequence of labels at the ends of the paths from root vertex to terminal vertex in $G_n$. The entropy $H(G_n)$ of the graph $G_n$ is defined as a nonnegative real number computed in terms of the number of incident edges to each vertex of $G_n$. An algorithm is given which assigns to $G_n$ a binary codeword from which $G_n$ can be reconstructed, such that the length of the codeword is approximately equal to $H(G_n)$. It is shown that if the number of edges of $G_n$ is $o(n)$, then the sequence $\{H(G_n)/n\}$ converges almost surely to the entropy of the process.
DOI : 10.1090/S1079-6762-97-00018-8

Kieffer, John 1 ; Yang, En-hui 2

1 Department of Electrical Engineering, University of Minnesota, 200 Union Street SE, Minneapolis, MN 55455
2 Department of Mathematics, Nankai University, Tianjin 300071, P. R. China 
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Kieffer, John; Yang, En-hui. Ergodic behavior of graph entropy. Electronic research announcements of the American Mathematical Society, Tome 03 (1997), pp. 11-16. doi: 10.1090/S1079-6762-97-00018-8

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