Ergodic behavior of graph entropy
Authors:
John Kieffer and Enhui Yang
Journal:
Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 1116
MSC (1991):
Primary 28D99; Secondary 60G10, 94A15
Published electronically:
March 12, 1997
MathSciNet review:
1433180
Fulltext PDF Free Access
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Abstract: For a positive integer , let be the vector formed by the first samples of a stationary ergodic finite alphabet process. The vector is hierarchically represented via a finite rooted acyclic directed graph . Each terminal vertex of carries a label from the process alphabet, and can be reconstituted as the sequence of labels at the ends of the paths from root vertex to terminal vertex in . The entropy of the graph is defined as a nonnegative real number computed in terms of the number of incident edges to each vertex of . An algorithm is given which assigns to a binary codeword from which can be reconstructed, such that the length of the codeword is approximately equal to . It is shown that if the number of edges of is , then the sequence converges almost surely to the entropy of the process.
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Additional Information
John Kieffer
Affiliation:
Department of Electrical Engineering, University of Minnesota, 200 Union Street SE, Minneapolis, MN 55455
Email:
kieffer@ee.umn.edu
Enhui Yang
Affiliation:
Department of Mathematics, Nankai University, Tianjin 300071, P. R. China
Email:
ehyang@irving.usc.edu
DOI:
http://dx.doi.org/10.1090/S1079676297000188
PII:
S 10796762(97)000188
Keywords:
Graphs,
entropy,
compression,
stationary ergodic process
Received by editor(s):
December 12, 1996
Published electronically:
March 12, 1997
Additional Notes:
This work was supported in part by the National Science Foundation under Grants NCR9304984 and NCR9627965
Communicated by:
Douglas Lind
Article copyright:
© Copyright 1997
American Mathematical Society
