Correlations in binary sequences and a generalized Zipf analysis

Andras Czirók, Rosario N. Mantegna, Shlomo Havlin, H. Eugene Stanley

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64 Scopus citations


We investigate correlated binary sequences using an n-tuple Zipf analysis, where we define "words" as strings of length n, and calculate the normalized frequency of occurrence ω(R) of "words" as a function of the word rank R. We analyze sequences with short-range Markovian correlations, as well as those with long-range correlations generated by three different methods: inverse Fourier transformation, Lévy walks, and the expansion-modification system. We study the relation between the exponent α characterizing long-range correlations and the exponent ζ characterizing power-law behavior in the Zipf plot. We also introduce a function P(ω), the frequency density, which is related to the inverse Zipf function R(ω), and find a simple relationship between ζ and ψ, where ω(R)∼R-ζ and P(ω)∼ω-ψ. Further, for Markovian sequences, we derive an approximate form for P(ω). Finally, we study the effect of a coarse-graining "renormalization" on sequences with Markovian and with long-range correlations.

Original languageEnglish
Pages (from-to)446-452
Number of pages7
JournalPhysical Review E
Issue number1
StatePublished - 1995


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