Abstract
For DNA sequences of various species we construct the Google matrix of Markov transitions between nearby words composed of several letters. The statistical distribution of matrix elements of this matrix is shown to be described by a power law with the exponent being close to those of outgoing links in such scale-free networks as the World Wide Web (WWW). At the same time the sum of ingoing matrix elements is characterized by the exponent being significantly larger than those typical for WWW networks. This results in a slow algebraic decay of the PageRank probability determined by the distribution of ingoing elements. The spectrum of is characterized by a large gap leading to a rapid relaxation process on the DNA sequence networks. We introduce the PageRank proximity correlator between different species which determines their statistical similarity from the view point of Markov chains. The properties of other eigenstates of the Google matrix are also discussed. Our results establish scale-free features of DNA sequence networks showing their similarities and distinctions with the WWW and linguistic networks.
Highlights
The theory of Markov chains [1] finds impressive modern applications to information retrieval and ranking of directed networks including the World Wide Web (WWW) where the number of nodes is counted by tens of billions
The PageRank algorithm (PRA) [2] uses the concept of the Google matrix G and allows to rank all WWW nodes in an efficient way
For DNA we find an opposite situation where the Google matrix is almost full and zero matrix elements are practically absent. ingoing links is the
Summary
The theory of Markov chains [1] finds impressive modern applications to information retrieval and ranking of directed networks including the World Wide Web (WWW) where the number of nodes is counted by tens of billions. The image of the DNA Google matrix of HS is shown in Fig. 1 for words of 5 and 6 letters. The DNA Google matrix is more similar to the case of Twitter which is characterized by a strong connectivity of top PageRank nodes [21].
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