PageRank of integers
This work introduces a novel ordering of integers based on PageRank, which is incremental as it applies an existing method to a new mathematical domain.
The authors tackled the problem of applying PageRank to a network of integers linked by divisibility, showing that the PageRank probability is inversely proportional to its index, similar to Zipf's law, and derived a semi-analytical expression enabling computation for matrices of up to a billion size.
We build up a directed network tracing links from a given integer to its divisors and analyze the properties of the Google matrix of this network. The PageRank vector of this matrix is computed numerically and it is shown that its probability is inversely proportional to the PageRank index thus being similar to the Zipf law and the dependence established for the World Wide Web. The spectrum of the Google matrix of integers is characterized by a large gap and a relatively small number of nonzero eigenvalues. A simple semi-analytical expression for the PageRank of integers is derived that allows to find this vector for matrices of billion size. This network provides a new PageRank order of integers.