PageRank of integers

TL;DR

This paper constructs a directed network of integers based on divisibility, analyzes its Google matrix, and finds that the PageRank distribution follows a Zipf-like law, offering a new ordering of integers.

Contribution

It introduces a novel network model of integers based on divisibility and derives a semi-analytical formula for their PageRank, enabling analysis of very large matrices.

Findings

PageRank of integers follows an inverse proportionality to its index.

The spectrum of the Google matrix has a large spectral gap.

A semi-analytical expression for PageRank allows analysis of billion-sized matrices.

Abstract

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.