Divisibility patterns of natural numbers on a complex network

TL;DR

This paper explores the divisibility of natural numbers through a complex network model, revealing scale-free properties, a new 'stretching similarity' pattern, and smooth variation of network metrics with size, supported by analytical and numerical analysis.

Contribution

It introduces a novel network-based approach to study number divisibility, uncovering unique patterns and asymptotic behaviors not previously documented.

Findings

Network is scale-free with non-stationary degree distribution

Discovery of 'stretching similarity' pattern in local clustering

Network metrics vary smoothly with size and match analytical estimates

Abstract

Investigation of divisibility properties of natural numbers is one of the most important themes in the theory of numbers. Various tools have been developed over the centuries to discover and study the various patterns in the sequence of natural numbers in the context of divisibility. In the present paper, we study the divisibility of natural numbers using the framework of a growing complex network. In particular, using tools from the field of statistical inference, we show that the network is scale-free but has a non-stationary degree distribution. Along with this, we report a new kind of similarity pattern for the local clustering, which we call "stretching similarity", in this network. We also show that the various characteristics like average degree, global clustering coefficient and assortativity coefficient of the network vary smoothly with the size of the network. Using analytical arguments we estimate the asymptotic behavior of global clustering and average degree which is validated using numerical analysis.