Families and clustering in a natural numbers network

TL;DR

This paper constructs a network based on natural numbers connected through shared prime factors, revealing how selective linking criteria influence the network's small-world properties and degree distribution.

Contribution

It introduces a prime-based natural numbers network model and analyzes how connection rules affect its small-world characteristics and degree distribution scaling.

Findings

Degree distribution scales linearly with network size

Selective linking criteria induce small-world effects

Network properties differ from random graphs under certain conditions

Abstract

We develop a network in which the natural numbers are the vertices. We use the decomposition of natural numbers by prime numbers to establish the connections. We perform data collapse and show that the degree distribution of these networks scale linearly with the number of vertices. We compare the average distance of the network and the clustering coefficient with the distance and clustering coefficient of the corresponding random graph. In case we set connections among vertices each time the numbers share a common prime number the network is not a small-world type. If the criterium for establishing links becomes more selective, only prime numbers greater than $p_l$ are used to establish links, the network shows small-world effect, it means, it has high clustering coefficient and low distance.