Integer Networks

TL;DR

This paper studies the topological properties of integer networks where vertices are integers connected by divisibility, revealing bounded diameter, high clustering, and power-law degree distribution, contrasting with real-world networks.

Contribution

It provides the first rigorous proofs of topological properties of integer networks, including bounded diameter and specific degree distribution.

Findings

Diameter has a constant upper bound independent of network size

Integer networks are highly clustered with clustering coefficient around 0.34

Degree distribution follows a power-law with exponent approximately 2.4

Abstract

Inspired by Pythagoras's belief that numbers are the absolute reality, we obtain some demonstrational results about topological properties of integer networks, in which the vertices represent integers and two vertices are neighbors if and only if there exists a divisibility relation between them. We strictly prove that the diameter of networks has a constant upper bound independent to the network size $N$, which is completely different from the extensively studied real-life networks with their average distance increasing logarithmically to $N$ as $L\sim \texttt{ln}N$ or $L\sim \texttt{lnln}N$. Further more, the integer networks is high clustered, with clustered coefficient $C\approx 0.34$, and display power-law degree distribution of exponent $\gamma\approx 2.4$.