Diophantine Networks

TL;DR

This paper introduces a novel class of deterministic networks based on Diophantine equations, linking algebraic properties to network topology, and analyzes their structural features, revealing power-law behaviors in degree distribution and clustering.

Contribution

It presents a new method of constructing networks from Diophantine equations, exploring their properties and contrasting them with traditional scale-free networks.

Findings

Degree distribution approximates a power law with exponential cutoff.

Network properties differ from preferential attachment scale-free networks.

Clustering coefficient also follows a power law.

Abstract

We introduce a new class of deterministic networks by associating networks with Diophantine equations, thus relating network topology to algebraic properties. The network is formed by representing integers as vertices and by drawing cliques between M vertices every time that M distinct integers satisfy the equation. We analyse the network generated by the Pythagorean equation $x^{2}+y^{2}= z^{2}$ showing that its degree distribution is well approximated by a power law with exponential cut-off. We also show that the properties of this network differ considerably from the features of scale-free networks generated through preferential attachment. Remarkably we also recover a power law for the clustering coefficient.