## Diophantine Networks

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BedognÃ©, C.

Masucci, A. P.

Rodgers, G. J.

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##### Abstract

##### Description

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.

##### Keywords

Physics - Physics and Society, Condensed Matter - Disordered Systems and Neural Networks, Physics - Data Analysis, Statistics and Probability