Diophantine networks

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 x2+y2=z2 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. We then study the network associated with the equation x2+y2=z showing that the degree distribution is consistent with a power law for several decades of values of k and that, after having reached a minimum, the distribution begins rising again. The power-law exponent, in this case, is given by γ 4.5 We then analyse clustering and ageing and compare our results to the ones obtained in the Pythagorean case