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|Citation:||Physica A: Statistical Mechanics and its Applications|
|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 rep- resenting integers as vertices and by drawing cliques between M vertices every time that M dis- tinct 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 exponen- tial cut-o®. We also show that the properties of this network di®er considerably from the features of scale-free networks generated through preferential attachment. Remarkably we also recover a power law for the clustering coe±cient. 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 ° » 4:5 We then analyse clustering and ageing and compare our results to the ones obtained in the Pythagorean case.|
|Appears in Collections:||Dept of Mathematics Research Papers|
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