Effect of degree correlations above the first shell on the percolation transition

Abstract: The use of degree-degree correlations to model realistic networks which are characterized by their Pearson's coefficient, has become widespread. However the effect on how different correlation algorithms produce different results on processes on top of them, has not yet been discussed. In this letter, using different correlation algorithms to generate assortative networks, we show that for very assortative networks the behavior of the main observables in percolation processes depends on the algorithm used to build the network. The different alghoritms used here introduce different inner structures that are missed in Pearson's coefficient. We explain the different behaviors through a generalization of Pearson's coefficient that allows to study the correlations at chemical distances l from a root node. We apply our findings to real networks.