Structural robustness of networks with degree-degree correlations between second-nearest neighbors

We numerically investigate the robustness of networks with degree-degree correlations between nodes separated by distance $l=2$ in terms of shortest path length. The degree-degree correlation between the $l$-th nearest neighbors can be quantified by Pearson's correlation coefficient $r_l$ for the degrees of two nodes at distance $l$. We introduce $l$-th nearest-neighbor correlated random networks ($l$-NNCRNs) that are degree-degree correlated at less than or equal to the $l$-th nearest neighbor scale and maximally random at farther scales. We generate $2$-NNCRNs with various $r_1$ and $r_2$ using two steps of random edge rewiring based on the Metropolis-Hastings algorithm and compare their robustness against failures of nodes and edges. As typical cases of homogeneous and heterogeneous degree distributions, we adopted Poisson and power law distributions. Our results show that the range of $r_2$ differs depending on the degree distribution and the value of $r_1$. Moreover, comparing $2$-NNCRNs sharing the same degree distribution and $r_1$, we demonstrate that a higher $r_2$ makes a network more robust against random node/edge failures as well as degree-based targeted attacks, regardless of whether $r_1$ is positive or negative.