Cascading failures in isotropic and anisotropic spatial networks induced by localized attacks and overloads

In this paper we study the Motter-Lai model of cascading failures induced by overloads in both isotropic and anisotropic spatial networks, generated placing nodes a square lattice using various distributions link lengths angles. Anisotropy has not been earlier considered is real feature that may affect failures. This could reflect existence preferred direction which given attribute system manifests, such as power lines follow city built parallel to coast. We show anisotropy causes greater spread damage along preferential links. also identify critical linear size, $l_c$, for shaped localized attack, satisfies with high probability above $l_c$ disrupts giant component functional nodes, while below does spread. find that, networks any characteristic length, their robustness decreases strength anisotropy. value finite independent size (for large systems), networks. Thus, contrast random attacks, where fraction survive initial $p_c$, usually 1, here $p_c = 1$. Note analogy 1$ found attacks interdependent Finally, measure final distribution cluster sizes power-law behavior, exponents similar regular percolation. indicates after cascade destroys component, at percolation point. Additionally, observe crossover exponent, from two-dimensional strong embedding, mean-field weak embedding.