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In this paper we study the Motter-Lai model of cascading failures induced byoverloads in both isotropic and anisotropic spatial networks, generated byplacing nodes in a square lattice and using various distributions of linklengths and angles. Anisotropy has not been earlier considered in theMotter-Lai model and is a real feature that may affect the cascading failures.This could reflect the existence of a preferred direction in which a givenattribute of the system manifests, such as power lines that follow a city builtparallel to the coast. We show that the anisotropy causes a greater spread ofdamage along the preferential direction of links. We also identify the criticallinear size, $l_c$, for a square shaped localized attack, which satisfies withhigh probability that above $l_c$ the cascading disrupts the giant component offunctional nodes, while below $l_c$ the damage does not spread. We find that,for networks with any characteristic link length, their robustness decreaseswith the strength of the anisotropy. We show that the value of $l_c$ is finiteand independent of the system size (for large systems), both for isotropic andanisotropic networks. Thus, in contrast to random attacks, where the criticalfraction of nodes that survive the initial attack, $p_c$, is usually below 1,here $p_c = 1$. Note that the analogy to $p_c = 1$ is also found for localizedattacks in interdependent spatial networks. Finally, we measure the finaldistribution of functional cluster sizes and find a power-law behavior, withexponents similar to regular percolation. This indicates that, after thecascade which destroys the giant component, the system is at a percolationcritical point. Additionally, we observe a crossover in the value of thedistribution exponent, from critical percolation in a two-dimensional latticefor strong spatial embedding, to mean-field percolation for weak embedding.

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