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

Ignacio A. Perez, Dana Vaknin Ben Porath, Cristian E. La Rocca, Sergey V. Buldyrev, Lidia A. Braunstein, Shlomo Havlin

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


Cascading failures are catastrophic processes that can destroy the functionality of a system, thus, understanding their development in real infrastructures is of vital importance. This may lead to a better management of everyday complex infrastructures relevant to modern societies, e.g., electrical power grids, communication and traffic networks. In this paper we examine the Motter-Lai model (2002 Phys. Rev. E 66 065102) of cascading failures induced by overloads in both isotropic and anisotropic spatial networks, generated by placing nodes in a square lattice and using various distributions of link lengths and angles. Anisotropy has not been earlier considered in the Motter-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 given attribute of the system manifests, such as power lines that follow a city built parallel to the coast. We analyze the evolution of the cascading failures for systems with different strengths of anisotropy and show that the anisotropy causes a greater spread of damage along the preferential direction of links. We identify the critical linear size, l c, for a square shaped localized attack, which satisfies with high probability that above l c the cascading disrupts the giant component of functional nodes, while below l c the damage does not spread. We find that, for networks with any characteristic link length, their robustness decreases with the strength of the anisotropy. We show that the value of l c is finite and independent of the system size (for large systems), both for isotropic and anisotropic networks. Thus, in contrast to random attacks, where the critical fraction 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 localized attacks in interdependent spatial networks (Berezin et al 2015 Sci. Rep. 5 8934). Finally, we measure the final distribution of functional cluster sizes and find a power-law behavior, with exponents similar to regular percolation. This indicates that, after the cascade which destroys the giant component, the system is at a percolation critical point. Additionally, we observe a crossover in the value of the distribution exponent, from critical percolation in a two-dimensional lattice for strong spatial embedding, to mean-field percolation for weak embedding.

Original languageEnglish
Article number043045
JournalNew Journal of Physics
Issue number4
StatePublished - 1 Apr 2022

Bibliographical note

Publisher Copyright:
© 2022 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft.


IAP, CEL and LAB wish to thank to UNMdP (EXA 956/20), FONCyT (PICT 1422/2019) and CONICET, Argentina, for financial support. SH wishes to thank the Israel Science Foundation, the Binational Israel-China Science Foundation (Grant No. 3132/19), the BIU Center for Research in Applied Cryptography and Cyber Security, NSF-BSF (Grant No. 2019740), the EU H2020 project RISE (Project No. 821115), the EU H2020 DIT4TRAM, and DTRA (Grant No. HDTRA-1-19-1-0016) for financial support. DV thanks the PBC of the Council for Higher Education of Israel for the Fellowship Grant. SVB acknowledges the partial support of this research through the Dr Bernard W Gamson Computational Science Center at Yeshiva College. This research was supported by a Grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel (Grant No. 2020255).

FundersFunder number
Binational Israel-China Science Foundation3132/19
EU H2020821115
EU H2020 DIT4TRAMHDTRA-1-19-1-0016
Yeshiva University
United States-Israel Binational Science Foundation2020255
Consejo Nacional de Investigaciones Científicas y Técnicas
Israel Science Foundation
Council for Higher Education
Fondo para la Investigación Científica y TecnológicaPICT 1422/2019
Universidad Nacional de Mar del PlataEXA 956/20


    • anisotropy
    • cascading failures
    • complex networks
    • overloads


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