Scaling in Local Optimal Paths Cracks

TL;DR

This paper investigates how local cracks influence the overall crack network in complex systems, revealing that the size of cracks along local optimal paths scales predictably and independently of disorder levels.

Contribution

It introduces a modified OPC model analyzing crack scaling behavior and demonstrates a universal power-law relation for crack mass along local paths.

Findings

Cracked link mass scales as a power-law with local path length.

Scaling behavior is independent of the disorder parameter.

Finite-size scaling confirms the universality of the crack growth pattern.

Abstract

How local cracks can contribute to the global cracks landscape is a goal of several scientific topics, for example, how bottlenecks can impact the robustness of traffic into a city? In one direction, cracks from cascading failures into networks were generated using a modified Optimal Path-Cracking (OPC) model proposed by Andrade et al \cite{Andrade2009}. In this model, we broke links of maximum energies from optimal paths between two sites with internal (euclidean) distances $l$ in networks with linear size $L$. Each link of this network has an energy value that scales with a power-law that can be controlled using a parameter of the disorder $\beta$. Using finite-size scaling and the exponents from percolation theory we found that the mass of the cracked links on local optimal paths scales with a power-law $l^{0.4}$ as a separable equation from $L$ and that can be independent of the disorder parameter.