Percolation on interdependent one-dimensional long-range networks
Abstract
We study the emergence of a mutual giant component in a multiplex network comprised of two one-dimensional long-range networks on a lattice. The probability that two nodes on one lattice, with a distance r between them, are connected with an edge falls algebraically with distance as Cr−(1+σ), where 0 < σ< 1 and 0 < C ≤ 1. A mutual giant component exists if C > Ccmul, where Ccmul(σ) is a critical value of C that is dependent on σ. We find rigorous lower bounds on Ccmul, and a tree approximation whose estimate of Ccmul agrees with simulations within 0.6% for σ≤ 0.5. The multiplex network is significantly less stable than the single network case, as is evident by the larger value of Ccmul, which is up to 2.4 times the single-layer value. Our simulations indicate that the order of the transition depends on σ. For σ≤ 0.35 the mutual giant component collapses abruptly, in a first-order phase transition, while for σ≥ 0.4 the transition is continuous. When the dependency links connect random pairs of nodes, the transition is first order for all σ.
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BibTeX
@misc{sharabi2026percolation,
title = {{Percolation on interdependent one-dimensional long-range networks}},
author = {Adar Sharabi and Amir Bashan and Sergey V. Buldyrev and Guy Amit},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10097},
eprint = {2610.10097},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10097},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Percolation on interdependent one-dimensional long-range networks
AU - Adar Sharabi
AU - Amir Bashan
AU - Sergey V. Buldyrev
AU - Guy Amit
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10097
UR - https://arxiv.org/abs/2610.10097
AB - We study the emergence of a mutual giant component in a multiplex network comprised of two one-dimensional long-range networks on a lattice. The probability that two nodes on one lattice, with a distance r between them, are connected with an edge falls algebraically with distance as Cr^−(1+σ), where 0 < σ< 1 and 0 < C ≤ 1. A mutual giant component exists if C > C_c^mul, where C_c^mul(σ) is a critical value of C that is dependent on σ. We find rigorous lower bounds on C_c^mul, and a tree approximation whose estimate of C_c^mul agrees with simulations within 0.6% for σ≤ 0.5. The multiplex network is significantly less stable than the single network case, as is evident by the larger value of C_c^mul, which is up to 2.4 times the single-layer value. Our simulations indicate that the order of the transition depends on σ. For σ≤ 0.35 the mutual giant component collapses abruptly, in a first-order phase transition, while for σ≥ 0.4 the transition is continuous. When the dependency links connect random pairs of nodes, the transition is first order for all σ.
ER -