Equivalent-neighbor k-core percolation in two dimensions
Abstract
We perform large-scale numerical simulations to investigate the critical behavior of k-core percolation in two dimensions with an extended interaction range r. By systematically varying both the core index k and the interaction range r, we construct a comprehensive phase diagram in the (k,r) plane. In contrast to k-core percolation in infinite dimensions, no hybrid transition is observed in two dimensions: the phase diagram contains only a continuous transition regime and a strictly first-order regime, separated by a tricritical or critical-end point (ks,rs). For k<ks and r<rs, the transition is continuous and belongs to the universality class of standard two-dimensional (2D) percolation. For k>ks and finite r>rs, the transition is discontinuous, with no hybrid features or critical singularities. In this first-order regime, the pseudocritical point approaches the critical point as 1/ln L, where L is the linear system size, distinct from the L−d scaling typical of conventional thermodynamic first-order transitions in d dimensions. This logarithmic finite-size drift is consistent with a nucleation-driven mechanism, in which rare voids trigger the collapse of the finite-range k-core. These results demonstrate that geometric constraints can fundamentally alter the nature of k-core percolation found in finite dimensions.
Versions
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BibTeX
@misc{shi2026equivalentneighbor,
title = {{Equivalent-neighbor $k$-core percolation in two dimensions}},
author = {Qiyuan Shi and Ming Li and Youjin Deng},
howpublished = {arXiv},
year = {2026},
month = may,
doi = {10.48550/arxiv.2605.25812},
eprint = {2605.25812},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.25812},
}
RIS
TY - UNPB
TI - Equivalent-neighbor k-core percolation in two dimensions
AU - Qiyuan Shi
AU - Ming Li
AU - Youjin Deng
PY - 2026
DA - 2026/05/25
PB - arXiv
DO - 10.48550/arxiv.2605.25812
UR - https://arxiv.org/abs/2605.25812
AB - We perform large-scale numerical simulations to investigate the critical behavior of k-core percolation in two dimensions with an extended interaction range r. By systematically varying both the core index k and the interaction range r, we construct a comprehensive phase diagram in the (k,r) plane. In contrast to k-core percolation in infinite dimensions, no hybrid transition is observed in two dimensions: the phase diagram contains only a continuous transition regime and a strictly first-order regime, separated by a tricritical or critical-end point (k_s,r_s). For k<k_s and r<r_s, the transition is continuous and belongs to the universality class of standard two-dimensional (2D) percolation. For k>k_s and finite r>r_s, the transition is discontinuous, with no hybrid features or critical singularities. In this first-order regime, the pseudocritical point approaches the critical point as 1/ln L, where L is the linear system size, distinct from the L^−d scaling typical of conventional thermodynamic first-order transitions in d dimensions. This logarithmic finite-size drift is consistent with a nucleation-driven mechanism, in which rare voids trigger the collapse of the finite-range k-core. These results demonstrate that geometric constraints can fundamentally alter the nature of k-core percolation found in finite dimensions.
ER -