Seed thresholds and degree variance in heterogeneous bootstrap percolation with growing degrees
Abstract
We study bootstrap percolation with independent vertex thresholds taking values one and two, with threshold-one probability $(1-c/d)/d$ for fixed c>0. The seed set is chosen uniformly among sets of a prescribed deterministic size, independently of the graph and thresholds. We prove threshold statements at fixed relative margins. For uniform simple graphs with prescribed nonnegative integer degrees of even sum and exact mean d, assume maxi|di−d|≤ C√d and c+1−vn≥κ>0, where vn=Var(D)/d and C,κ are fixed. When d→∞ and d=o(n1/7), the leading seed scale is n(c+1−vn)2/(2d4), without requiring a limit of vn. At fixed relative margins below and above this scale, the final active set has size O(n/d3) and n−o(n), respectively, with high probability. For regular graphs, the supercritical conclusion strengthens to full activation. For G(n,d/n), the same O(n/d3) and n−o(n) conclusions hold when d→∞ and d5/n→0, with seed scale nc2/(2d4). Thus regular and independent-edge graphs have different coefficients at the same asymptotic mean degree. Local exploration estimates yield explicit inactive remainders and survive conditioning on simplicity in the prescribed model. We also quantify the precision obstruction to static inclusion transfer and compute deterministic response-barrier corrections, without identifying a shrinking random critical window.
Related
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BibTeX
@misc{rodionov2026seed,
title = {{Seed thresholds and degree variance in heterogeneous bootstrap percolation with growing degrees}},
author = {Aleksandr Rodionov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04420},
eprint = {2610.04420},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04420},
}
RIS
TY - UNPB
TI - Seed thresholds and degree variance in heterogeneous bootstrap percolation with growing degrees
AU - Aleksandr Rodionov
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.04420
UR - https://arxiv.org/abs/2610.04420
AB - We study bootstrap percolation with independent vertex thresholds taking values one and two, with threshold-one probability $(1-c/d)/d$ for fixed c>0. The seed set is chosen uniformly among sets of a prescribed deterministic size, independently of the graph and thresholds. We prove threshold statements at fixed relative margins. For uniform simple graphs with prescribed nonnegative integer degrees of even sum and exact mean d, assume max_i|d_i−d|≤ C√d and c+1−v_n≥κ>0, where v_n=Var(D)/d and C,κ are fixed. When d→∞ and d=o(n^1/7), the leading seed scale is n(c+1−v_n)^2/(2d^4), without requiring a limit of v_n. At fixed relative margins below and above this scale, the final active set has size O(n/d^3) and n−o(n), respectively, with high probability. For regular graphs, the supercritical conclusion strengthens to full activation. For G(n,d/n), the same O(n/d^3) and n−o(n) conclusions hold when d→∞ and d^5/n→0, with seed scale nc^2/(2d^4). Thus regular and independent-edge graphs have different coefficients at the same asymptotic mean degree. Local exploration estimates yield explicit inactive remainders and survive conditioning on simplicity in the prescribed model. We also quantify the precision obstruction to static inclusion transfer and compute deterministic response-barrier corrections, without identifying a shrinking random critical window.
ER -