Permutation theory governs long-term dynamics of critical Boolean networks
Abstract
Boolean networks are widely used to model gene regulatory Attractors of Boolean networks model stable gene-expression patterns, yet deriving their properties from network structure remains an open problem. We solve this problem for critical K=1 networks by showing that their feedback loops induce a permutation whose order bounds the average attractor length both above and below by universal constant factors. This correspondence allows classical results from combinatorics and number theory to be applied directly to Boolean network dynamics. We find three distinct asymptotic scales for both average and maximum attractor lengths: typical networks scale as exp[(1/8+o(1))(ln N)2], the ensemble means grow as exp[N1/3+o(1)], and extremal networks attain exp[(1+o(1))√(Nln N)]. Thus, ensemble averages are governed by rare network realizations and are unrepresentative of typical dynamics.
Related
- BIRDNet: Mining and Encoding Boolean Implication Knowledge Graphs as Interpretable Deep Neural Networks
- Nonequivalence of expected free energy and revised system integrated information in noisy permutation networks
- Stochastic evolutionary dynamics in structured populations: Network degree governs evolutionary stability
- A single attractor governs the population dynamics of marine fishes worldwide
- The gut-brain vagal axis governs mesolimbic dopamine dynamics and reward events
BibTeX
@misc{bavisetty2026permutation,
title = {{Permutation theory governs long-term dynamics of critical Boolean networks}},
author = {Venkata Sai Narayana Bavisetty and Matthew Wheeler and Julian Vignes and Matthew Stephen Jones Jr. and Ruodan Liu and Sean Campbell and Claus Kadelka},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.07790},
eprint = {2608.07790},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.07790},
}
RIS
TY - UNPB
TI - Permutation theory governs long-term dynamics of critical Boolean networks
AU - Venkata Sai Narayana Bavisetty
AU - Matthew Wheeler
AU - Julian Vignes
AU - Matthew Stephen Jones Jr.
AU - Ruodan Liu
AU - Sean Campbell
AU - Claus Kadelka
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.07790
UR - https://arxiv.org/abs/2608.07790
AB - Boolean networks are widely used to model gene regulatory Attractors of Boolean networks model stable gene-expression patterns, yet deriving their properties from network structure remains an open problem. We solve this problem for critical K=1 networks by showing that their feedback loops induce a permutation whose order bounds the average attractor length both above and below by universal constant factors. This correspondence allows classical results from combinatorics and number theory to be applied directly to Boolean network dynamics. We find three distinct asymptotic scales for both average and maximum attractor lengths: typical networks scale as exp[(1/8+o(1))(ln N)^2], the ensemble means grow as exp[N^1/3+o(1)], and extremal networks attain exp[(1+o(1))√(Nln N)]. Thus, ensemble averages are governed by rare network realizations and are unrepresentative of typical dynamics.
ER -