Fixed points of Boolean networks with sparse connections
Abstract
We study fixed points (FPs) of cellular automata with N sites on random sparse graphs. In the large N limit such models are known to exhibit phase transitions, from a 'frozen' phase, where at most a finite number of sites fluctuate at long times, to a 'fluctuating' phase where a finite fraction of sites fluctuate. We consider several models, calculating the first and second moments of the number of FPs, and find that these moments remain finite in the large N limit, except at the transitions where they become singular. The singularities can take several forms, including divergence of the mean or variance of the number of FPs, on one or both sides of the transition. The type of singularity is related to properties of the mean field dynamics or dynamics of the distance between copies of the system. In configuration space, we find that FPs are organized into clusters, each consisting of sets of FPs that agree with one another except for on a finite number of sites. In the frozen phase there is only one cluster, while in the fluctuating phase there may be multiple clusters. If there are multiple clusters, the distance between FPs in different clusters is extensive. We show that the differences within the clusters correspond to local changes near short cycles in the directed graph of connections whose influence is eventually limited. In the frozen phase, we calculate the full distribution of the number of FPs.
Related
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BibTeX
@article{marcus2026fixed,
title = {{Fixed points of Boolean networks with sparse connections}},
author = {Stav Marcus and Ari M Turner and Guy Bunin and Bernard Derrida},
journal = {Journal of Statistical Mechanics: Theory and Experiment},
year = {2026},
month = jun,
doi = {10.1088/1742-5468/ae727c},
url = {https://hal.science/hal-05753343v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Fixed points of Boolean networks with sparse connections
AU - Stav Marcus
AU - Ari M Turner
AU - Guy Bunin
AU - Bernard Derrida
PY - 2026
DA - 2026/06/12
JO - Journal of Statistical Mechanics: Theory and Experiment
DO - 10.1088/1742-5468/ae727c
UR - https://hal.science/hal-05753343v1
AB - We study fixed points (FPs) of cellular automata with N sites on random sparse graphs. In the large N limit such models are known to exhibit phase transitions, from a 'frozen' phase, where at most a finite number of sites fluctuate at long times, to a 'fluctuating' phase where a finite fraction of sites fluctuate. We consider several models, calculating the first and second moments of the number of FPs, and find that these moments remain finite in the large N limit, except at the transitions where they become singular. The singularities can take several forms, including divergence of the mean or variance of the number of FPs, on one or both sides of the transition. The type of singularity is related to properties of the mean field dynamics or dynamics of the distance between copies of the system. In configuration space, we find that FPs are organized into clusters, each consisting of sets of FPs that agree with one another except for on a finite number of sites. In the frozen phase there is only one cluster, while in the fluctuating phase there may be multiple clusters. If there are multiple clusters, the distance between FPs in different clusters is extensive. We show that the differences within the clusters correspond to local changes near short cycles in the directed graph of connections whose influence is eventually limited. In the frozen phase, we calculate the full distribution of the number of FPs.
SN - 1742-5468
ER -