Boolean networks with reliable dynamics

TL;DR

This paper studies Boolean networks with reliable trajectories, revealing their structural properties, motif distributions, and dynamics, especially the dominance of fixed points as system size increases, indicating proximity to a frozen phase.

Contribution

It introduces a class of Boolean networks with reliable trajectories constructed with minimal links and simple functions, analyzing their topology, motifs, and state space structure.

Findings

Clustering coefficient is higher than in random networks.

Three-node motif distribution resembles gene regulation networks.

Fixed points dominate as system size increases.

Abstract

We investigated the properties of Boolean networks that follow a given reliable trajectory in state space. A reliable trajectory is defined as a sequence of states which is independent of the order in which the nodes are updated. We explored numerically the topology, the update functions, and the state space structure of these networks, which we constructed using a minimum number of links and the simplest update functions. We found that the clustering coefficient is larger than in random networks, and that the probability distribution of three-node motifs is similar to that found in gene regulation networks. Among the update functions, only a subset of all possible functions occur, and they can be classified according to their probability. More homogeneous functions occur more often, leading to a dominance of canalyzing functions. Finally, we studied the entire state space of the networks. We observed that with increasing systems size, fixed points become more dominant, moving the networks close to the frozen phase.