Fixed-points in Random Boolean Networks: The impact of parallelism in the scale-free topology case

TL;DR

This paper investigates the behavior of fixed points in random Boolean networks with scale-free topology, highlighting the robustness of such networks under asynchronous dynamics and contrasting it with parallel dynamics.

Contribution

It proves that fixing source node values yields an expected one fixed point regardless of topology and demonstrates the robustness of scale-free networks under asynchronous updates.

Findings

Scale-free networks are extremely robust.

Asynchronous dynamics lead to universal fixed points from any initial state.

Parallel dynamics have a much lower likelihood of reaching fixed points.

Abstract

Fixed points are fundamental states in any dynamical system. In the case of gene regulatory networks (GRNs) they correspond to stable genes profiles associated to the various cell types. We use Kauffman's approach to model GRNs with random Boolean networks (RBNs). We start this paper by proving that, if we fix the values of the source nodes (nodes with in-degree 0), the expected number of fixed points of any RBN is one (independently of the topology we choose). For finding such fixed points we use the {\alpha}-asynchronous dynamics (where every node is updated independently with probability 0 < {\alpha} < 1). In fact, it is well-known that asynchrony avoids the cycle attractors into which parallel dynamics tends to fall. We perform simulations and we show the remarkable property that, if for a given RBN with scale-free topology and {\alpha}-asynchronous dynamics an initial configuration reaches a fixed point, then every configuration also reaches a fixed point. By contrast, in the parallel regime, the percentage of initial configurations reaching a fixed point (for the same networks) is dramatically smaller. We contrast the results of the simulations on scale-free networks with the classical Erdos-Renyi model of random networks. Everything indicates that scale-free networks are extremely robust. Finally, we study the mean and maximum time/work needed to reach a fixed point when starting from randomly chosen initial configurations.