Complex Network Analysis of State Spaces for Random Boolean Networks

TL;DR

This paper applies complex network analysis to the state spaces of random Boolean networks, revealing criticality and complexity features, especially at K=2, through local and global topological measures.

Contribution

It introduces a novel application of complex network analysis to RBN state spaces, highlighting fluctuations and structural properties related to criticality.

Findings

K=2 RBNs show the largest fluctuations and non-self-averaging behavior.

Non-GoE node in-degrees vary widely for K>1, but are uniform and power-of-two for K=1.

SSN fluctuations indicate the critical and complex nature of K=2 RBNs.

Abstract

We apply complex network analysis to the state spaces of random Boolean networks (RBNs). An RBN contains $N$ Boolean elements each with $K$ inputs. A directed state space network (SSN) is constructed by linking each dynamical state, represented as a node, to its temporal successor. We study the heterogeneity of an SSN at both local and global scales, as well as sample-to-sample fluctuations within an ensemble of SSNs. We use in-degrees of nodes as a local topological measure, and the path diversity [Phys. Rev. Lett. 98, 198701 (2007)] of an SSN as a global topological measure. RBNs with $2 \leq K \leq 5$ exhibit non-trivial fluctuations at both local and global scales, while K=2 exhibits the largest sample-to-sample, possibly non-self-averaging, fluctuations. We interpret the observed ``multi scale'' fluctuations in the SSNs as indicative of the criticality and complexity of K=2 RBNs. ``Garden of Eden'' (GoE) states are nodes on an SSN that have in-degree zero. While in-degrees of non-GoE nodes for $K>1$ SSNs can assume any integer value between 0 and $2^N$, for K=1 all the non-GoE nodes in an SSN have the same in-degree which is always a power of two.