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 these SSNs 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 (Shreim A et al 2007 Phys. Rev. Lett. 98 198701) of an SSN as a global topological measure. RBNs with 2 ⩽ K ⩽ 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 2N, for K = 1 all the non-GoE nodes in a given SSN have the same in-degree which is always a power of two.
Highlights
In this paper we apply complex network analysis to discrete, disordered, deterministic dynamical systems
For K = 1 we distinguish between random Boolean networks (RBNs) constructed using all four boolean functions and the critical K = 1 RBN discussed earlier
We have studied the topology of state space networks (SSNs) for ensembles of random Boolean networks
Summary
In this paper we apply complex network analysis to discrete, disordered, deterministic dynamical systems. We exploit complex network theory to characterize disordered dynamical systems by examining SSNs for ensembles of random Boolean networks. Many real world networks such as regulatory networks [22], the world-wide web [23], or the correlation structure of earthquakes [24, 25]) differ markedly from a random graph – where the degree distribution is Poisson and clustering is absent. They often display “fat-tailed” or even scale-free degree distributions. We speculate that SSN fluctuations at these three different scales (local, global, and sample to sample) for K = 2 RBNs are associated with criticality in the thermodynamic limit N → ∞.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
