Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2007-10-02
New J. Phys. 10 (2008) 013028
Physics
Condensed Matter
Statistical Mechanics
Scientific paper
10.1088/1367-2630/10/1/013028
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.
Berdahl Andrew
Grassberger Peter
Paczuski Maya
Shreim Amer
Sood Vishal
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