Quantifying the complexity of random Boolean networks

TL;DR

This paper investigates measures of complexity in random Boolean networks, revealing how different metrics capture spatial and dynamical inhomogeneity, and identifying the conditions under which complexity peaks.

Contribution

It introduces a modified complexity measure for Boolean networks that accounts for node-specific complexity, highlighting the importance of information transfer in complex nodes.

Findings

Complexity peaks in the disordered regime.

Node complexity correlates with information transfer.

Original measure fails to distinguish phases.

Abstract

We study two measures of the complexity of heterogeneous extended systems, taking random Boolean networks as prototypical cases. A measure defined by Shalizi et al. for cellular automata, based on a criterion for optimal statistical prediction [Shalizi et al., Phys. Rev. Lett. 93, 118701 (2004)], does not distinguish between the spatial inhomogeneity of the ordered phase and the dynamical inhomogeneity of the disordered phase. A modification in which complexities of individual nodes are calculated yields vanishing complexity values for networks in the ordered and critical regimes and for highly disordered networks, peaking somewhere in the disordered regime. Individual nodes with high complexity are the ones that pass the most information from the past to the future, a quantity that depends in a nontrivial way on both the Boolean function of a given node and its location within the network.