  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.
