  Complex systems are often modeled as Boolean networks in attempts to capture
their logical structure and reveal its dynamical consequences. Approximating
the dynamics of continuous variables by discrete values and Boolean logic gates
may, however, introduce dynamical possibilities that are not accessible to the
original system. We show that large random networks of variables coupled
through continuous transfer functions often fail to exhibit the complex
dynamics of corresponding Boolean models in the disordered (chaotic) regime,
even when each individual function appears to be a good candidate for Boolean
idealization. A suitably modified Boolean theory explains the behavior of
systems in which information does not propagate faithfully down certain chains
of nodes. Model networks incorporating calculated or directly measured transfer
functions reported in the literature on transcriptional regulation of genes are
described by the modified theory.
