  We present a rigorous mathematical framework for analyzing dynamics of a
broad class of Boolean network models. We use this framework to provide the
first formal proof of many of the standard critical transition results in
Boolean network analysis, and offer analogous characterizations for novel
classes of random Boolean networks. We precisely connect the short-run dynamic
behavior of a Boolean network to the average influence of the transfer
functions. We show that some of the assumptions traditionally made in the more
common mean-field analysis of Boolean networks do not hold in general.
  For example, we offer some evidence that imbalance, or expected internal
inhomogeneity, of transfer functions is a crucial feature that tends to drive
quiescent behavior far more strongly than previously observed.
