  We study the stability of orbits in large Boolean networks with given complex
topology. We impose no restrictions on the form of the update rules, which may
be correlated with local topological properties of the network. While recent
past work has addressed the separate effects of nontrivial network topology and
certain special classes of update rules on stability, only crude results exist
about how these effects interact. We present a widely applicable solution to
this problem. Numerical experiments confirm our theory and show that local
correlations between topology and update rules can have profound effects on the
qualitative behavior of these systems.
