Stability of Boolean networks: the joint effects of topology and update rules.

We study the stability of orbits in large Boolean networks. We treat the case in which the network has a given complex topology, and we do not assume a specific form for the update rules, which may be correlated with local topological properties of the network. While recent past work has addressed the separate effects of complex network topology and certain 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 simulations confirm our theory and show that local correlations between topology and update rules can have profound effects on the qualitative behavior of these systems.