  Boolean networks have been proposed as potentially useful models for genetic
control. An important aspect of these networks is the stability of their
dynamics in response to small perturbations. Previous approaches to stability
have assumed uncorrelated random network structure. Real gene networks
typically have nontrivial topology significantly different from the random
network paradigm. In order to address such situations, we present a general
method for determining the stability of large Boolean networks of any specified
network topology and predicting their steady-state behavior in response to
small perturbations. Additionally, we generalize to the case where individual
genes have a distribution of `expression biases,' and we consider
non-synchronous update, as well as extension of our method to non-Boolean
models in which there are more than two possible gene states. We find that
stability is governed by the maximum eigenvalue of a modified adjacency matrix,
and we test this result by comparison with numerical simulations. We also
discuss the possible application of our work to experimentally inferred gene
networks.
