The Robustness of Stochastic Switching Networks

Many natural systems, including chemical and biological systems, can be modeled using stochastic switching circuits. These circuits consist of stochastic switches, called pswitches, which operate with a fixed probability of being open or closed. We study the effect caused by introducing an error of size epsilon to each pswitch in a stochastic circuit. We analyze two constructions-simple series-parallel and general series-parallel circuits-and prove that simple series-parallel circuits are robust to small error perturbations, while general series-parallel circuits are not. Specifically, the total error introduced by perturbations of size less than epsilon is bounded by a constant multiple of e in a simple series-parallel circuit, independent of the size of the circuit. However, the same result does not hold in the case of more general series-parallel circuits. In the case of a general stochastic circuit, we prove that the overall error probability is bounded by a linear function of the number of pswitches.