  We describe how noise propagates through a network by calculating the
variance of the outputs. Using stochastic calculus and dynamical systems
theory, we study the network topologies that accentuate or alleviate the effect
of random variance in the network for both directed and undirected graphs.
Given a linear tree network, the variance in the output is a convex function of
the poles of the individual nodes. Cycles create correlations which in turn
increase the variance in the output. Feedforward and feedback have a limited
effect on noise propagation when the respective cycles is sufficiently long.
Crosstalk between the elements of different pathways helps reduce the output
noise, but makes the network slower. Next, we study the differences between
disturbances in the inputs and disturbances in the network parameters, and how
they propagate to the outputs. Finally, we show how noise correlations can
affect the steady state of the system in chemical reaction networks with
reactions of two or more reactants, each of which may be affected by
independent or correlated noise sources.
