We consider dynamical networks whose agents have general linear time-invariant dynamics and are disturbed by the Gaussian white noise. The network graph evolves in time according to a random process. We demonstrate that under certain continuous-time switching regimes, the stability and performance of the network can be characterized using the spectra of the expected Laplacian in the steady-state. More specifically, we show that in these networks, the stability of the network is equivalent to the stability of a linear time-invariant network whose graph Laplacian is this expected Laplacian. Moreover, we indicate that our performance measure, the long-run output variance, is equal to its value for the described linear time-invariant network. This lets us express the measure as a spectral function of that Laplacian. We illustrate the effectiveness of our framework by investigating various random networks, including a consensus network over Erdos-Reyni graphs, a network of triple-integrators with random link failures, and a platoon of vehicles with random feedbacks decided by each car.