  In this paper, we consider the $\mathcal{H}_2$-norm of networked systems with
multi-time scale consensus dynamics and vector-valued agent states. This allows
us to explore how measurement and process noise affect consensus on
matrix-weighted graphs by examining edge-state consensus. In particular, we
highlight an interesting case where the influences of the weighting and scaling
on the $\mathcal{H}_2$ norm can be separated in the design problem. We then
consider optimization algorithms for updating the time scale parameters and
matrix weights in order to minimize network response to injected noise.
Finally, we present an application to formation control for multi-vehicle
systems.
