  We consider network structures that optimize the $\mathcal{H}_2$ norm of
weighted, time scaled consensus networks, under a minimal representation of
such consensus networks described by the edge Laplacian. We show that a greedy
algorithm can be used to find the minimum-$\mathcal{H}_2$ norm spanning tree,
as well as how to choose edges to optimize the $\mathcal{H}_2$ norm when edges
are added back to a spanning tree. In the case of edge consensus with a
measurement model considering all edges in the graph, we show that adding edges
between slow nodes in the graph provides the smallest increase in the
$\mathcal{H}_2$ norm.
