  Orthogonal projection-based reduced order models (PROM) are the output of
widely-used model reduction methods. In this work, a novel product form is
derived for the reduction error system of these reduced models, and it is shown
that any such PROM can be obtained from a sequence of 1-dimensional projection
reductions. Investigating the error system product form, we then define
interface-invariant PROMs, model order reductions with projection-invariant
input and output matrices, and it is shown that for such PROMs the error
product systems are strictly proper. Furthermore, exploiting this structure, an
analytic $\mathcal{H}_{\infty}$ reduction error bound is obtained and an
$\mathcal{H}_{\infty}$ bound optimization problem is defined.
Interface-invariant reduced models are natural to graph-based model reduction
of multi-agent systems where subsets of agents function as the input and output
of the system. In the second part of this study, graph contractions are used as
a constructive solution approach to the $\mathcal{H}_{\infty}$ bound
optimization problem for multi-agent systems. Edge-based contractions are then
utilized in a greedy-edge reduction algorithm and are demonstrated for the
model reduction of a first-order Laplacian controlled consensus protocol.
