  We investigate robust linear consensus over networks under
capacity-constrained communication. The capacity of each edge is encoded as an
upper bound on the number of state variables that can be communicated
instantaneously. When the edge capacities are small compared to the
dimensionality of the state vectors, it is not possible to instantaneously
communicate full state information over every edge. We investigate how robust
consensus (small steady state variance of the states) can be achieved within a
linear time-invariant setting by optimally assigning edges to state-dimensions.
We show that if a finite steady state variance of the states can be achieved,
then both the minimum cut capacity and the total capacity of the network should
be sufficiently large. Optimal and approximate solutions are provided for some
special classes of graphs. We also consider the related problem of optimally
allocating additional capacity on a feasible initial solution. We show that
this problem corresponds to the maximization of a submodular function subject
to a matroid constraint, which can be approximated via a greedy algorithm.
