  We study the problem of distributed consensus in networks where the local
agents have high-order ($n\ge 3$) integrator dynamics, and where all feedback
is localized in that each agent has a bounded number of neighbors. We prove
that no consensus algorithm based on relative differences between states of
neighboring agents can then achieve consensus in networks of any size. That is,
while a given algorithm may allow a small network to converge to consensus, the
same algorithm will lead to instability if agents are added to the network so
that it grows beyond a certain finite size. This holds in classes of network
graphs whose algebraic connectivity, that is, the smallest non-zero Laplacian
eigenvalue, is decreasing towards zero in network size. This applies, for
example, to all planar graphs. Our proof, which relies on Routh-Hurwitz
criteria for complex-valued polynomials, holds true for directed graphs with
normal graph Laplacians. We survey classes of graphs where this issue arises,
and also discuss leader-follower consensus, where instability will arise in any
growing, undirected network as long as the feedback is localized.
