  We consider distributed consensus in networks where the agents have
integrator dynamics of order two or higher ($n\ge 2$). We assume all feedback
to be localized in the sense that each agent has a bounded number of neighbors
and consider a scaling of the network through the addition of agents in a
modular manner, i.e., without re-tuning controller gains upon addition. We show
that standard consensus algorithms, which rely on relative state feedback, are
subject to what we term scale fragilities, meaning that stability is lost as
the network scales. For high-order agents ($n\ge 3$), we prove that no
consensus algorithm with fixed gains can achieve consensus in networks of any
size. That is, while a given algorithm may allow a small network to converge,
it causes instability if the network grows beyond a certain finite size. This
holds in families of network graphs whose algebraic connectivity, that is, the
smallest non-zero Laplacian eigenvalue, is decreasing towards zero in network
size (e.g. all planar graphs). For second-order consensus ($n = 2$) we prove
that the same scale fragility applies to directed graphs that have a complex
Laplacian eigenvalue approaching the origin (e.g. directed ring graphs). The
proofs for both results rely on Routh-Hurwitz criteria for complex-valued
polynomials and hold true for general directed network graphs. We survey
classes of graphs subject to these scale fragilities, discuss their scaling
constants, and finally prove that a sub-linear scaling of nodal neighborhoods
can suffice to overcome the issue.
