  We investigate the performance of linear consensus algorithms subject to a
scaling of the underlying network size. Specifically, we model networked
systems with $n^{\text{th}}$ order integrator dynamics over families of
undirected, weighted graphs with bounded nodal degrees. In such networks, the
algebraic connectivity affects convergence rates, sensitivity, and, for
high-order consensus ($n \ge 3$), stability properties. This connectivity
scales unfavorably in network size, except in expander families, where
consensus performs well regardless of network size. We show, however, that
consensus over expander families is fragile to a grounding of the network
(resulting in leader-follower consensus). We show that grounding may
deteriorate system performance by orders of magnitude in large networks, or
cause instability in high-order consensus. Our results, which we illustrate
through simulations, also point to a fundamental limitation to the scalability
of consensus networks with leaders, which does not apply to leaderless
networks.
