  In this paper, we study controllability of a network of linear
single-integrator agents when the network size goes to infinity. We first
investigate the effect of increasing size by injecting an input at every node
and requiring that network controllability Gramian remain well-conditioned with
the increasing dimension. We provide theoretical justification to the intuition
that high degree nodes pose a challenge to network controllability. In
particular, the controllability Gramian for the networks with bounded maximum
degrees is shown to remain well-conditioned even as the network size goes to
infinity. In the canonical cases of star, chain and ring networks, we also
provide closed-form expressions which bound the condition number of the
controllability Gramian in terms of the network size. We next consider the
effect of the choice and number of leader nodes by actuating only a subset of
nodes and considering the least eigenvalue of the Gramian as the network size
increases. Accordingly, while a directed star topology can never be made
controllable for all sizes by injecting an input just at a fraction $f<1$ of
nodes; for path or cycle networks, the designer can actuate a non-zero fraction
of nodes and spread them throughout the network in such way that the least
eigenvalue of the Gramians remain bounded away from zero with the increasing
size. The results offer interesting insights on the challenges of control in
large networks and with high-degree nodes.
