  The extreme eigenvalues of connectivity matrices govern the influence of the
network structure on a number of network dynamical processes. A fundamental
open question is whether the eigenvalues of large networks are well represented
by ensemble averages. Here we investigate this question explicitly and validate
the concept of ensemble averageability in random scale-free networks by showing
that the ensemble distributions of extreme eigenvalues converge to peaked
distributions as the system size increases. We discuss the significance of this
result using synchronization and epidemic spreading as example processes.
