  Network modeling based on ensemble averages tacitly assumes that the networks
meant to be modeled are typical in the ensemble. Previous research on network
eigenvalues, which govern a range of dynamical phenomena, has shown that this
is indeed the case for uncorrelated networks with minimum degree $\ge 3$. Here
we focus on real networks, which generally have both structural correlations
and low-degree nodes. We show that: (i) the ensemble distribution of the
dynamically most important eigenvalues can be not only broad and far apart from
the real eigenvalue but also highly structured, often with a multimodal rather
than bell-shaped form; (ii) these interesting properties are found to be due to
low-degree nodes, mainly those with degree $< 3$, and network communities,
which is a common form of structural correlation found in real networks. In
addition to having implications for ensemble-based approaches, this shows that
low-degree nodes may have a stronger influence on collective dynamics than
previously anticipated from the study of computer-generated networks.
