  It is common in the study of networks to investigate meso-scale features to
try to gain an understanding of network structure and function. For example,
numerous algorithms have been developed to try to identify "communities," which
are typically construed as sets of nodes with denser connections internally
than with the remainder of a network. In this paper, we adopt a complementary
perspective that "communities" are associated with bottlenecks of
locally-biased dynamical processes that begin at seed sets of nodes, and we
employ several different community-identification procedures (using
diffusion-based and geodesic-based dynamics) to investigate community quality
as a function of community size. Using several empirical and synthetic
networks, we identify several distinct scenarios for ``size-resolved community
structure'' that can arise in real (and realistic) networks. Depending on which
scenario holds, one may or may not be able to successfully identify ``good''
communities in a given network, the manner in which different small communities
fit together to form meso-scale network structures can be very different, and
processes such as viral propagation and information diffusion can exhibit very
different dynamics.In addition, our results suggest that, for many large
realistic networks, the output of locally-biased methods that focus on
communities that are centered around a given seed node might have better
conceptual grounding and greater practical utility than the output of global
community-detection methods. They also illustrate subtler structural properties
that are important to consider in the development of better benchmark networks
to test methods for community detection.
  [Note: Because of space limitations in the arXiv's abstract field, this is an
abridged version of the paper's abstract.]
