  A defining feature of many large empirical networks is their intrinsic
complexity. However, many networks also contain a large degree of structural
repetition. An immediate question then arises: can we characterize essential
network complexity while excluding structural redundancy?
  In this article we utilize inherent network symmetry to collapse all
redundant information from a network, resulting in a coarse-graining which we
show to carry the essential structural information of the `parent' network. In
the context of algebraic combinatorics, this coarse-graining is known as the
\emph{quotient}. We systematically explore the theoretical properties of
network quotients and summarize key statistics of a variety of `real-world'
quotients with respect to those of their parent networks. In particular, we
find that quotients can be substantially smaller than their parent networks yet
typically preserve various key functional properties such as complexity
(heterogeneity and hubs vertices) and communication (diameter and mean geodesic
distance), suggesting that quotients constitute the essential structural
skeleton of their parent network. We summarize with a discussion of potential
uses of quotients in analysis of biological regulatory networks and ways in
which using quotients can reduce the computational complexity of network
algorithms.
