  As complex networks find applications in a growing range of disciplines, the
diversity of naturally occurring and model networks being studied is exploding.
The adoption of a well-developed collection of network taxonomies is a natural
method for both organizing this data and understanding deeper relationships
between networks. Most existing metrics for network structure rely on classical
graph-theoretic measures, extracting characteristics primarily related to
individual vertices or paths between them, and thus classify networks from the
perspective of local features. Here, we describe an alternative approach to
studying structure in networks that relies on an algebraic-topological metric
called persistent homology, which studies intrinsically mesoscale structures
called cycles, constructed from cliques in the network. We present a
classification of 14 commonly studied weighted network models into four groups
or classes, and discuss the structural themes arising in each class. Finally,
we compute the persistent homology of two real-world networks and one network
constructed by a common dynamical systems model, and we compare the results
with the three classes to obtain a better understanding of those networks.
