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Classification of weighted networks through mesoscale homological features

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.