  Hierarchies permeate the structure of real networks, whose nodes can be
ranked according to different features. However, networks are far from
tree-like structures and the detection of hierarchical ordering remains a
challenge, hindered by the small-world property and the presence of a large
number of cycles, in particular clustering. Here, we use geometric
representations of undirected networks to achieve an enriched interpretation of
hierarchy that integrates features defining popularity of nodes and similarity
between them, such that the more similar a node is to a less popular neighbor
the higher the hierarchical load of the relationship. The geometric approach
allows us to measure the local contribution of nodes and links to the hierarchy
within a unified framework. Additionally, we propose a link filtering method,
the similarity filter, able to extract hierarchical backbones containing the
links that represent statistically significant deviations with respect to the
maximum entropy null model for geometric heterogeneous networks. We applied our
geometric approach to the detection of similarity backbones of real networks in
different domains and found that the backbones preserve local topological
features at all scales. Interestingly, we also found that similarity backbones
favor cooperation in evolutionary dynamics modelling social dilemmas.
