Most complex systems can be captured by graphs or networks. Networks connect nodes (e.g. airports) through edges (flights), thus summarizing the system’s structure. A popular way of interrogating graphs is community detection, which uncovers sets of close and densely connected nodes. Alternatively, we may ask what {} a node occupies within a network, and cluster them according to topological or structural similarity, disregarding of the distance and density of connections between them. We exploit {} to study a series of networks that capture relevant complex systems. In our framework, each graph guides our attention to its key features, in order of salience. Thanks to this, we uncover informative network decompositions that complement community-based analyses. We illustrate this framework for the global airport network, for which we discover a non-trivial decomposition based on airport functionality. We also apply the approach to the US House bill co-sponsorship graph, in which we discover a drastic reorganization of cooperation patterns likely driven by polarization. Other brief analyses on more networks highlight the framework’s versatility.