Structural changes in a network representation of a system, due to different experimental conditions, different connectivity across layers, or to its time evolution, can provide insight on its organization, function, and on how it responds to external perturbations. The deeper understanding of how gene networks cope with diseases and treatments is maybe the most incisive demonstration of the gains obtained through this differential network analysis point of view, which led to an explosion of new numeric techniques in the last decade. However, where to focus one's attention, or how to navigate through the differential structures in the context of large networks, can be overwhelming even for a few experimental conditions. In this paper, we propose a theory and a methodological implementation for the characterization of shared "structural roles" of nodes simultaneously within and between networks. Inspired by recent methodological advances in chaotic phase synchronization analysis, we show how the information about the shared structures of a set of networks can be split and organized in an automatic fashion, in scenarios with very different (i) community sizes, (ii) total number of communities, and (iii) even for a large number of 100 networks compared using numerical benchmarks generated by a stochastic block model. Then, we investigate how the network size, number of networks, and mean size of communities influence the method performance in a series of Monte Carlo experiments. To illustrate its potential use in a more challenging scenario with real-world data, we show evidence that the method can still split and organize the structural information of a set of four gene coexpression networks obtained from two cell types × two treatments (interferon-β stimulated or control). Aside from its potential use as for automatic feature extraction and preprocessing tool, we discuss that another strength of the method is its "story-telling"-like characterization of the information encoded in a set of networks, which can be used to pinpoint unexpected shared structure, leading to further investigations and providing new insights. Finally, the method is flexible to address different research-field-specific questions, by not restricting what scientific-meaningful characteristic (or relevant feature) of a node shall be used.