  A network can be analyzed at different topological scales, ranging from
single nodes to motifs, communities, up to the complete structure. We propose a
novel intermediate-level topological analysis that considers non-overlapping
subgraphs (connected components) and their interrelationships and distribution
through the network. Though such subgraphs can be completely general, our
methodology focuses the cases in which the nodes of these subgraphs share some
special feature, such as being critical for the proper operation of the
network. Our methodology of subgraph characterization involves two main
aspects: (i) a distance histogram containing the distances calculated between
all subgraphs, and (ii) a merging algorithm, developed to progressively merge
the subgraphs until the whole network is covered. The latter procedure
complements the distance histogram by taking into account the nodes lying
between subgraphs, as well as the relevance of these nodes to the overall
interconnectivity. Experiments were carried out using four types of network
models and four instances of real-world networks, in order to illustrate how
subgraph characterization can help complementing complex network-based studies.
