Characterization of Subgraphs Relationships and Distribution in Complex   Networks
  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. 
Lucas Antiqueira and Luciano da Fontoura Costa