D-chain tomography of networks: a new structure spectrum and an application to the SIR process

The analysis of the dynamics on complex networks is closely connected to structural features of the networks. Features like, for instance, graph-cores and node degrees have been studied ubiquitously. Here we introduce the D-spectrum of a network, a novel new framework that is based on a collection of nested chains of subgraphs within the network. Graph-cores and node degrees are merely from two particular such chains of the D-spectrum. Each chain gives rise to a ranking of nodes and, for a fixed node, the collection of these ranks provides us with the D-spectrum of the node. Besides a node deletion algorithm, we discover a connection between the D-spectrum of a network and some fixed points of certain graph dynamical systems (MC systems) on the network. Using the D-spectrum we identify nodes of similar spreading power in the susceptible-infectious-recovered (SIR) model on a collection of real world networks as a quick application. We then discuss our results and conclude that D-spectra represent a meaningful augmentation of graph-cores and node degrees.