  We introduce a new network statistic that measures diverse structural
properties at the micro-, meso-, and macroscopic scales, while still being easy
to compute and easy to interpret at a glance. Our statistic, the onion
spectrum, is based on the onion decomposition, which refines the k-core
decomposition, a standard network fingerprinting method. The onion spectrum is
exactly as easy to compute as the k-cores: It is based on the stages at which
each vertex gets removed from a graph in the standard algorithm for computing
the k-cores. But the onion spectrum reveals much more information about a
network, and at multiple scales; for example, it can be used to quantify node
heterogeneity, degree correlations, centrality, and tree- or lattice-likeness
of the whole network as well as of each k-core. Furthermore, unlike the k-core
decomposition, the combined degree-onion spectrum immediately gives a clear
local picture of the network around each node which allows the detection of
interesting subgraphs whose topological structure differs from the global
network organization. This local description can also be leveraged to easily
generate samples from the ensemble of networks with a given joint degree-onion
distribution. We demonstrate the utility of the onion spectrum for
understanding both static and dynamic properties on several standard graph
models and on many real-world networks.
