  Networks are ubiquitous in various fields, representing systems where nodes
and their interconnections constitute their intricate structures. We introduce
a network decomposition scheme to reveal multiscale core-periphery structures
lurking inside, using the concept of locally defined nodal hub centrality and
edge-pruning techniques built upon it. We demonstrate that the
hub-centrality-based edge pruning reveals a series of breaking points in
network decomposition, which effectively separates a network into its backbone
and shell structures. Our local-edge decomposition method iteratively
identifies and removes locally least important nodes, and uncovers an
onion-like hierarchical structure as a result. Compared with the conventional
$k$-core decomposition method, our method based on relative information
residing in local structures exhibits a clear advantage in terms of discovering
locally crucial substructures. Furthermore, we introduce the core-periphery
score to properly separate the core and periphery with our decomposition
scheme. By extending the method combined with the network community structure,
we successfully detect multiple core-periphery structures by decomposition
inside each community. Moreover, the application of our decomposition to
supernode networks defined from the communities reveals the intricate relation
between the two representative mesoscale structures.
