Cores, shell indices and the degeneracy of a graph limit

The $k$-core of a graph is the maximal subgraph in which every node has degree at least~$k$, the shell index of a node is the largest $k$ such that the $k$-core contains the node, and the degeneracy of a graph is the largest shell index of any node. After a suitable normalization, these three concepts generalize to limits of dense graphs (also called graphons). In particular, the degeneracy is continuous with respect to the cut metric.