Nested subgraphs of complex networks

We analytically explore the scaling properties of a general class of nested subgraphs in complex networks, which includes the K-core and the K-scaffold, among others. We name such a class of subgraphs K-nested subgraphs since they generate families of subgraphs such that ...SK+1(G) subset or equal SK(G) subset or equal SK-1(G).... Using the so-called configuration model it is shown that any family of nested subgraphs over a network with diverging second moment and finite first moment has infinite elements (i.e. lacking a percolation threshold). Moreover, for a scale-free network with the above properties, we show that any nested family of subgraphs is self-similar by looking at the degree distribution. Both numerical simulations and real data are analyzed and display good agreement with our theoretical predictions