Nested Subgraphs of Complex Networks

Details Nested Subgraphs of Complex Networks Nested Subgraphs of Complex Networks

2007-12-04

arxiv.org/abs/0712.0512v1

Physics

Condensed Matter

Disordered Systems and Neural Networks

6 pages, 4 figures

Scientific paper

10.1088/1751-8113/41/38/385003

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 class of subgraphs $K$-nested subgraphs due to the fact that they generate families of subgraphs such that $...S_{K+1}({\cal G})\subseteq S_K({\cal G})\subseteq S_{K-1}({\cal G})...$. Using the so-called {\em 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.

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