Nested Subgraphs of Complex Networks

TL;DR

This paper analytically investigates the scaling properties of nested subgraphs in complex networks, revealing their self-similarity and percolation behavior, supported by simulations and real data analysis.

Contribution

It introduces a unified framework for nested subgraphs like the K-core and K-scaffold, analyzing their properties in scale-free networks.

Findings

Nested subgraphs lack a percolation threshold in certain networks.

Scale-free networks exhibit self-similarity in nested subgraph degree distributions.

Numerical and real data confirm theoretical predictions.

Abstract

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.