Deciphering the global organization of clustering in real complex networks

TL;DR

This paper investigates the global clustering structure of real complex networks, revealing they are closer to random models than ordered ones, using novel $m$-core analysis and visualization tools.

Contribution

It introduces the $m$-core landscape analysis and LaNet-vi 3.0 tool to distinguish hierarchical and modular clustering structures in networks.

Findings

Real networks are closer to maximally random clustering architectures.

$m$-core analysis reveals hierarchical versus modular clustering patterns.

Developed visualization tool LaNet-vi 3.0 for $m$-core decomposition.

Abstract

We uncover the global organization of clustering in real complex networks. As it happens with other fundamental properties of networks such as the degree distribution, we find that real networks are neither completely random nor ordered with respect to clustering, although they tend to be closer to maximally random architectures. We reach this conclusion by comparing the global structure of clustering in real networks with that in maximally random and in maximally ordered clustered graphs. The former are produced with an exponential random graph model that maintains correlations among adjacent edges at the minimum needed to conform with the expected clustering spectrum; the later with a random model that arranges triangles in cliques inducing highly ordered structures. To compare the global organization of clustering in real and model networks, we compute $m$-core landscapes, where the $m$-core is defined, akin to the $k$-core, as the maximal subgraph with edges participating at least in $m$ triangles. This property defines a set of nested subgraphs that, contrarily to $k$-cores, is able to distinguish between hierarchical and modular architectures. To visualize the $m$-core decomposition we developed the LaNet-vi 3.0 tool.