Strongly clustered random graphs via triadic closure: Degree correlations and clustering spectrum

TL;DR

This paper introduces a model for strongly clustered random graphs that captures key features like degree correlations and clustering spectra, providing exact analytical expressions and numerical analysis.

Contribution

It presents a tractable model for highly clustered networks with exact formulas for clustering and degree correlations, addressing a gap in theoretical network modeling.

Findings

Positive degree assortativity accompanies high transitivity.

Non-trivial structure observed in the clustering spectrum.

Exact asymptotic results match numerical simulations.

Abstract

Real-world networks often exhibit strong transitivity with nontrivial local clustering spectra and degree correlations. Such features are not easily modelled in tractable network models, creating an obstacle to the theoretical understanding of such complex network structures. Here, we address this problem using a model for strongly clustered random graphs in which each triad of a random backbone is closed with a certain probability. Despite the intricate loopy local structure of the graphs obtained, we provide exact expressions for the local clustering spectrum and the degree correlations, filling the gap in the theoretical description of this model for random graphs. In particular, we find positive degree assortativity accompanies high transitivity, and non-trivial structure in the clustering spectrum. Exact asymptotic analytical results are complemented with extensive numerical characterization of finite size effects.