Limit theorems for assortativity and clustering in null models for scale-free networks

TL;DR

This paper establishes limit theorems for assortativity and clustering in null models of scale-free networks, explaining how edge removal affects network correlations and comparing two different models.

Contribution

It provides the first central limit theorems for clustering and assortativity in the erased configuration model with infinite variance degrees, and compares these results to the rank-1 inhomogeneous random graph.

Findings

Edge removal significantly alters clustering coefficient scaling.

Central limit theorems are proven for Pearson's correlation and clustering coefficients.

Results are consistent across the erased configuration model and the rank-1 inhomogeneous random graph.

Abstract

An important problem in modeling networks is how to generate a randomly sampled graph with given degrees. A popular model is the configuration model, a network with assigned degrees and random connections. The erased configuration model is obtained when self-loops and multiple edges in the configuration model are removed. We prove an upper bound for the number of such erased edges for regularly-varying degree distributions with infinite variance, and use this result to prove central limit theorems for Pearson's correlation coefficient and the clustering coefficient in the erased configuration model. Our results explain the structural correlations in the erased configuration model and show that removing edges leads to different scaling of the clustering coefficient. We then prove that for the rank-1 inhomogeneous random graph, another null model that creates scale-free simple networks, the results for Pearson's correlation coefficient as well as for the clustering coefficient are similar to the results for the erased configuration model.