Clustering in random line graphs

TL;DR

This paper analyzes the degree distribution and clustering coefficient of line graphs derived from various network models, showing that these properties are preserved and tend to specific values as mean degree increases.

Contribution

It provides a detailed analysis of clustering and degree distribution in line graphs of different network types, extending understanding of their structural properties.

Findings

Degree distributions in line graphs mirror original networks (Poisson, exponential, power law).

Clustering coefficient approaches specific values as mean degree increases.

Results align with theoretical predictions assuming negligible degree correlations.

Abstract

We investigate the degree distribution $P(k)$ and the clustering coefficient $C$ of the line graphs constructed on the Erd\"os-R\'enyi networks, the exponential and the scale-free growing networks. We show that the character of the degree distribution in these graphs remains Poissonian, exponential and power law, respectively, i.e. the same as in the original networks. When the mean degree $<k>$ increases, the obtained clustering coefficient $C$ tends to 0.50 for the transformed Erd\"os-R\'enyi networks, to 0.53 for the transformed exponential networks and to 0.61 for the transformed scale-free networks. These results are close to theoretical values, obtained with the model assumption that the degree-degree correlations in the initial networks are negligible.