Clustering function: a measure of social influence

TL;DR

This paper introduces the clustering function as an extension of the clustering coefficient to measure social influence in networks, analyzing its behavior in various models and establishing asymptotic properties.

Contribution

It defines the clustering function, compares it across networks, and derives its asymptotic behavior in random intersection graph models with non-vanishing clustering.

Findings

Clustering functions show similar patterns across different networks.

Surprising regularity observed in clustering functions.

Asymptotic behavior of clustering function established for certain graph models.

Abstract

A commonly used characteristic of statistical dependence of adjacency relations in real networks, the clustering coefficient, evaluates chances that two neighbours of a given vertex are adjacent. An extension is obtained by considering conditional probabilities that two randomly chosen vertices are adjacent given that they have r common neighbours. We denote such probabilities cl(r) and call r-> cl(r) the clustering function. We compare clustering functions of several networks having non-negligible clustering coefficient. They show similar patterns and surprising regularity. We establish a first order asymptotic (as the number of vertices tends to infinity) of the clustering function of related random intersection graph models admitting nonvanishing clustering coefficient and asymptotic degree distribution having a finite second moment.