Correlation between clustering and degree in affiliation networks

TL;DR

This paper investigates the relationship between clustering and degree in affiliation networks modeled by power law random intersection graphs, revealing a scaling law for adjacency probability among neighbors.

Contribution

It provides a rigorous analysis of how the probability of neighbor adjacency scales with degree in power law intersection graphs, linking it to weight distribution tail indices.

Findings

Probability scales as k^{- extdelta} for large k

The scaling exponent etermines clustering behavior

Results are mathematically rigorous and based on power law distributions

Abstract

We are interested in the probability that two randomly selected neighbors of a random vertex of degree (at least) $k$ are adjacent. We evaluate this probability for a power law random intersection graph, where each vertex is prescribed a collection of attributes and two vertices are adjacent whenever they share a common attribute. We show that the probability obeys the scaling $k^{-\delta}$ as $k\to+\infty$. Our results are mathematically rigorous. The parameter $0\le \delta\le 1$ is determined by the tail indices of power law random weights defining the links between vertices and attributes.