Random Networks with Heterogeneous Reciprocity

TL;DR

This paper analyzes a preferential attachment model for social networks incorporating heterogeneous reciprocity levels, revealing its growth dynamics and complex degree distribution properties, including multivariate and hidden regular variation.

Contribution

It introduces a new preferential attachment model with heterogeneous reciprocity and proves its degree distribution converges with complex regular variation properties.

Findings

Edge count growth rate is characterized.

Empirical degree frequencies converge to a limiting distribution.

The limiting distribution exhibits multivariate and hidden regular variation.

Abstract

Users of social networks display diversified behavior and online habits. For instance, a user's tendency to reply to a post can depend on the user and the person posting. For convenience, we group users into aggregated behavioral patterns, focusing here on the tendency to reply to or reciprocate messages. The reciprocity feature in social networks reflects the information exchange among users. We study the properties of a preferential attachment model with heterogeneous reciprocity levels, give the growth rate of model edge counts, and prove convergence of empirical degree frequencies to a limiting distribution. This limiting distribution is not only multivariate regularly varying, but also has the property of hidden regular variation.