A comprehensive generalization of the Friendship Paradox to weights and attributes

TL;DR

This paper unifies and extends the Friendship Paradox to weighted edges and numeric node attributes in undirected graphs, providing theoretical insights, practical rules, and empirical validation on real and synthetic networks.

Contribution

It offers a comprehensive framework that generalizes the Friendship Paradox to weighted edges and attributes, clarifying when these extensions hold or fail.

Findings

Original and weighted friendship paradoxes hold universally.

Attribute-based extensions fail about 50% of the time with random attributes.

Correlation rules can predict when the paradox applies.

Abstract

The Friendship Paradox is a simple and powerful statement about node degrees in a graph (Feld 1991). However, it only applies to undirected graphs with no edge weights, and the only node characteristic it concerns is degree. Since many social networks are more complex than that, it is useful to generalize this phenomenon, if possible, and a number of papers have proposed different generalizations. Here, we unify these generalizations in a common framework, retaining the focus on undirected graphs and allowing for weighted edges and for numeric node attributes other than degree to be considered, since this extension allows for a clean characterization and links to the original concepts most naturally. While the original Friendship Paradox and the Weighted Friendship Paradox hold for all graphs, considering non-degree attributes actually makes the extensions fail around 50% of the time, given random attribute assignment. We provide simple correlation-based rules to see whether an attribute-based version of the paradox holds. In addition to theory, our simulation and data results show how all the concepts can be applied to synthetic and real networks. Where applicable, we draw connections to prior work to make this an accessible and comprehensive paper that lets one understand the math behind the Friendship Paradox and its basic extensions.