The Triangle Friendship Paradox

TL;DR

This paper investigates the generalized friendship paradox focusing on the number of triangles at a vertex, revealing that the average bias can be negative and identifying classes of graphs with non-negative bias.

Contribution

It introduces the triangle-based friendship paradox, analyzes conditions for non-negative bias, and computes bias scaling in large random graphs.

Findings

Bias can be negative for triangle attributes.

Certain deterministic graphs have non-negative bias.

Bias scales predictably in large sparse and dense random graphs.

Abstract

We consider the generalised friendship paradox, focussing on the number of triangles at a vertex as the relevant attribute. We show that, contrary to the setting where the attribute is the number of edges at a vertex or the number of wedges at a vertex, the average friendship-bias of the number of triangles at a vertex is not always non-negative. We identify classes of finite deterministic graphs for which the bias is non-negative, and provide examples of finite deterministic graphs for which it is not. For certain classes of sparse and dense random graphs, we compute the scaling of the bias in the limit as the number of vertices tends to infinity.