The Triangle Friendship Paradox

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.