Generalized Friendship Paradoxes in Network Science

Abstract

Generalized friendship paradoxes occur when, on average, our friends have more of some attribute than us. These paradoxes are relevant to many aspects of human interaction, notably in social science and epidemiology. Here, we derive new theoretical results concerning the inevitability of a paradox arising, using a linear algebra perspective. Following the seminal 1991 work of Scott L. Feld, we consider two distinct ways to measure and compare averages, which may be regarded as global and local. For global averaging, we show that a generalized friendship paradox holds for a large family of walk-based centralities, including Katz centrality and total subgraph communicability, and also for nonbacktracking eigenvector centrality. However, we also find counterexamples for centralities based on walks of even length. For local averaging we establish a paradox for nonbacktracking eigenvector centrality and we characterize the cases where the paradox holds with equality for the walk-based case. Defining loneliness as the reciprocal of the number of friends, we show that for this attribute the generalized and local friendship paradoxes always hold in reverse. In this sense, we are always more lonely, on average, than our friends. We also derive global and local averaging paradoxes for the case where the arithmetic mean is replaced by the geometric mean. As well as unifying and adding to the literature in this area, we highlight some open questions.