Three More Generalized Friendship Paradoxes

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 local and global. For local averaging, we show that a generalized friendship paradox holds for nonbacktracking eigenvector centrality. In this sense, our friends are always more important as us, on average. For global averaging, we define loneliness as the reciprocal of the number of friends and show that for this attribute the generalized friendship paradox always holds in reverse. In this sense, we are always more lonely, on average, than our friends. We also derive a global averaging paradox result for the case where the arithmetic mean is replaced by the geometric mean.