The Extended Directed Friendship Paradox

Abstract

The friendship paradox says that on average your friends have more friends than you. Equivalently, in an undirected graph, the average degree of the nodes is no greater than the average degree of the terminal node of a random walk of length one. We generalize this result in two ways: (1) by considering directed graphs, which also allow one-sided relations, such as between followers and leaders; and (2) by characterizing the relations between the expected values of the in and out degrees of the terminal nodes of random alternating walks of length 2k and 2k+1, where k is nonnegative. The limiting value of these averages is proportional to the largest singular value of the associated adjacency matrix, and to its largest eigenvalue in the special case of an undirected graph. We interpret the results for one-sided relations (e.g., between leaders and followers) and two-sided relations (e.g., between friends). We further relate such extension to centrality measures. We show that beta centrality approaches eigenvector centrality when the inverse of the weighting parameter becomes arbitrarily close to the limiting value of the expected degree of the terminal node of a random walk of length k. When relations are asymmetric, this observation extends to the singular vectors of the associated directed graph.