Friendship paradox disappears under degree biased network sampling

Abstract

We show that in an undirected graph under degree biased sampling the expected degree of vertices is equal to the expected degree of their neighbors. In consequence, under the biased sampling the social network result known as the friendship paradox disappears. The identity is equivalent to the existence of a stationary state of a random walk on the graph or to the conservation of the total flow defined by the difference of the degrees of the vertices.