Friendship paradox disappears under degree biased network sampling

TL;DR

This paper demonstrates that degree-biased sampling in undirected networks eliminates the friendship paradox by showing the expected degree of vertices equals that of their neighbors, linking to stationary states of random walks.

Contribution

It reveals that degree-biased sampling removes the friendship paradox and connects this phenomenon to random walk stationary states and flow conservation in networks.

Findings

Friendship paradox disappears under degree-biased sampling.

Expected degree of vertices equals that of their neighbors.

Relation to stationary states of random walks.

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.