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Friendship paradox disappears under degree biased network sampling

Wojciech Roga

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.

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.
Paper Structure (6 sections, 17 equations, 3 figures)

This paper contains 6 sections, 17 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Degree biased sampling identity
  3. Random walk
  4. Conservation of flow
  5. Simulations
  6. Discussion

Figures (3)

  • Figure 1: Random walk on random graph (Erdős–Rényi) with $n=1000$ nodes and probability of each pair of nodes being connected $p=0.05$.
  • Figure 2: Random walk on Zachary's Karate Club graph with 34 nodes and 78 edges.
  • Figure 3: Random walk on SNAP Facebook graph (${\rm facebook\_combined}$) with 4039 nodes and 88234 connections.