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The weak law of large numbers for the friendship paradox index

Mingao Yuan

Abstract

The friendship paradox index is a network summary statistic used to quantify the friendship paradox, which describes the tendency for an individual's friends to have more friends than the individual. In this paper, we utilize Markov's inequality to derive the weak law of large numbers for the friendship paradox index in a random geometric graph, a widely-used model for networks with spatial dependence and geometry. For uniform random geometric graph, where the nodes are uniformly distributed in a space, the friendship paradox index is asymptotically equal to $1/4$. On the contrary, in nonuniform random geometric graphs, the nonuniform node distribution leads to distinct limiting properties for the index. In the relatively sparse regime, the friendship paradox index is still asymptotically equal to $1/4$, the same as in the uniform case. In the intermediate sparse regime, however, the index converges in probability to $1/4$ plus a constant that is explicitly dependent on the node distribution. Finally, in the relatively dense case, the index diverges to infinity as the graph size increases. Our results highlight the sharp contrast between the uniform case and its nonuniform counterpart.

The weak law of large numbers for the friendship paradox index

Abstract

The friendship paradox index is a network summary statistic used to quantify the friendship paradox, which describes the tendency for an individual's friends to have more friends than the individual. In this paper, we utilize Markov's inequality to derive the weak law of large numbers for the friendship paradox index in a random geometric graph, a widely-used model for networks with spatial dependence and geometry. For uniform random geometric graph, where the nodes are uniformly distributed in a space, the friendship paradox index is asymptotically equal to . On the contrary, in nonuniform random geometric graphs, the nonuniform node distribution leads to distinct limiting properties for the index. In the relatively sparse regime, the friendship paradox index is still asymptotically equal to , the same as in the uniform case. In the intermediate sparse regime, however, the index converges in probability to plus a constant that is explicitly dependent on the node distribution. Finally, in the relatively dense case, the index diverges to infinity as the graph size increases. Our results highlight the sharp contrast between the uniform case and its nonuniform counterpart.
Paper Structure (18 sections, 16 theorems, 262 equations, 1 table)

This paper contains 18 sections, 16 theorems, 262 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The expectation of the friendship paradox
  4. The variance of the friendship paradox
  5. The WLLN for the friendship paradox
  6. Proof of lemmas
  7. Proof of Lemma \ref{['propmain']}
  8. Proof of Lemma \ref{['lem2']}
  9. Proof of Lemma \ref{['lnem3']}
  10. Proof of Lemma \ref{['lem1']}:
  11. Proof of Lemma \ref{['lem3']}
  12. Proof of Lemma \ref{['addlem1']}
  13. Proof of Lemma \ref{['adlemnew1']}
  14. Proof of Lemma \ref{['lemnew1']}
  15. Proof of Lemma \ref{['lemnew3']}
  16. ...and 3 more sections

Key Result

Lemma 2.3

Under Assumption assumptiona, we have Here, the remainder terms $O(r_n^5)$ and $O(r_n^6)$ in (prope0)-(2prope3) do not depend on $X_1$.

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 8 more