The weak law of large numbers for the friendship paradox index

TL;DR

This paper establishes the weak law of large numbers for the friendship paradox index in random geometric graphs, revealing how node distribution and graph density affect its limiting behavior.

Contribution

It derives the asymptotic properties of the friendship paradox index in both uniform and nonuniform random geometric graphs, highlighting differences based on network density.

Findings

In uniform graphs, the index approaches 1/4 asymptotically.

In nonuniform graphs, the index converges to 1/4 plus a distribution-dependent constant.

In dense graphs, the index diverges to infinity.

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.