The weak law of large numbers for the friendship paradox index

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.

💡 Research Summary

The paper investigates the asymptotic behavior of the Friendship Paradox Index (FPI), a statistic that measures how much, on average, a node’s neighbors have more connections than the node itself. The authors focus on Random Geometric Graphs (RGGs), a class of spatially embedded random graphs where nodes are placed according to a probability density f on the unit interval (or circle) and edges are formed between nodes whose distance is below a threshold rₙ. Two regimes are considered: the uniform case (f(x)=1) and the non‑uniform case (f varies smoothly, is bounded away from zero, and possesses bounded fourth derivatives). Under the mild technical condition rₙ=o(1) and n rₙ=ω(1) (the graph is neither too sparse nor too dense), the paper derives a weak law of large numbers (WLLN) for the average FPI, denoted Fₙ, using Markov’s inequality.

The core technical work consists of a series of lemmas that approximate conditional edge probabilities and higher‑order joint events (edges, triangles, four‑node subgraphs) up to terms of order rₙ⁵ or rₙ⁶. Lemma 2.3 provides expansions such as
E