Friendship-paradox paradox: Do most people's friends really have more friends than they do?

TL;DR

This paper clarifies the distinction between the mean-based friendship paradox and nodewise majority relations by introducing the global mean-based fraction $phi_global$ and the local median-based fraction $phi_local$, along with hub centrality $h_i$. It demonstrates that these quantities are logically independent from the classical FP and can diverge depending on neighborhood structure and degree distribution shape. Through a toy example and two empirical networks (Zachary's karate club and the American football network), it shows that left- or right-skewed neighbor-degree distributions can yield opposite conclusions for mean- and median-based comparisons. The framework provides a foundation for analyzing local advantage, disadvantage, and perception asymmetry in complex networks and outlines directions for future analytical and applied research.

Abstract

The classical friendship paradox asserts that, on average, an individual's neighbors have a higher degree than the individual. This statement concerns network-level means and does not describe how often a typical node is locally dominated by its neighbors. Motivated by this distinction, we develop a framework that separates mean-based friendship paradox inequalities from two majority-type quantities: a global fraction measuring how many nodes have a degree smaller than the mean degree of their neighbors, and a local fraction based on hub centrality that measures how many nodes are dominated in a median-based sense. We show that neither fraction is constrained by the classical friendship paradox and that they can behave independently of each other. A simple example and two empirical networks illustrate how quadrant patterns in the joint distribution of a node's degree and its neighbors' degree determine the signs and magnitudes of the two fractions, and how left- or right-skewed degree distributions of neighboring nodes can yield opposite conclusions for mean-based and median-based comparisons. The resulting framework offers a clearer distinction between population averages and local majority relations and provides a foundation for future analyses of local advantage, disadvantage, and perception asymmetry in complex networks.

Figures (3)

• Figure 1: A five-node toy network with $\phi_{\mathrm{global}}=\phi_{\mathrm{local}}=2/5<1/2$. Nodes $1$, $2$, and $3$ form a triangle, and nodes $4$ and $5$ are connected to all of $1$, $2$, and $3$. Degrees are $k_1=k_2=k_3=4$ and $k_4=k_5=3$. The high-degree nodes $1$--$3$ are not locally dominated by their neighbors, whereas the low-degree nodes $4$ and $5$ are.
• Figure 2: Density plots for the ZKC network Zachary1977, illustrating node-level relationships among degree $k_i$, mean neighbor degree $k_{nn}(i)$, hub centrality $h_i$, and the difference $k_i - k_{nn}(i)$ between a node's degree and the average value of its neighbors' degrees. The thresholds for the mean-based or median-based comparison, $k_{nn}(i)=k_i$ and $h_i = 1/2$, are plotted as the white dashed lines. Panel (a) shows $k_i$ versus $k_{nn}(i)$, providing a node-level view of the ego-based FP; most points lie above the diagonal $k_{nn}(i)=k_i$, consistent with the large value $\phi_{\mathrm{global}} \approx 0.85$. Panel (b) shows $k_i$ versus $h_i$, emphasizing the median-based notion of local dominance. Panel (c) plots $h_i$ against $k_i - k_{nn}(i)$, revealing that nodes with negative mean-based contrast tend to have low hub centrality, indicating simultaneous mean-based and median-based disadvantage. Together, the panels illustrate why both $\phi_{\mathrm{global}}$ and $\phi_{\mathrm{local}}$ are high in this network.
• Figure 3: Density plots for the AFB network GirvanNewman2002football_data, illustrating the same three relationships shown in Fig. \ref{['fig:karate_density']}. Panel (a) displays $k_i$ versus $k_{nn}(i)$ and shows that a majority of points lie below the diagonal $k_{nn}(i)=k_i$, consistent with $\phi_{\mathrm{global}} \approx 0.43 < 1/2$, i.e., a majority of teams have degree larger than the mean degree of their neighbors, even though the classical FP in Eqs. \ref{['eq:FPclassic_v1']} and \ref{['eq:FPclassic_v2']} holds. Panel (b) shows $k_i$ versus $h_i$, and panel (c) shows $h_i$ versus $k_i - k_{nn}(i)$; both plots reveal that a large fraction of nodes satisfy $h_i < 1/2$, yielding $\phi_{\mathrm{local}} \approx 0.84$. These contrasting patterns demonstrate that mean-based and median-based majority-type inequalities in Eqs. \ref{['eq:phi_global_condition']} and \ref{['eq:phi_local_condition']} can also be quite different, in contrast to the toy network in Fig. \ref{['fig:toy_local']} with $\phi_{\mathrm{global}} = \phi_{\mathrm{local}} < 1/2$.