Strong Friendship Paradox in Social Networks

TL;DR

The paper addresses why most of your friends have more friends than you do and how higher-order network structure biases local observations and trait perceptions. It formalizes the strong friendship paradox (SFP) and its generalized form, deriving that the effect depends on the degree distribution via $E\{d(Y)\}-E\{d(X)\} = \frac{\mathrm{Var}\{d(X)\}}{E\{d(X)\}}$, and on degree–trait correlations via $E\{f(Y)\}-E\{f(X)\} = \frac{\mathrm{Cov}(f(X),d(X))}{E\{d(X)\}}$, with transsortativity playing a key role. The analysis shows SFP is prevalent in real networks (roughly 70–90%), amplified by disassortativity and degree–trait correlations, and that shuffling trait assignments can remove SFP while leaving FP intact. These biases have implications for measurement, polling, and contagion dynamics, underscoring the need to account for higher-order network structure to mitigate misperceptions and guide interventions.

Abstract

The friendship paradox in social networks states that your friends have more friends than you do, on average. Recently, a stronger variant of the paradox was shown to hold for most people within a network: `most of your friends have more friends than you do.' Unlike the original paradox, which arises trivially because a few very popular people appear in the social circles of many others and skew their average friend popularity, the strong friendship paradox depends on features of higher-order network structures. Similar to the original paradox, the strong friendship paradox generalizes beyond popularity. When individuals have traits, many will observe that most of their friends have more of that trait than they do. This can lead to the Majority illusion, in which a rare trait will appear highly prevalent within a network. Understanding how the strong friendship paradox biases local observations within networks can inform better measurements of network structure and our understanding of collective phenomena in networks.

Figures (3)

• Figure 1: Friendship paradox in a social network representing friendship relationships between members of a university karate club zachary1977information. Darker nodes represent members who observe that they are less popular (have fewer friends) than their friends are on average.
• Figure 2: Illustrations of network paradoxes in the karate club social network. (left) Strong friendship paradox: Darker nodes represent members that observe that they have fewer friends than most of their friends do. (right) Majority illusion: Although only eight of the 34 nodes are red, all but two of the remaining nodes (yellow) observe that at least half of their friends are red.
• Figure 3: Strong FP in real-world networks (data from Wu2017neighbor). (a) The plot shows the fraction of nodes in a citation network of physics papers with $k$ citations that experience the strong friendship paradox. To experience the paradox, a node with degree $k$ must observe that most of its neighbors have degree at least $k$. The lines show predictions of the model that includes degree distribution and degree assortativity (dashed), and also transsortativity (solid) capturing degree correlations between node's neighbors. Both models and observational measurements ignore direction of edges and consider as neighbors both cited and citing papers. (b) Observed fraction of nodes in real-world networks that experience the strong friendship paradox, compared to predicted prevalence of the paradox with the transsortativity mode.