The Generalized Friendship Paradox for Spectral Centralities
Rajat Subhra Hazra, Evgeny Verbitskiy
TL;DR
This work establishes a unified neighbor-averaged centrality paradox for a broad class of spectral centralities on connected graphs, showing that the average centrality of a node's neighbors, computed via $\langle \mathbf{1}, C\mathbf{r}\rangle$, dominates the node's own score $\langle \mathbf{1}, \mathbf{r}\rangle$ for centralities such as degree, eigenvector, walk-counts, Katz, and PageRank. The authors develop a variational framework around the random-walk matrix $C=D^{-1}A$ and leverage Perron–Frobenius theory to prove these inequalities across multiple centrality definitions, including the directed PageRank case with teleportation. They compare neighbor-averaging to edge-sampled averages, demonstrating that the two notions capture different, non-dominant aspects of the paradox and can yield different orderings in practice. The results illuminate a general sampling bias in network centrality assessments, with implications for understanding perceived popularity and influence, and they outline open directions for distance-based measures and sparse random-graph limits. Overall, the paper extends the classical friendship paradox into a cohesive spectral centrality framework, providing precise inequalities, conditions for equality, and guidance for future extensions in directed and sparse settings.
Abstract
We revisit the classical friendship paradox which states that on an average ones friends have at least as many friends as oneself and generalize it to a variety of network centrality indices. For a broad class of spectral centralities on connected undirected graphs degree, eigenvector centrality, walk counts, Katz centrality and PageRank, we show that the average centrality of a nodes neighbours always exceeds the global average centrality.