The Centrality Paradox: Why Your Friends Are Always More Important

Abstract: We revisit the classical friendship paradox which states that on an average one's friends have at least as many friends as oneself and generalize it to a variety of network centrality measures. In particular, we show that for any irreducible, undirected graph $G$, the "friends-average" of degree, eigenvector-centrality, walk-count, Katz, and PageRank centralities exceeds the global average. We show that the result follows from the variational characterisation of the eigenvector corresponding to the Perron eigenvalue.