Centrality-Friendship Paradoxes: When Our Friends Are More Important Than Us

TL;DR

This paper proves that, similar to degree, eigenvector and certain matrix-function centralities also exhibit a paradox where our friends are more central than us, providing theoretical backing for empirical observations in social networks.

Contribution

It establishes a rigorous proof of the friendship paradox for eigenvector and matrix-function centralities, extending the concept beyond degree centrality.

Findings

Eigenvector centrality also exhibits the friendship paradox.

Conditions are identified under which the paradox holds for matrix-function centralities.

The results are extended to directed and weighted networks.

Abstract

The friendship paradox states that, on average, our friends have more friends than we do. In network terms, the average degree over the nodes can never exceed the average degree over the neighbours of nodes. This effect, which is a classic example of sampling bias, has attracted much attention in the social science and network science literature, with variations and extensions of the paradox being defined, tested and interpreted. Here, we show that a version of the paradox holds rigorously for eigenvector centrality: on average, our friends are more important than us. We then consider general matrix-function centrality, including Katz centrality, and give sufficient conditions for the paradox to hold. We also discuss which results can be generalized to the cases of directed and weighted edges. In this way, we add theoretical support for a field that has largely been evolving through empirical testing.