  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.
