Desmond Higham: Generalized Friendship Paradoxes in Network Science

Abstract: The original Friendship Paradox dates back to the work of the sociologist Scott Feld, who showed that, on average, our friends have at least as many friends as us. In network science terms, for any network, neighbours have a degree at least as high as nodes, on average. Feld discussed two types of averaging, which may be viewed as global and local. These paradoxes have attracted attention in academia and beyond. They are relevant to many aspects of human interaction, notably in on-line behaviour, epidemiology and healthcare intervention. Generalized friendship paradoxes occur when degree is replaced by some other
attribute. I will focus on the case of network centrality, where we ask whether, on average, our friends are more influential than us. I will present some new results concerning the inevitability of network centrality paradoxes arising. The analysis brings together a surprisingly wide range of ideas; from linear algebra, matrix polynomials, number theory, mathematical physics and chemical graph theory. I will also point out some open questions. This is based on joint work with Francesco Hrobat (Oxford) and Francesco Tudisco (Edinburgh).