Consider a group of individuals who form a social network. For each individual in the group compute its friendship-bias, i.e., the difference between the average number of friends of its friends and the number of its friends (all friendships are mutual) and average these numbers over all the individuals in the group. It turns out that the latter average is always non-negative and is strictly positive as soon as not all individuals have exactly the same number of friends. This fact, which at first glance seems counterintuitive, goes under the name of friendship paradox. In this talk we model the social network as a graph and explain where the friendship paradox comes from. For sequences of random graphs that converge locally in an appropriate sense, we quantify the friendship paradox by identifying the limit of the empirical distribution of the friendship-biases of all the individuals. For two examples of random graphs, we work out the properties of this limit in detail. Based on joint work with R.S. Hazra (Leiden), N. Litvak (Eindhoven) and A. Parvaneh (Bielefeld).
Prof. Frank received his PhD in Mathematical Physics at the University of Leiden in 1985. From 1985 to 2024 he held positions at the universities of Delft, Utrecht, Nijmegen, Eindhoven and Leiden. In 2024 he retired and became emeritus professor at Leiden University. From 2000 to 2005, Frank was scientific director of the European research institute EURANDOM in Eindhoven, The Netherlands. From 2002 to 2007, he was chair of a Scientific Programme of the European Science Foundation, involving 13 European countries. In 2005, he was elected to the Royal Netherlands Academy of Sciences. From 2008 to 2016, he was chair of the Advisory Council for the Natural and Technical Sciences of the Royal Netherlands Academy of Sciences. In 2016, he was named Knight in the Order of the Dutch Lion. In 2018, he received a Humboldt Research Award from the Alexander Humboldt Foundation.