The Friendship Paradox for Social Networks

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. We briefly discuss variants of the model where we look at the number of friends at deeper levels and at quantities other than the number of friends.

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JH Mitchell, Peter Hall building, Parkville, Australia

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