Reinterpretaion of the friendship paradox

Abstract

The friendship paradox (FP) is a sociological phenomenon stating that most people have fewer friends than their friends do. It is to say that in a social network, the number of friends that most individuals have is smaller than the average number of friends of friends. This has been verified by Feld. We call this interpreting method mean value version. But is it the best choice to portray the paradox? In this paper, we propose a probability method to reinterpret this paradox, and we illustrate that the explanation using our method is more persuasive. An individual satisfies the FP if his (her) randomly chosen friend has more friends than him (her) with probability not less than [Formula: see text]. Comparing the ratios of nodes satisfying the FP in networks, [Formula: see text], we can see that the probability version is stronger than the mean value version in real networks both online and offline. We also show some results about the effects of several parameters on [Formula: see text] in random network models. Most importantly, [Formula: see text] is a quadratic polynomial of the power law exponent [Formula: see text] in Price model, and [Formula: see text] is higher when the average clustering coefficient is between [Formula: see text] and [Formula: see text] in Petter–Beom (PB) model. The introduction of the probability method to FP can shed light on understanding the network structure in complex networks especially in social networks.