Reinterpretaion of the friendship paradox

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 1/2. Comparing the ratios of nodes satisfying the FP in networks, r(p), we can see that the probability version is stronger than the mean value version in real networks both online and or line. We also show some results about the effects of several parameters on r(p) in random network models. Most importantly, r(p) is a quadratic polynomial of the power law exponent gamma in Price model, and r(p) is higher when the average clustering coefficient is between 0.4 and 0.5 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.