Limit Distributions of Friendship Index in Scale-Free Networks

The friendship index measures a node’s popularity relative to its friends on a social network. The friendship index is calculated by dividing the average degree of a node’s friends by its own degree, i.e. it is the ratio of the sum of the degrees of its neighbors to the square of the degree of the node itself. Under some assumptions, the numerator of this fraction can be viewed as the sum of some random variables distributed according to the cumulative degree distribution function in the given network. It is known that for the vast majority of real complex networks, their degree distributions follow a power-law with some exponent \(\gamma \) . We examine the dependence of the average value of the friendship index among nodes of the same degree k in the network on k. We will explore scale-free networks with degree-degree neutral mixing and find the limit distributions of the friendship index with the network size tending to infinity in the configuration model. Moreover, we compare our findings with the behavior of empirical friendship index distributions for several real networks.