Dynamics of Friendship Index in Complex Networks

We study the evolution of the friendship index in complex social networks over time. The elements of the networks are the users, and the connections correspond to the interactions between them. The friendship index of a node is defined as the ratio of the average degree of its neighbors to the degree of the node itself. Obviously, in the process of network growth, the value of the friendship index for any node in the network may change due to the fact that new edges can be attached to this node or its neighbors. In this paper, we study the dynamics of the friendship index of a single node over time for growth networks generated on the basis of the preferential attachment mechanism. We find both the asymptotics of their expected values and the variances over time. In addition, we analyze the behavior of the friendship index for five real networks.