Local Degree Asymmetry for Preferential Attachment Model

One of the well-known phenomena of the sociological experience is the friendship paradox which states that your friends are more popular than you, on average. The friendship paradox is widely detected empirically in various complex networks including social, coauthorship, citation, collaboration, online social media networks. A local network metric called “the friendship index” has been recently introduced in order to evaluate some aspects of the friendship paradox for complex networks. The value of the friendship index for a node is defined as the ratio of the average degree of the neighbors of the node to the degree of this node. In this paper we examine the theoretical properties of the friendship index in growth networks generated by the Barabási-Albert model by deriving the equation that describes the evolution of the expected value of the friendship index of a single node over time. Results show that there is a clear presence of the friendship paradox for networks evolved in accordance with the Barabási-Albert model in which each new node acquires a single edge. Moreover, the distributions of the friendship index for such networks are heavy-tailed and comparable with the empirical distribution obtained for some real networks.