Friendship paradox in growth networks: analytical and empirical analysis

Abstract

Many empirical studies have shown that in social, citation, collaboration, and other types of networks in real world, the degree of almost every node is less than the average degree of its neighbors. This imbalance is well known in sociology as the friendship paradox and states that your friends are more popular than you on average. If we introduce a value equal to the ratio of the average degree of the neighbors for a certain node to the degree of this node (which is called the ‘friendship index’, FI), then the FI value of more than 1 for most nodes indicates the presence of the friendship paradox in the network. In this paper, we study the behavior of the FI over time for networks generated by growth network models. We will focus our analysis on two models based on the use of the preferential attachment mechanism: the Barabási–Albert model and the triadic closure model. Using the mean-field approach, we obtain differential equations describing the dynamics of changes in the FI over time, and accordingly, after obtaining their solutions, we find the expected values of this index over iterations. The results show that the values of FI are decreasing over time for all nodes in both models. However, for networks constructed in accordance with the triadic closure model, this decrease occurs at a much slower rate than for the Barabási–Albert graphs. In addition, we analyze several real-world networks and show that their FI distributions follow a power law. We show that both the Barabási–Albert and the triadic closure networks exhibit the same behavior. However, for networks based on the triadic closure model, the distributions of FI are more heavy-tailed and, in this sense, are closer to the distributions for real networks.