Friendship paradox in growth networks: analytical and empirical analysis

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.

Correlation between Triadic Closure and Homophily Formed over Location-Based Social Networks

Social Internet of Things (SIoT) is a variation of social networks that adopt the property of peer-to-peer networks, in which connections between the things and social actors are automatically established. SIoT is a part of various organizations that inherit the social interaction, and these organizations include industries, institutions, and other establishments. Triadic closure and homophily are the most commonly used measures to investigate social networks’ formation and nature, where both measures are used exclusively or with statistical models. The triadic closure patterns are mapped for actors’ communication behavior over a location-based social network, affecting the homophily. In this study, we investigate triads emergence in homophilic social networks. This evaluation is based on the empirical review of triads within social networks (SNs) formed on Big Data. We utilized a large location-based dataset for an in-depth analysis, the Chinese telecommunication-based anonymized call detail records (CDRs). Two other openly available datasets, Brightkite and Gowalla, were also studied. We identified and proposed three social triad classes in a homophilic network to feature the correlation between social triads and homophily. The study opened a promising research direction that relates the variation of homophily based on closure triads nature. The homophilic triads are further categorized into transitive and intransitive groups. As our concluding research objective, we examined the relative triadic throughput within a location-based social network for the given datasets. The research study attains significant results highlighting the positive connection between homophily and a specific social triad class.