On distribution and dependence of extremes in PageRank-type processes

Social networks contain clusters of nodes centered at high-degree nodes and surrounded by low-degree nodes. Such a cluster structure of the networks is caused by the dependence (social relationships and interests) between nodes and possibly by heavytailed distributions of the node degrees. We consider the degree sequences generated by PageRank type sampling processes and investigate clusters of exceedances of such sequences over large degrees. It is important to investigate its stochastic nature since it allows us to disseminate advertisement or collect opinions more effectively within the clusters. The dependence structure of sampled degrees exceeding sufficiently high thresholds is measured using a parameter called extremal index, θ. It is defined as follows.