Distribution and Dependence of Extremes in Network Sampling Processes

We explore the dependence structure in the sampled sequence of complex networks. We consider randomized algorithms to sample the nodes and study extremal properties in any associated stationary sequence of characteristics of interest like node degrees, number of followers, or income of the nodes in online social networks, which satisfy two mixing conditions. Several useful extremes of the sampled sequence like the kth largest value, clusters of exceedances over a threshold, and first hitting time of a large value are investigated. We abstract the dependence and the statistics of extremes into a single parameter that appears in extreme value theory called extremal index (EI). In this work, we derive this parameter analytically and also estimate it empirically. We propose the use of EI as a parameter to compare different sampling procedures. As a specific example, degree correlations between neighboring nodes are studied in detail with three prominent random walks as sampling techniques.