Extremes control of complex systems with applications to social networks

The control and risk assessment in complex information systems require to take into account extremes arising from nodes with large node degrees. Various sampling techniques like a Page Rank random walk, a Metropolis-Hastings Markov chain and others serve to collect information about the nodes. The paper contributes to the comparison of sampling techniques in complex networks by means of the first hitting time, that is the minimal time required to reach a large node. Both the mean and the distribution of the first hitting time is shown to be determined by the so called extremal index. The latter indicates a dependence measure of extremes and also reflects the cluster structure of the network. The clustering is caused by dependence between nodes and heavy-tailed distributions of their degrees. Based on extreme value theory we estimate the mean and the distribution of the first hitting time and the distribution of node degrees by real data from social networks. We demonstrate the heaviness of the tails of these data using appropriate tools. The same methodology can be applied to other complex networks like peer-to-peer telecommunication systems.