We investigate exceedances of the process over a sufficiently high
threshold. The exceedances determine the risk of hazardous events like climate
catastrophes, huge insurance claims, the loss and delay in telecommunication
networks. Due to dependence such exceedances tend to occur in clusters. The
cluster structure of social networks is caused by dependence (social
relationships and interests) between nodes and possibly heavy-tailed
distributions of the node degrees. A minimal time to reach a large node
determines the first hitting time. We derive an asymptotically equivalent
distribution and a limit expectation of the first hitting time to exceed the
threshold un as the sample size n tends to infinity. The results can be extended
to the second and, generally, to the kth (k > 2) hitting times. Applications in
large-scale networks such as social, telecommunication and recommender
systems are discussed.
