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We investigate exceedances of the process over a sufficiently high threshold.The exceedances determine the risk of hazardous events like climatecatastrophes, huge insurance claims, the loss and delay in telecommunicationnetworks.  Due to dependence such exceedances tend to occur in clusters. The clusterstructure of social networks is caused by dependence (social relationships andinterests) between nodes and possibly heavy-tailed distributions of the nodedegrees. A minimal time to reach a large node determines the first hittingtime. We derive an asymptotically equivalent distribution and a limitexpectation of the first hitting time to exceed the threshold $u_n$ as thesample size $n$ tends to infinity. The results can be extended to the secondand, generally, to the $k$th ($k> 2$) hitting times. Applications inlarge-scale networks such as social, telecommunication and recommender systemsare discussed.

Clustering and Hitting Times of Threshold Exceedances and Applications