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.
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机译：我们调查过程中是否超出足够高的阈值，从而确定危险事件的风险，例如气候灾难，巨额保险索赔，电信网络的损失和延迟。由于依赖性，这种超出往往会在集群中发生。社交网络的集群结构是由节点之间的依赖关系（社会关系和兴趣）以及节点度的可能是重尾分布引起的。到达大型节点的最短时间确定了第一次命中时间。当样本大小$ n $趋于无穷大时，我们得出渐近等效分布和首次命中时间超过阈值$ u_n $的极限期望。结果可以扩展到第二个，通常扩展到$ k $ th（$ k> 2 $）的命中时间。讨论了在大型网络中的应用，例如社交，电信和推荐系统。
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