Clustering and Hitting Times of Threshold Exceedances and Applications

2017/09/30 by Natalia M. Markovich, Natalia Markovich, Markovich, Natalia
Mathematics · Physics and Astronomy · Psychology · #Complex Network Analysis Techniques #FOS: Mathematics #Mental Health Research Topics #Opinion Dynamics and Social Influence #Statistics Theory (math.ST) #math.ST #stat.TH

paper · pdf · doi:10.48550/arxiv.1710.00229

19 pages, 4 figures

arxiv created 2017/09/30 · openalex publication_date 2017/09/30 · arxiv updated 2017/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 u_n 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.

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