Multivariate Regular Variation of Preferential Attachment Models
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911975745781760 |
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| author | Janßen, Anja Ziegenbalg, Max |
| author_facet | Janßen, Anja Ziegenbalg, Max |
| contents | We use the framework of multivariate regular variation to analyse the extremal behaviour of preferential attachment models. To this end, we follow a directed linear preferential attachment model for a random, heavy-tailed number of steps in time and treat the incoming edge count of all existing nodes as a random vector of random length. By combining martingale properties, moment bounds and a Breiman type theorem we show that the resulting quantity is multivariate regularly varying, both as a vector of fixed length formed by the edge counts of a finite number of oldest nodes, and also as a vector of random length viewed in sequence space. A Pólya urn representation allows us to explicitly describe the extremal dependence between the degrees with the help of Dirichlet distributions. As a by-product of our analysis we establish new results for almost sure convergence of the edge counts in sequence space as the number of nodes goes to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_02785 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Multivariate Regular Variation of Preferential Attachment Models Janßen, Anja Ziegenbalg, Max Probability 60G70, 05C80, 60F25 We use the framework of multivariate regular variation to analyse the extremal behaviour of preferential attachment models. To this end, we follow a directed linear preferential attachment model for a random, heavy-tailed number of steps in time and treat the incoming edge count of all existing nodes as a random vector of random length. By combining martingale properties, moment bounds and a Breiman type theorem we show that the resulting quantity is multivariate regularly varying, both as a vector of fixed length formed by the edge counts of a finite number of oldest nodes, and also as a vector of random length viewed in sequence space. A Pólya urn representation allows us to explicitly describe the extremal dependence between the degrees with the help of Dirichlet distributions. As a by-product of our analysis we establish new results for almost sure convergence of the edge counts in sequence space as the number of nodes goes to infinity. |
| title | Multivariate Regular Variation of Preferential Attachment Models |
| topic | Probability 60G70, 05C80, 60F25 |
| url | https://arxiv.org/abs/2310.02785 |