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A time-invariant random graph with splitting events
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Georgakopoulos, Agelos and Haslegrave, John (2021) A time-invariant random graph with splitting events. Electronic Communications in Probability, 26 . pp. 1-15. 66. doi:10.1214/21-ECP436 ISSN 1083-589X.
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Official URL: https://doi.org/10.1214/21-ECP436
Abstract
We introduce a process where a connected rooted multigraph evolves by splitting events on its vertices, occurring randomly in continuous time. When a vertex splits, its incoming edges are randomly assigned between its offspring and a Poisson random number of edges are added between them. The process is parametrised by a positive real λ which governs the limiting average degree. We show that for each value of λ there is a unique random connected rooted multigraph M(λ) invariant under this evolution. As a consequence, starting from any finite graph G the process will almost surely converge in distribution to M(λ), which does not depend on G. We show that this limit has finite expected size. The same process naturally extends to one in which connectedness is not necessarily preserved, and we give a sharp threshold for connectedness of this version.
| Item Type: | Journal Article | |||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
| Library of Congress Subject Headings (LCSH): | Random graphs , Combinatorial analysis , Programming (Mathematics) , Operations research | |||||||||
| Journal or Publication Title: | Electronic Communications in Probability | |||||||||
| Publisher: | Institute of Mathematical Statistics | |||||||||
| ISSN: | 1083-589X | |||||||||
| Official Date: | 6 December 2021 | |||||||||
| Dates: |
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| Volume: | 26 | |||||||||
| Page Range: | pp. 1-15 | |||||||||
| Article Number: | 66 | |||||||||
| DOI: | 10.1214/21-ECP436 | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Access rights to Published version: | Open Access (Creative Commons) | |||||||||
| Date of first compliant deposit: | 3 November 2021 | |||||||||
| Date of first compliant Open Access: | 18 January 2022 | |||||||||
| RIOXX Funder/Project Grant: |
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