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 invariant under this evolution. As a consequence, starting from any finite graph G the process will almost surely converge in distribution to , 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.
This is an asynchronous version, which is more realistic from the real-world network point of view, of a process we studied in [8, 9].
Funding Statement
Both authors were supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 639046). J.H. was also partially supported by the UK Research and Innovation Future Leaders Fellowship MR/S016325/1.
Acknowledgments
We are grateful to the anonymous referee for their very helpful comments.
Citation
Agelos Georgakopoulos. John Haslegrave. "A time-invariant random graph with splitting events." Electron. Commun. Probab. 26 1 - 15, 2021. https://doi.org/10.1214/21-ECP436
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