A time-invariant random graph with splitting events

We introduce a process where connected rooted multigraph evolves by splitting events on its vertices, occurring randomly in continuous time. When vertex splits, incoming edges are assigned between offspring and Poisson random number of added them. The is parametrised positive real λ which governs the limiting average degree. show that for each value there unique M(λ) invariant under this evolution. As consequence, starting from any finite graph G will almost surely converge distribution to M(λ), does not depend G. limit has expected size. same naturally extends one connectedness necessarily preserved, we give sharp threshold version. This an asynchronous version, more realistic real-world network point view, studied [8, 9].