A time-invariant random graph with splitting events

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 lambda which governs the limiting average degree. We show that for each value of lambda there is a unique random connected rooted multigraph M(lambda) invariant under this evolution. As a consequence, starting from any finite graph G the process will almost surely converge in distribution to M(lambda), 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].