A time-invariant random graph with splitting events

TL;DR

This paper introduces a novel stochastic process modeling the evolution of connected rooted multigraphs through splitting events, establishing a unique invariant graph distribution and analyzing its properties, including connectedness thresholds.

Contribution

It presents a new continuous-time splitting process for multigraphs, proves the existence and uniqueness of an invariant distribution, and analyzes the connectedness threshold for a non-preserving variant.

Findings

Unique invariant multigraph $M()$ for each $$

Almost sure convergence to $M()$ from any finite graph

Finite expected size of the limiting graph

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 $\lambda$ 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 arXiv:1506.02697, arXiv:1703.09011.