In this talk we will consider an inhomogeneous random graph process where the vertices have sizes and the edges appear with a rate proportional to the product of the sizes of the vertices. In addition, we do not restrict the graph to be simple, i.e. we allow the existence of multi-edges and self-loops. It is well-known that the size of the connected component of this graph follows the standard multiplicative coalescent dynamics. Besides, when accounting for the additional information consisting in the number of surplus edges, the resulting process follows the so-called standard augmented multiplicative coalescent dynamic.
In this talk, we will explain how we can extend a graph exploration process called simultaneous breadth-first walk (introduced by Limic (2019)) so we encode the augmented multiplicative coalescent dynamics. This encoding is then used to prove in a simple way the scaling limits of the standard augmented multiplicative coalescent. Even if this result is already known, our proof is much more simple and direct than those encounters in the bibliography. Furthermore, this framework could potentially be extended to the study of general non-standard augmented multiplicative coalescent as scaling limits of inhomogeneous random graphs, which is a work in progress.
This talk is based on a work in collaboration with Vlada Limic (CNRS, Université de Strasbourg) contained in the preprints:
- J. C., Vlada Limic (2023) A dynamical approach to spanning and surplus edges of random graphs arXiv:2305.04716
- J. C., Vlada Limic (2023) The standard augmented multiplicative coalescent revisited arXiv:2304.07545
