Branching processes are known to be useful tools in studying random graphs, for instance in understanding the phase transition phenomenon in the asymptotics sizes of their connected components. In this talk, I’d like to discuss some applications of Galton—Watson trees (genealogy of branching process) in studying the geometrical aspects of random graphs. In particular, we will look at the so-called Poisson random graph, a random graph model which generalises the Erdos-Renyi graph G(n, p) and which has been previously studied by Aldous and Limic for its close connection with the multiplicative coalescents. Relying upon an embedding of the graph into a Galton-Watson forest, we can identify the scaling limits of these graphs inside the critical window.
Based on a joint work with Nicolas Broutin and Thomas Duquesne.