Local weak limit of dynamical inhomogeneous random graphs

Abstract

We consider dynamical graphs, namely graphs that evolve over time, and investigate a notion of local weak convergence that extends naturally the usual Benjamini-Schramm local weak convergence for static graphs. One of the well-known results of Benjamini-Schramm local weak convergence is that of the inhomogeneous random graph $IRG_n(\kappa)$ on $n$ vertices with connection kernel $\kappa$. When the kernel satisfies the mild technical condition of being a graphical kernel, the $IRG_n(\kappa)$ converges locally in probability to the unimodular multi-type Poisson-Galton-Watson tree $MPGW(\kappa)$, see the book of van der Hofstad for a recent detailed exposure of this result. We extend this to dynamical settings, by introducing the dynamical inhomogeneous random graph $DIRG_n(\kappa,\beta)$, with connection kernel $\kappa$ and updating kernel $\beta$, and its limit the growth-and-segmentation multi-type Poisson-Galton-Watson tree $GSMPGW(\kappa,\beta)$. We obtain similarly the local limit of variation of our model, namely the vertex updating inhomogeneous random graph. Our framework provides a natural tool for the study of processes defined on these graphs, that evolve simultaneously as the graph itself and with local dynamics. We discuss briefly the case of the contact process, where we obtain a slight reinforcement of the results of Jacob, Linker and M\"{o}rters (see arXiv:1807.09863 and arXiv:2206.01073).