Limits of multiplicative inhomogeneous random graphs and Levy trees

101 pagesWe consider a natural model of inhomogeneous random graphs that extends the classical Erdos-Renyi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous . In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Levy-type processes. We, instead, look at the metric structure of these components and prove their Gromov-Hausdorff-Prokhorov convergence to a class of random compact measured metric spaces. Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general "critical" regime, and relies upon two key ingredients: an encoding of the graph by some Levy process as well as an embedding of its connected components into Galton-Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of Levy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Levy-type process. As a consequence of our construction, we give a transparent and explicit condition for the compactness of the limit objects and determine their fractal dimensions. These results extend and complement several previous results that had obtained via model- or regime-specific proofs, for instance: the case of Erdos-Renyi random graphs obtained by Addario-Berry, Goldschmidt and B., the asymptotic homogeneous case as studied by Bhamidi, Sen and Wang, or the power-law case as considered by Bhamidi, Sen and van der Hofstad