Limits of multiplicative inhomogeneous random graphs and Lévy trees: the continuum graphs

Motivated by limits of critical inhomogeneous random graphs, we con- struct a family of sequences of measured metric spaces that we call continu- ous multiplicative graphs, that are expected to be the universal limit of graphs related to the multiplicative coalescent (the Erdo ̋s–Rényi random graph, more generally the so-called rank-one inhomogeneous random graphs of various types, and the configuration model). At the discrete level, the construction relies on a new point of view on (discrete) inhomogeneous random graphs that involves an embedding into a Galton–Watson forest. The new represen- tation allows us to demonstrate that a process that was already present in the pioneering work of Aldous [Ann. Probab., vol. 25, pp. 812–854, 1997] and Aldous and Limic [Electron. J. Probab., vol. 3, pp. 1–59, 1998] about the multiplicative coalescent actually also essentially encodes the limiting met- ric. The discrete embedding of random graphs into a Galton–Watson forest is paralleled by an embedding of the encoding process into a Lévy process which is crucial in proving the very existence of the local time functionals on which the metric is based; it also yields a transparent approach to com- pactness and fractal dimensions of the continuous objects. In a companion paper, we show that the continuous Lévy graphs are indeed the scaling limit of inhomogeneous random graphs