Limits of multiplicative inhomogeneous random graphs and L\'evy trees: The continuum graphs

Motivated by limits of critical inhomogeneous random graphs, we construct a family of sequences of measured metric spaces that we call continuous multiplicative graphs, that are expected to be the universal limit of graphs related to the multiplicative coalescent (the Erd\H{o}s--R\'enyi 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 representation allows us to demonstrate that a processus that was already present in the pionnering 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 metric: The discrete embedding of random graphs into a Galton--Watson forest is paralleled by an embedding of the encoding process into a L\'evy 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 compactness and fractal dimensions of the continuous objects. In a companion paper, we show that the continuous L\'evy graphs are indeed the scaling limit of inhomogeneous random graphs.