  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.
