LIMITS OF MULTIPLICATIVE INHOMOGENEOUS RANDOM GRAPHS AND LEVY TREES: THE CONTINUUM GRAPHS

Motivated by limits of critical inhomogeneous random graphs, we construct a family 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 Erdos-Renyi 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 process that was already present in the pioneering work of Aldous [Ann. Probab. 25 (1997) 812-854] and Aldous and Limic [Electron. J. Probab. 3 (1998) 1-59] 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 Levy 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 multiplicative graphs are indeed the scaling limit of inhomogeneous random graphs.