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

TL;DR

This paper constructs continuous multiplicative graphs as universal limits of inhomogeneous random graphs, using a novel embedding into Galton--Watson forests and Lévy processes, advancing understanding of their metric properties and scaling limits.

Contribution

It introduces a new representation of inhomogeneous random graphs via embeddings into Galton--Watson forests and Lévy processes, enabling analysis of their continuum limits and metric structures.

Findings

Constructed continuous multiplicative graphs as universal limits.

Linked the encoding process to Lévy processes for metric analysis.

Provided a framework for studying fractal dimensions of the limits.

Abstract

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.