  Motivated by applications, the last few years have witnessed tremendous
interest in understanding the structure as well as the behavior of dynamics for
inhomogeneous random graph models. In this study we analyze the maximal
components at criticality of one famous class of such models, the rank-one
inhomogeneous random graph model. Viewing these components as measured random
metric spaces, under finite moment assumptions for the weight distribution, we
show that the components in the critical scaling window with distances scaled
by $n^{-1/3}$ converge in the Gromov-Haussdorf-Prokhorov metric to rescaled
versions of the limit objects identified for the Erd\H{o}s-R\'enyi random graph
components at criticality Addario-Berry, Broutin and Goldschmidt (2012). A key
step is the construction of connected components of the random graph through an
appropriate tilt of a famous class of random trees called $\mathbf{p}$-trees
(studied previously by Aldous, Miermont and Pitman (2004) and by Camarri and
Pitman (2000)). This is the first step in rigorously understanding the scaling
limits of objects such as the Minimal spanning tree and other strong disorder
models from statistical physics (see Braunstein et al., 2003) for such graph
models. By asymptotic equivalence (Janson, 2010), the same results are true for
the Chung-Lu model and the Britton-Deijfen-Lof model. A crucial ingredient of
the proof of independent interest is tail bounds for the height of
$\mathbf{p}$-trees. The techniques developed in this paper form the main
technical bedrock for proving continuum scaling limits in the critical regime
for a wide array of other random graph models (Bhamidi, Broutin, Sen and Wang,
2014) including the configuration model and inhomogeneous random graphs with
general kernels which were introduced by Bollobas, Janson and Riordan (2007).
