Since the work of Bollobas, Janson and Riordan (2007, Random Struct. Alg.), there has been increasing interest in the study of inhomogeneous random graphs. Consider the random graph on $n$ vertices constructed from i.i.d. positive random variables $w_1, \ldots,w_n$ by placing an edge between vertex $i$ and vertex $j$ with probability $1-\exp(-w_i w_j / \sum_{k=1}^n w_k)$ independently for each $i\neq j$. This model corresponds to the rank-1 case of the general inhomogeneous random graphs. The random graph is critical if $E(w_1)=E(w_1^2)$. The study of the critical behavior for this model was initiated by van der Hofstad (2012, Random Struct. Alg.).

In the present work, we show that after assigning mass $w_i/n^{2/3}$ to vertex $i$ and scaling the graph distance by $n^{1/3}$, the components of this graph (at criticality) viewed as measured metric spaces converge in Gromov-Hausdorff-Prokhorov topology to some limiting (random) compact, measured metric spaces if $w_1$ satisfies appropriate conditions. This result also addresses the question of universality of the continuum limit of the components (viewed as measured metric spaces) of critical Erdos-Renyi random graphs.

Joint work with Shankar Bhamidi, Amarjit Budhiraja and Xuan Wang.