Growing random graphs with a preferential attachment structure

The aim of this paper is to develop a method for proving almost sure convergence in Gromov-Hausodorff-Prokhorov topology class models growing random graphs that generalises Remy's algorithm binary trees. We describe the obtained limits using some iterative gluing construction famous line-breaking Aldous' Brownian tree. In order do that, we framework which metric space constructed by smaller spaces, called \emph{blocks}, along structure (possibly infinite) discrete Our seen as spaces can be understood framework, is, evolving blocks glued tree structure. Their scaling limit then separately every block and verifying relative compactness property whole For particular study, behind has distribution an affine preferential attachment or weighted recursive strongly rely on results concerning those two trees their connection, companion paper.