  The Erd\H{o}s-R\'enyi random graph is the fundamental random graph model. In
this paper we consider its continuous-time version, where multi-edges and
self-loops are also allowed. It is well-known that the sizes of its connected
components evolve according to the multiplicative coalescent dynamics.
Moreover, with the additional information on the number of surplus edges, the
resulting process follows the augmented multiplicative coalescent dynamic,
constructed by Bhamidi, Budhiraja and Wang in 2014. The same authors exhibit
the scaling limit, which can be specified in terms of the infinite vector of
excursions (in particular their lengths, and the areas enclosed by the
excursion curves) above past infima of a reflected Brownian motion with linear
infinitesimal drift. We use some recent results, using a graph exploration
process called the simultaneous breadth-first walk, to study the same scaling
limit, called the standard augmented multiplicative coalescent (SAMC). We
present a self-contained, simpler and more direct approach than that of any
previous construction of the SAMC. Furthermore, we believe that the method
described here is convenient for generalizations, one of which would be the
study of general non-standard eternal augmented multiplicative coalescents.
