  We introduce a new model of correlated randomly growing graphs and study the
fundamental questions of detecting correlation and estimating aspects of the
correlated structure. The model is simple and starts with any model of randomly
growing graphs, such as uniform attachment (UA) or preferential attachment
(PA). Given such a model, a pair of graphs $(G_1, G_2)$ is grown in two stages:
until time $t_{\star}$ they are grown together (i.e., $G_1 = G_2$), after which
they grow independently according to the underlying growth model.
  We show that whenever the seed graph has an influence in the underlying graph
growth model---this has been shown for PA and UA trees and is conjectured to
hold broadly---then correlation can be detected in this model, even if the
graphs are grown together for just a single time step. We also give a general
sufficient condition (which holds for PA and UA trees) under which detection is
possible with probability going to $1$ as $t_{\star} \to \infty$. Finally, we
show for PA and UA trees that the amount of correlation, measured by
$t_{\star}$, can be estimated with vanishing relative error as $t_{\star} \to
\infty$.
