$P$.

A [**rooted graph**]{} $(\gh, x)$ is a graph $\gh$ with a distinguished vertex $x$ of $\gh$, called the [**root**]{}. A [**rooted isomorphism**]{} of rooted graphs is an isomorphism of the underlying graphs that takes the root of one to the root of the other. We shall use the following notion of random weak convergence introduced by \[b\].BS:rdl/ and studied further by \[b\].AS:obj/ and \[b\].AL:uni/. Given a positive integer $R$, a finite rooted graph $H$, and a probability distribution $\rtd$ on rooted graphs, let $p(R, H, \rtd)$ denote the probability that $H$ is rooted isomorphic to the ball of radius $R$ about the root of a graph chosen with distribution $\rtd$. If $(\gh, \rtd)$ is a graph with probability distribution $\rtd$ on its vertices, then $\rtd$ induces naturally a distribution on rooted graphs, which we also denote by $\rtd$; namely, the probability of $(\gh, x)$ is $\rtd(x)$. For a finite graph $\gh$, let $U(\gh)$ denote the distribution of rooted graphs obtained by choosing a uniform random vertex of $\gh$ as root of $\gh$. Suppose that $\Seq{\gh_n}$ is a sequence of finite graphs and that $\rtd$ is a probability measure on rooted infinite graphs; in most practical cases, $\rtd$ will be 