  A countable, bounded degree graph is almost finite if it has a tiling with
isomorphic copies of finitely many F\o lner sets, and we call it strongly
almost finite, if the tiling can be randomized so that the probability that a
vertex is on the boundary of a tile is uniformly small. We give various
equivalents for strong almost finiteness. In particular, we prove that Property
A together with a uniform version of amenability is equivalent to strong almost
finiteness. Using these characterizations, we show that graphs of
subexponential growth and Schreier graphs of amenable groups are always
strongly almost finite, generalizing the celebrated result of Downarowicz,
Huczek and Zhang about amenable Cayley graphs, based on graph theoretic rather
than group theoretic principles. We give various equivalents to Property A for
graphs, and show that if a sequence of graphs of Property A (in a uniform
sense) converges to a graph $G$ in the neighborhood distance (a purely
combinatorial analogue of the classical Benjamini-Schramm distance), then their
Laplacian spectra converge to the Laplacian spectrum of $G$ in the Hausdorff
distance. We apply the previous theory to construct a new and rich class of
classifiable $C^{\star}$-algebras. Namely, we show that for any minimal strong
almost finite graph $G$ there are naturally associated simple, nuclear, stably
finite $C^{\star}$-algebras that are classifiable by their Elliott invariants.
