  The colored neighborhood metric for sparse graphs was introduced by
Bollob\'as and Riordan. The corresponding convergence notion refines a
convergence notion introduced by Benjamini and Schramm. We prove that even in
this refined sense, the limit of a convergent graph sequence (with uniformly
bounded degree) can be represented by a graphing. We study various topics
related to this convergence notion such as: Bernoulli graphings, factor of
i.i.d. processes and hyperfiniteness.
