  Following a general program of studying limits of discrete structures, and
motivated by the theory of limit objects of converge sequences of dense simple
graphs, we study the limit of graph sequences such that every edge is labeled
by an element of a compact second-countable Hausdorff space K. The "local
structure" of these objects can be explored by a sampling process, which is
shown to be equivalent to knowing homomorphism numbers from graphs whose edges
are decorated by continuous functions on K. The model includes multigraphs with
bounded edge multiplicities, graphs whose edges are weighted with real numbers
from a finite interval, edge-colored graphs, and other models. In all these
cases, a limit object can be defined in terms of 2-variable functions whose
values are probability distributions on K.
