  We consider sequences of graphs and define various notions of convergence
related to these sequences: ``left convergence'' defined in terms of the
densities of homomorphisms from small graphs into the graphs of the sequence,
and ``right convergence'' defined in terms of the densities of homomorphisms
from the graphs of the sequence into small graphs; and convergence in a
suitably defined metric.
  In Part I of this series, we show that left convergence is equivalent to
convergence in metric, both for simple graphs, and for graphs with nodeweights
and edgeweights. One of the main steps here is the introduction of a
cut-distance comparing graphs, not necessarily of the same size. We also show
how these notions of convergence provide natural formulations of Szemeredi
partitions, sampling and testing of large graphs.
