  Recently, Bollob\'as, Janson and Riordan introduced a family of random graph
models producing inhomogeneous graphs with $n$ vertices and $\Theta(n)$ edges
whose distribution is characterized by a kernel, i.e., a symmetric measurable
function $\ka:[0,1]^2 \to [0,\infty)$. To understand these models, we should
like to know when different kernels $\ka$ give rise to `similar' graphs, and,
given a real-world network, how `similar' is it to a typical graph $G(n,\ka)$
derived from a given kernel $\ka$.
  The analogous questions for dense graphs, with $\Theta(n^2)$ edges, are
answered by recent results of Borgs, Chayes, Lov\'asz, S\'os, Szegedy and
Vesztergombi, who showed that several natural metrics on graphs are equivalent,
and moreover that any sequence of graphs converges in each metric to a graphon,
i.e., a kernel taking values in $[0,1]$.
  Possible generalizations of these results to graphs with $o(n^2)$ but
$\omega(n)$ edges are discussed in a companion paper [arXiv:0708.1919]; here we
focus only on graphs with $\Theta(n)$ edges, which turn out to be much harder
to handle. Many new phenomena occur, and there are a host of plausible metrics
to consider; many of these metrics suggest new random graph models, and vice
versa.

Probability;Combinatorics