  In a recent paper, Caron and Fox suggest a probabilistic model for sparse
graphs which are exchangeable when associating each vertex with a time
parameter in $\mathbb{R}_+$. Here we show that by generalizing the classical
definition of graphons as functions over probability spaces to functions over
$\sigma$-finite measure spaces, we can model a large family of exchangeable
graphs, including the Caron-Fox graphs and the traditional exchangeable dense
graphs as special cases. Explicitly, modelling the underlying space of features
by a $\sigma$-finite measure space $(S,\mathcal{S},\mu)$ and the connection
probabilities by an integrable function $W\colon S\times S\to [0,1]$, we
construct a random family $(G_t)_{t\geq 0}$ of growing graphs such that the
vertices of $G_t$ are given by a Poisson point process on $S$ with intensity
$t\mu$, with two points $x,y$ of the point process connected with probability
$W(x,y)$. We call such a random family a graphon process. We prove that a
graphon process has convergent subgraph frequencies (with possibly infinite
limits) and that, in the natural extension of the cut metric to our setting,
the sequence converges to the generating graphon. We also show that the
underlying graphon is identifiable only as an equivalence class over graphons
with cut distance zero. More generally, we study metric convergence for
arbitrary (not necessarily random) sequences of graphs, and show that a
sequence of graphs has a convergent subsequence if and only if it has a
subsequence satisfying a property we call uniform regularity of tails. Finally,
we prove that every graphon is equivalent to a graphon on $\mathbb{R}_+$
equipped with Lebesgue measure.
