  We study random subgraphs of an arbitrary finite connected transitive graph
$\mathbb G$ obtained by independently deleting edges with probability $1-p$.
Let $V$ be the number of vertices in $\mathbb G$, and let $\Omega$ be their
degree. We define the critical threshold $p_c=p_c(\mathbb G,\lambda)$ to be the
value of $p$ for which the expected cluster size of a fixed vertex attains the
value $\lambda V^{1/3}$, where $\lambda$ is fixed and positive. We show that
for any such model, there is a phase transition at $p_c$ analogous to the phase
transition for the random graph, provided that a quantity called the triangle
diagram is sufficiently small at the threshold $p_c$. In particular, we show
that the largest cluster inside a scaling window of size
$|p-p_c|=\Theta(\cn^{-1}V^{-1/3})$ is of size $\Theta(V^{2/3})$, while below
this scaling window, it is much smaller, of order
$O(\epsilon^{-2}\log(V\epsilon^3))$, with $\epsilon=\cn(p_c-p)$. We also obtain
an upper bound $O(\cn(p-p_c)V)$ for the expected size of the largest cluster
above the window. In addition, we define and analyze the percolation
probability above the window and show that it is of order $\Theta(\cn(p-p_c))$.
Among the models for which the triangle diagram is small enough to allow us to
draw these conclusions are the random graph, the $n$-cube and certain Hamming
cubes, as well as the spread-out $n$-dimensional torus for $n>6$.
