Assume we are simulating percolation on a 2d lattice. While the system is of finite size, we say that the critical state appears when a cluster connects two opposing ends of the lattice. The bigger the lattice the better our approximation.
My question is: assume we are doing the same thing but instead of a lattice we have a random graph with a known degree distribution. As we are increasing the occupation probability, who do we know that we reached percolation?
In other words, how does the 'connecting opposing ends' assumption translate to random graph topologies?