Probabilities summing to 1 for random directed graph?

Suppose we have a directed graph $G$ with $n$ nodes. Suppose that the probability of an edge from node $i$ to node $j$ is $\pi_{ij}$.

Probability distributions must must assign probability 1 to the entire sample space, but what is the relevant probability distribution/sample space here?

Is there a probability distribution for any two nodes? Or perhaps two probability distributions for any two nodes (one for each direction)? And then each probability distribution has two events: there is a directed edge; there isn't a directed edge?

Or is there perhaps a probability distribution over all nodes? If this is the case, then what probabilities must sum to 1? Would it be $$\sum_i\sum_j \pi_{ij}$$ where $i,j$ are nodes?

Or is the probability distribution something else I have not mentioned?

One way to think of this: for each pair of nodes, flip an independent coin which has heads probability $\pi_{ij}$. Then put an edge if the coin came up heads. Thus your question reduces to understanding the probability space underlying a coin flip experiment, which you can find in any just about any probability textbook. I would start by making sure you understand that construction.