The problem you are describing is called link prediction. Here is a short tutorial explaining about the problem and some simple heuristics that can be used to solve it.
Since this is an open-ended problem, these simple solutions can be improved a lot by using more complicated techniques. Another approach for predicting the probability for an edge is to use Machine Learning rather than rule-based heuristics.
A recent article called node2vec, proposed an algorithm that maps each node in a graph to a dense vector (aka embedding). Then, by applying some binary operator on a pair of nodes, we get an edge representation (another vector). This vector is then used as input features to some classifier that predicts the edge-probability. The paper compared a few such binary operators over a few different datasets, and significantly outperformed the heuristic benchmark scores across all of these datasets.
The code to compute embeddings given your graph can be found here.