In elastic Graph single edges represent a collection of documents that contain a pair of terms. An edge could represent a billion documents, or as in your case, it could be just one.
When you want to view properties of an edge you can use the search API to retrieve individual documents or perhaps use the "aggregations" feature to summarise properties of many docs e.g. show a line graph of volumes of transactions between 2 accounts over time. The next version of the Graph UI  provides "drill-down" features which make these calls for you and provide visualizations.
I'm not sure what data/approach you use to compute your scores in advance but it is worth noting that elastic Graph can derive scores on-the-fly from raw data e.g. looking at StackOverflow posts it can summarise who is strongly connected with who based on the volumes of exchanges on StackOverflow over time. We use this to prioritise which connections to explore first.