Let's assume your node is called "myBusyNode". The attribute you're looking to check is called, "thisAttr".
// Selects all nodes that are outgoing connections of myBusyNode.thisAttr
select -r `listConnections -s false -d true myBusyNode.thisAttr` ;
Alternately you can select the incoming connections by swapping source and destination.
// Selects all nodes that are incoming connections of myBusyNode.thisAttr
select -r `listConnections -s true -d false myBusyNode.thisAttr` ;
And, of course you could set both to true and select both incoming and outgoing nodes.
The return value of listConnections will give you all of the outgoing connected nodes from that attribute. You can then focus on your selection in the Hypergraph and you'll be able to see only the network of your selected nodes, not every single connection coming out of myBusyNode. I use this trick about four hundred and eighty seven times a day.