With luck, there will also be some more data quite soon. On the analysis score, notably, this and also this seem useful. The first estimates the value of a node based on its edges, which is fundamentally what I'm trying to achieve, and the second finds the cliques in the network a given node belongs to.
Regarding visualisation issues, I think one of my mistakes last time out was to visualise the data as a multi-graph - i.e. a structure with zero or more links between each node, permitting the existence of multiple links between the same pair of nodes. This invariably means a lot of links. The nature of the data - multiple meetings are absolutely central to the whole project, and lobbies meet ministers at different times and on different issues - enforces an underlying multigraph structure. But it would be possible to condense it for visualisation purposes - if we rolled up all links between the same nodes into one, we could tot up their weights and perhaps show that in the visualisation, as a thicker line for example.