Odd thought, working on NetworkX visualisations of lobbying activity. So much of whether the finished image is any use or not is an aesthetic decision, nothing to do with the actual graph metrics in any formal or programmatic way.
Randomly fiddle with the scaling factor, or increase the multiplier you apply to the nodes' link degree, and things look different - and possibly better. Also, more information may not be better - I can't help but think it was a lot easier to derive conclusions before I gave the various ministries node weights to reflect their relative power and divided the ministry weighting on each link by the number of organisations taking part in the meeting, which should be an improvement.
Also, after sweating quite a bit over this, does anyone know if NetworkX even lets you position nodes using polar coordinates? This looks useful.
This is far from ideal, but it should give some idea. This one is using the standard force-directed spring layout and just the data I have from June, with the nodes coloured by their degree in the chart and sized by their meeting count. Downing Street (the large purple plook) is fixed in place and the others adjusted. I rather like the fact that the lobbyists look like bacteria swarming into a cell, or possibly sperm trying to fertilise David Cameron.