How to animate growth of a social network?
Fancy animations are cool
I was very impressed when I saw this animation of the discourse git repository. They used Gourse which is specifically for git. But it may give ideas about how to represent the dynamics of growth.
You can create animations with matplotlib
This stackoverflow answer seems to point at a python/networkx/matplotlib solution.
But D3.js provides interaction
If you're looking for a web-based solution then d3.js is excellent. See this, this and this for example.
See also this stackoverflow question, the accepted answer points to D3.js again.
Conclusion
I would be drawn towards the python/networkx options for network analysis (possibly to add attributes to your raw data file for example). Then, for visualisation and dissemination D3.js is perfect. You might be surprised how easy it can be to write d3.js once you get into it. I believe it even works within an ipython notebook!