Simulating and visualizing the diffusion of information would be two different tasks, probably best handled by different tools. If you use R, you could try the following packages:
Those can handle dynamic network data and track - or model - the dissemination process.
In terms of visualizing the spread of information, Gephi (gephi.org) is another option. You would have to add a dynamic node attribute to the data with a value of, say, 0 during the time period before the node gets "infected" (tweets the link/hashtag) - and 1 after. Gephi's timeline would let you visualize your Twitter network over time. You can use "Auto apply" for node color based on the dynamic attribute mentioned above. When a user tweets out the information, the node in the Gephi visualization will change color. This way you can show the diffusion process over time.