Oh, you're on the right track... but it's worse than you imagine. Those "leaders" and "organizers" are unimportant. You're familiar with the 'Six Degrees of Kevin Bacon', right? Say you were omnipotent and wanted to make it so that instead of only needing 6 links to get to Mr. Bacon, you needed 15 links on average. And you want to do as little disruption of the network as necessary.
What nodes in the graph of connections would you target? Intuition says go after the nodes with tons of connections to tons of people. Intuition is, as it almost always is, dead wrong. Removing those massively-connected people in a social network is basically useless. They are almost exclusively connected to people who are already connected to one another. They're the center of clusters, and you will almost never route through one in an attempt to find the path to Mr. Bacon.
Instead, what you want to look for are node which bridge clusters. People who are usually not strongly connected to one of the clusters they are associated with, but provide a conduit through which connections can be made. Because they are inherently a bit 'different' from the people in each of the clusters they bridge (few others bridge those two clusters), they tend not to be important figures. They're a member of a biker gang who hangs out with his great aunts knitting circle on Saturdays. They're the ones who facilitate the flow of ideas between groups that never speak to one another otherwise.
So you know who to target... but how many 'bridge' nodes like that do you need to really take out in order to significantly increase the 'distance' between nodes on the graph on average? Disturbingly few. Removing something like 25 nodes from the graph of tens of thousands of performers will make it so you need 15 or more connections on average to get to Mr. Bacon. (The book 'Linked: The New Science of Networks' details the research specifically, its been years since I read it though so I am fuzzy on exact numbers but it was definitely fewer than 30 you needed to remove)
Now, imagine a crazy scenario where you had access to the social graph of a country (somehow!) and knew who was talking to who. And your goal was just to maintain the status quo. Well, what does any large-scale social change require? An idea must spread to large numbers of extremely different, and almost entirely disconnected, groups. Any idea that remains sheltered in one or a few groups will die out of its own accord. A revolution doesn't happen because one minority wills it, there has to be buy-in from a wide array of groups. So remove those bridge nodes. Any widespread social change becomes very nearly impossible.
But what is "removing a node"? I'm not talking about black-bagging a person and dragging them off to some hole or blowing their brains out. Such things are entirely unnecessary and counter-productive. The more significant your action is, the more profound the unintended consequences will be and predicting the outcome quickly becomes intractable. Instead, notice that those bridges are usually connected more strongly to one cluster rather than the other. If their communication became more burdensome to the group they are less connected to... for how long would they persist in fighting to maintain it? If something nutty happened and they had to change their phone number, what're the odds they'll forget to give the new one to the contact in a group they're barely involved in?
You could have very quiet oppression through these means. I expect there would be unintended consequences, and have been trying to figure out for a few ways a good way to detect such changes in a social network, but its just a thing I keep in the back of my head, not something I actively work on. If you've got any ideas, I'd be happy to hear them.