Little of what they did was actually paid for advertising
I don't think anything influences how people vote "to a significant degree". Taking the presidential election as an example: a third of voters are locked in no matter what - they'd vote for Hitler promising gas chambers for kittens and puppies if he had the D or R by his name. Of the third remaining, a third each will vote for the D or R pretty much unless they are Hitler.
It is that remaining third of a third who decide the election, they are the ones you need to sway - the only ones you CAN sway. They are the reason you see all the negative ads, because it is a lot easier to make them vote AGAINST someone (or decide not to show up) than to make them vote FOR someone.
I don't think that Facebook should be worrying about ads, because based on the evidence we have now almost none of what the Russians did was paid for advertising. They had a lot of troll accounts that shared made up stuff dressed up to look like "news" on Facebook and Twitter. Then other trolls would like, share and comment on Facebook, favoring and retweeting on Twitter.
Those troll accounts had long since formed relationships with some US based people (i.e. finding people who had similar viewpoints to the ones they wanted to push and liking/sharing/commenting on their stuff and eventually friending them, and the equivalent on Twitter) so the stuff they were sharing amongst themselves was picked up and re-shared in the US on and on again through real US based friend networks. Those US based people who were unwittingly spreading what the Russian trolls had started weren't interested in fact checking whether Hillary really was going to be dead from cancer within a year of the election, if the story had a slant they liked they helped spread it. Even if some were skeptical that Hillary really was dying and didn't spread the story enough did that it and stories like it took on a life of their own.
Sure, the Russians could have bought advertising on Facebook but instead hired a hundred people who write English well and had them work all day every day doing this. They were able to amplify stories 1000x more per dollar than if they were run as paid for ads. If you wanted to advertise a business honestly you'd probably get more bang for your buck hiring a dozen college students to work a few hours a day posting/sharing/etc. making your product look good and the other product look bad. Imagine if some newcomer to the Android market did this, hyping up their product and slagging on Samsung as overpriced etc. I've heard claims that Samsung has paid people on social media to hype them and that's why they got so big in the US...dunno whether that's true or not but if it is I guess they were ahead of their time.
What I'm really worried about is that campaigns are going to hire social media teams who will basically be like US based troll farms. Campaign laws are backwards looking and concerned with traditional ads like you see on TV or via fliers received in the mail. They don't cover paying people to share false stories about your opponent that someone else wrote, and even if they were fixed so they did they'd simply use the third party soft-money groups to do it for them. Since Trump's administration sure isn't going to lift a finger to stop the Russian trolls they'll be around with us in 2018 and 2020, but they may be drowned out by the US run troll operations. I guess that's a good thing????