Here is the simple correlation between Facebook penetration and Transparency International's corruption perception index (CPI), which is one of the most used measures of corruption in the literature. I have used negative of CPI such that a higher value implies higher corruption.
The problem in identifying the causal impact, however, is the huge problem of endogeneity. The penetration of social media websites and even internet is potentially endogenous to the level of corruption in a country. Clearly, if a corrupt government knows that social media and technological progress are going to hurt their discretion and would limit their powers to extract money, they would try to restrict the usage and expansion of these websites. That's the reason, China and Russia do not allow usage of Facebook. There are several other examples where government attempted to monitor the use of internet and social media.
However, after spending a lot of time, we were able to find a way around that -- something that we call "instrument" in econometric language. So, we wrote an article, collected much more extensive data on Facebook, internet, GDP per capita, population, political rights and other variables. And indeed, a much careful analysis showed that countries that have a higher number of Facebook users have lower corruption. Moreover, the effect of the presence of social media (as proxied by Facebook penetration) on corruption is in addition to the effect that internet has on corruption.
If it interests you, here is the link to the article. Any feedback and comments are welcome.
Social Media, Internet and Corruption