As @norabora points out, the purpose of your app will probably dictate which you will want to use.
Facebook - Likely to be the one that most people will be signed up with, however the profile pictures that people use on the network are often, of them in social situations, with friends or similar. Since the network is about simply connecting people with people.
LinkedIn - If your site is aimed at professional or work based tasks, this one is a good choice. People tend to use professional portrait photos or images of them in work related situations. However is smaller than Facebook.
Twitter - Has a similar number of users to LinkedIn and the style of photos is probably balanced somewhere between social and professional. Although a chunk of accounts are probably brand/company ones.
Gravatar - Has a relatively small user base, mainly centred around the WordPress ecosystem. However is a great open project.
Which country your app is aimed at might also have an influence, as whilst Facebook is a leader in many countries, places such as Russia with VKontakte and China with Weibo or QZone, have different market leaders.
You could of course give you users the option to choose from a few of them, or upload a different image. More development time, however possibly the most flexible option.