“[C]an you give me all options that are possible”? (emphasis added) No, because HTML, CSS, JS, PHP, and WordPress are powerful and sophisticated. Below are are some different approaches, however.
[Affiliation: I am not affiliated with any plugins or programs listed here.]
is_user_logged_in()
In WP, the level closest to the code is to use the function is_user_logged_in(). You would put this function somewhere in your theme, and if the user is logged in, WP would not execute the code that serves your icons. For this specific goal, however, I do not think this method is a good idea.
Plugins to manage access to widgets
Some plugins, such as Restrict Widgets, allow you to selectively display widgets based on various parameters. There are multiple plugins that do this, so shop around. I linked to an example that allows you to restrict based on whether the user is logged in or not logged in. Widget-restriction plugins will only work, however, if your social icons are displayed by a widget.
Plugins to manage access to content
In some cases, icons are displayed by shortcode within the content of the post or page. In that case, you will need a plugin like User Specific Content. It works well for me on WP 4.1.1. There are many other plugins, however, including some incredibly powerful premium plugins that give you control over the entire installation–content, widgets, plugins, menus, everything–so do shop around.
Plugins to manage access to plugins
I have seen multiple plugins that manage the ability of end-users to see specific plugins, but for some reason I am only finding plugins that restrict back-end access to plugins. I know they exist. If the widget style plugin doesn’t work, then find a plugin-level manager.
Other
Some plugins are extremely narrow. Nav Menu Roles only manages your menus. I have seen plugins that are specifically for hiding portions of one other plugin. Some plugins have add-on components to disable features. Some social icon plugins also have the feature you are looking for.
Conclusion
There are many ways to achieve your goal. The best path for you depends on your knowledge of the various WP-associated technologies, how much money you want to spend, how much effort you want to expend, and how many features you need.