Only display link to author social media when it exists [closed]

This is a very simple PHP issue, unrelated to WordPress. You need to check if the value is empty before echoing it. So, store them as variables first:

$twitter  = get_the_author_meta('twitter', $author_id);
$facebook = get_the_author_meta('facebook', $author_id);
//etc

Then check them before displaying:

if(!empty($twitter)) {
    echo '<a title="Follow me on Twitter" href="'.$twitter.'"><img src="twitter.png" alt="" /></a>';
}
if(!empty($facebook)) {
    echo '<a title="Follow me on Facebook" href="'.$facebook.'"><img src="facebook.png" alt="" /></a>';
}
//etc