According to http://docs.reduxframework.com/redux-framework/arguments/required/, your conditional option should look like this:
array(
'id' => 'facebook-url',
'type' => 'text',
'required' => array('social-networks','equals','fb'),
'title' => __('Facebook URL', 'options'),
'subtitle' => __('Enter your Facebook URL.', 'options'),
),
Next option:
array(
'id' => 'twitter-url',
'type' => 'text',
'required' => array('social-networks','equals','tw'),
'title' => __('Twitter URL', 'options'),
'subtitle' => __('Enter your twitter URL.', 'options'),
),
and so on.