What determines the presence / absence of images in preview of links when you paste it in Facebook?

When writing a post in Vkontakte, inserted link. Then there is its definition and the editor shows preview of the image taken from the page. After the post published, there is only one link, no picture.
Should be something like this:

And it turns out like this (a link without an image):


Prescribing a particular image using link rel="image_src" in the head - the problem is not solved. The editor takes it, but then when sending does not display.